Stochastic Ferromagnetism: Analysis and Numerics 9783110307108, 9783110306996

Table of contents :
1 The role of noise in finite ensembles of nanomagnetic particles
1.1 Preliminaries
1.1.1 Geometric ergodicity of Markov chains
1.1.2 Ergodicity with rates for solutions of SDEs
1.1.3 Convergent discretizations of the deterministic LLG equation
1.2 Exponential Ergodicity and Asymptotic Rates
1.2.1 Low-dimensional noise for finitely many interacting spins
1.2.2 High-dimensional noise for finitely many interacting spins
1.2.3 L2-ergodicity with rate
1.2.4 Penalization with multiplicative noise
1.3 Discretizations of the stochastic Landau-Lifshitz-Gilbert equation
1.3.1 A structure-preserving discretization of (1.36): the geometric exponential ergodicity
1.3.2 Strong Convergence of Scheme 1.11
1.3.3 A linear implicit discretization scheme
1.4 Computational studies
1.4.1 Numerical schemes
1.4.2 Long-time dynamics
1.4.3 Interplay of penalization and noise
2 The stochastic Landau-Lifshitz-Gilbert equation
2.1 Preliminaries
2.1.1 Finite elements and temporal discretization
2.1.2 Fractional Sobolev spaces and related compact embeddings
2.1.3 Young integral
2.1.4 Wiener process and the approximating random walk
2.1.5 Convergence of random variables and representation theorems
2.1.6 Stability of solutions of the Landau-Lifshitz-Gilbert equation
2.2 Convergent discretization of SLLG
2.2.1 Unconditional Stability of Scheme 2.9
2.2.2 Convergence of iterates from Scheme 2.9
2.2.3 Existence of a solution to the SLLG equation
2.2.4 A convergent discretization of the SLLG equation which uses random walks
2.3 Computational studies
2.3.1 Numerical implementation
2.3.2 Effects of the space-time white noise in 1D and 2D
2.3.3 Discrete blow-up of the SLLG equation with space-time white noise
3 Effective equations for macrospin magnetization dynamics
3.1 Construction of local strong solutions for the augmented LLG
3.2 Convergence with optimal rates for Scheme A
3.3 Construction of a weak solutions via Scheme 3.5
3.3.1 Solving the nonlinear system in Scheme 3.5
3.4 Computational experiments
3.4.1 µMag standard problem no. 4 with thermal effects
3.4.2 Comparison of the macroscopic model with the SLLG equation
Bibliography

Citation preview

De Gruyter Studies in Mathematics 58 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

L’ ubomı´r Banˇas, Zdzisław Brzez´niak, Mikhail Neklyudov, Andreas Prohl

Stochastic Ferromagnetism Analysis and Numerics

De Gruyter

Mathematics Subject Classification 2010: 35R60, 60H15, 60H35, 65Cxx, 91B51, 37A25, 37A30, 37C40, 37M25, 60J05, 60J22, 60M12, 65N06, 65N30. Authors: L’ ubomı´r Banˇas Department of Mathematics Universität Bielefeld Postfach 100131 33501 Bielefeld Germany [email protected] Mikhail Neklyudov School of Mathematics and Statistics Carslaw Building (F07) University of Sydney NSW 2006 Australia [email protected]

Zdzisław Brzez´niak University of York Department of Mathematics Heslington York YO10 5DD UK [email protected] Andreas Prohl Universität Tübingen FB Mathematik Auf der Morgenstelle 10 72076 Tübingen Germany [email protected]

ISBN 978-3-11-030699-6 e-ISBN 978-3-11-030710-8 Set-ISBN 978-3-11-030711-5 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. ” 2014 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents

1

The role of noise in finite ensembles of nanomagnetic particles 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Geometric ergodicity of Markov chains . . . . . . . . . . . . . . . . 1.1.2 Ergodicity with rates for solutions of SDEs . . . . . . . . . . . . 1.1.3 Convergent discretizations of the deterministic LLG equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 21

1.2 Exponential Ergodicity and Asymptotic Rates . . . . . . . . . . . . . . . 1.2.1 Low-dimensional noise for finitely many interacting spins 1.2.2 High-dimensional noise for finitely many interacting spins 1.2.3 L2 -ergodicity with rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Penalization with multiplicative noise . . . . . . . . . . . . . . . . .

33 33 39 48 51

1.3 Discretizations of the stochastic Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A structure-preserving discretization of (1.36): the geometric exponential ergodicity . . . . . . . . . . . . . . . . . . . . . 1.3.2 Strong Convergence of Scheme 1.11 . . . . . . . . . . . . . . . . . . . 1.3.3 A linear implicit discretization scheme . . . . . . . . . . . . . . . . 1.4 Computational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Long-time dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Interplay of penalization and noise . . . . . . . . . . . . . . . . . . . 2

7

24

67 67 74 79 85 86 93 98

The stochastic Landau-Lifshitz-Gilbert equation

103

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Finite elements and temporal discretization . . . . . . . . . . . . 2.1.2 Fractional Sobolev spaces and related compact embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Young integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Wiener process and the approximating random walk . . . . 2.1.5 Convergence of random variables and representation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Stability of solutions of the Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106 106 111 114 115 117 123

vi

Contents

2.2 Convergent discretization of SLLG . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Unconditional Stability of Scheme 2.9 . . . . . . . . . . . . . . . . . 2.2.2 Convergence of iterates from Scheme 2.9 . . . . . . . . . . . . . . 2.2.3 Existence of a solution to the SLLG equation . . . . . . . . . . 2.2.4 A convergent discretization of the SLLG equation which uses random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

129 138 155 163 176

2.3 Computational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Effects of the space-time white noise in 1D and 2D . . . . . 2.3.3 Discrete blow-up of the SLLG equation with space-time white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186 186 188

Effective equations for macrospin magnetization dynamics

196

190

3.1 Construction of local strong solutions for the augmented LLG . . 200 3.2 Convergence with optimal rates for Scheme A . . . . . . . . . . . . . . . . 207 3.3 Construction of a weak solutions via Scheme 3.5 . . . . . . . . . . . . . . 209 3.3.1 Solving the nonlinear system in Scheme 3.5 . . . . . . . . . . . . 216 3.4 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 3.4.1 μMag standard problem no. 4 with thermal effects . . . . . . 220 3.4.2 Comparison of the macroscopic model with the SLLG equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Bibliography

236

Introduction

We study thermally activated magnetization dynamics of ferromagnetic nanostructures. A classical microscopic description of an interacting spin system which couples with the surrounding microscopic degrees of freedom (i.e. phonons, conducting electrons, nuclear spins, etc.) is based on the principles of Hamiltonian mechanics [61, Chapter 6]. A mesoscopic description of the statistical properties can be motivated from these equations to reduce the complexity of the model: a general Langevin type model which describes the interaction of atomistic ferromagnetic N -spin ensembles X ≡ (X1 , . . . , XN ) : R+ ×Ω → (S2 )N in a heat bath is the stochastic Landau-Lifshitz-Gilbert equation (SLLG), see [24, 62, 82, 22, 61],     ∂X = X × Heff + Hthm − α X × X × Heff . ∂t

(1)

The deterministic version of this equation (i.e. Hthm ≡ 0) has been introduced in 1935 by Landau and Lifshitz as a phenomenological equation to describe the magnetization at positive temperatures. It was extended to the form (1) by W.F. Brown [24] to account for thermal effects in the case of a single spin (N = 1). Here, Heff ≡ Heff (X) = −∇E(X) denotes the effective field which acts on spins in the ensemble and which is governed by the total energy of the system E : (S2 )N → R. This energy is the sum of the exchange energy Eexch to describe spin-spin interactions, the anisotropy energy Eani to model energetically favored alignment of spins with crystallographic axes with the help of the density φ : S2 → R+ 0 , and the external energy Eext to account for applied forces hext , E(X) =

N N   A  l K Jm Xl , Xm  + φ(Xi ) − hext , X . 2 2 m,l=1

(2)

i=1

  Here ·, · denotes the scalar product in (R3 )N , ·, · stands for the scalar l )N N ×N is some given symmetric positive product in R3 , and J = (Jm m,l=1 ∈ R definite matrix. The dynamics of magnetic nanostructures in a heat bath may not be described by classical thermodynamics which is used for macroscopic systems, and where the behavior is reproducible; instead, their modelling is based on non-equilibrium stochastic thermodynamics [102], where irreversible heat losses between the system and the surrounding heat bath are described

2

Introduction

by relevant thermal fluctuations far away from the equilibrium. In the above model (1), the stochastic field Hthm : R+ × Ω → (R3 )N accounts for the interaction of the spin system with thermal fluctuations which allows the system to overcome energy barriers, and to realize related relaxation dynamics. In order to model non-equilibrium thermodynamics, it is customarily assumed that 1 , . . . , H N ) is Gaussian noise which is uncorrelated in space and Hthm ≡ (Hthm thm time (t, s ≥ 0), i.e.,  i   i  j E Hthm (t) = 0 , E Hthm (t), Hthm (s) = ν 2 δij δ(t − s) , (3) for all 1 ≤ i, j ≤ N . Here ν ≡ ν(τ ) ∝ τ > 0 is a temperature dependent constant to scale the intensity of thermal fluctuations relative to dissipative effects. The intensity obeys a fluctuation-dissipation relation such that the coupled system converges towards a thermal equilibrium which is described by a Gibbs distribution; see Chapter 1 for further details. A practically relevant task is to study relaxation dynamics towards thermal equilibrium at elevated temperatures, which often goes along with a spontaneous magnetization reversal to migrate from a metastable magnetic state to another one with lower energy; the quantitative behavior then depends on the intensity ν ≡ ν(τ ) > 0 in (3). Different approaches by Neel and Brown for single spins [104] provide strong evidence that probabilities for a corresponding thermally induced magnetization reversal to overcome an energy barrier ΔE √ − ΔE follow the Arrhenius law, which to leading order is proportional to τ e kB τ . However, the thermodynamic properties of non-uniform magnetization reversal for general energies E from (2) are more involved, which is why less is known about corresponding energy barriers. In this case, computational studies may provide valuable insight in the coupling dynamics. A better understanding of the magnetization dynamics at elevated temperatures helps to develop improved nano-scale data storage devices, where too short relaxation times may result in a loss of initially stored data: the smaller memory elements are, the more relevant becomes thermal noise, and its ability to trigger noise-induced magnetization reversal. Another application is heat-assisted magnetic recording to alleviate magnetization reversal on hard-disks by laser pulses, and a corresponding study of the response of spins depending on the temperature. Chapter 1 addresses finitely many interacting spins and related long-time dynamics, which is inspired by the early work [24] for a single ferromagnetic spin. A question of considerable interest is whether these results also hold for a system which consists of infinitely many spins, cf. [21], and if e.g. the corresponding L2 (O, R3 )-valued noise may be correlated in space or not to allow for thermodynamically consistent long-time dynamics. For systems which consist of infinitely many spins occupying the ferromagnetic body O ⊂ Rn , n ≤ 3, the following mesoscopic continuum model describes the magnetization

3

Introduction

process m : R+ × O × Ω → S2 at elevated temperatures,   m ∂m m × Heff ) = m × Heff + Hthm − α m × (m ∂t m ∂m =0 ∂n m (0, ·) = m 0

on R+ × O × Ω on R+ × ∂O × Ω

(4)

on O × Ω ,

m) = −DE(m m), and where Heff (m A K m) = m|2 dx + m) dx − hext , m  dx . E(m |∇m φ(m 2 O 2 Rd O

(5)

There is again physical evidence [37] that the related deterministic LLG model (i.e. Hthm ≡ 0) which describes the dynamics of magnetizations m : R+ × O → S2 requires a modification at elevated temperatures: in this case, an enhanced damping property of the spin system is observed in experiments, as well as a non-constant (sample averaged) magnetization magnitude in both, space and time, which may not be explained by the deterministic model. As a consequence, a stochastic version of the deterministic LLG model is used to statistically describe small-scale effects which are too complex to be described in detail by a microscopic model. From a mathematical viewpoint, problem (4)–(5) is a stochastic nonlinear partial differential equation where the solution process is S2 -valued, see also [32]. The related deterministic LLG model has been analyzed in the literature for n = 2, 3: global weak solutions are known to exist, and the possible formation of singularities at finite times from smooth initial data (n ≥ 2) is motivated by the numerical studies in [17, 10]. In Chapter 2, we evidence a regularizing effect on solutions of (4)–(5) in the case of space-time white noise in (4) by means of computational experiments which are obtained from a convergent space-time discretization. Simulations to obtain relevant statistical information from (4)–(5) are in general based on Monte-Carlo methods, and are computationally intensive. Hence, a major goal is to derive effective macroscopic equations of motion for averaged magnetizations which accurately account for thermal effects. A m] which allows phenomenological description for a single macro-spin m = E[m for proper relaxation dynamics has been derived in [59] within a mean-field approximation, which is based on the following consequence of equation (4),     ∂m m × Heff ) = ΛN m + E m × Heff − α E m × (m ∂t

on R+ × O ,

(6)

where ΛN ≡ ΛN (τ ) ∝ τ in front of the Bloch relaxation term is known as the Neel time; see [59]. It is this term which allows for a varying length of the

4

Introduction

macro-spin m for different temperatures. An approximation of the nonlinear terms in (6) which yields a closed effective equation for the macro-spin m is then often referred to as Landau-Lifshitz-Bloch equation (LLB). It has been shown to properly describe domain wall motion in the presence of a non-constant magnetization length, macroscopic magnetization magnitudes, observed enhanced macroscopic damping, or longitudinal — next to transverse — relaxation dynamics at elevated temperatures. A different approach to construct effective magnetization models which account for thermal activation is proposed in [14], where mutual orthogonality of vectors m, m × Heff , and m × (m × Heff ) is used to describe the temperaturedependent damped gyroscopic precession by means of ∂m α

= κ m + m × Heff − m × (m × Heff ) ∂t m ∂m =0 ∂n m(0, ·) = m0

on R+ × O on R+ × ∂O

(7)

on O .

The leading Bloch relaxation term again allows for shrinking (κ < 0), extension (κ > 0), and conservation (κ = 0) of the magnetization length at finite temperatures, where the function κ is chosen to meet the following phenomenological power-law by Landau below the Curie temperature τC ,  τ β , m(τ  )=m 0 1 − τC for some β > 0. Both models, the one which comes from a moment closure approximation of (6), and (7) are phenomenological and lack a rigorous derivation from the mesoscopic model (4)–(5), so that it remains unclear whether these descriptions properly describe the magnetization dynamics at elevated temperatures which is governed by (4)–(5). A major advantage of both macrospin models is their capability to describe space-time multiscale magnetization, while the spin model (4)–(5) is practically restricted to nanometer scales; a major disadvantage, however, is that additional material functions are needed, such as e.g. α

≡α

(τ ) in order to reliably model microscopic dissipative effects on a macroscopic scale for temperatures τ ∈ [0, τC ). This work reports on recent developments concerning the analytical and the numerical treatment of the SLLG equation, addressing in particular the following questions: (i) Finite ensembles: long-time behavior. The fluctuation-dissipation relation from physics determines the noise in dissipative non-equilibrium systems. For one spin and a simplified field Heff , formal arguments in [62] show that the stationary distribution of (1) is Gibbsian. We remark that (1) is not

Introduction

5

a gradient system with additive noise, for which the invariant measure is known to be of this type. In Chapter 1, we show uniqueness, and exponential ergodicity of an invariant measure of Gibbs type for (1). A structurepreserving numerical discretization is proposed which yields S2 -valued iterates, inherits the Lyapunov structure, as well as the discrete ergodicity property, and thus converges to the solution of the SLLG equation both, at finite and infinite times. We remark that to construct a convergent discretization is non-trivial because of the Stratonovich stochastic integral and the weak coercivity properties of the nonlinear drift function in (1), which is why general time-explicit integrators may even not converge at finite times [84, 62], or may fail to be ergodic; see e.g. [91, 88, 92]. Another subject which is addressed is the interplay of stochasticity and (R3 )N -valued solutions which approximate (S2 )N by a penalization strategy: a main observation here is that such an approximation also requires a modification of the noise term in order to ensure a proper long-time dynamics. (ii) Infinite ensembles: blow-up behavior and long-time dynamics at elevated temperatures. Possible finite-time finite-energy blow-up behavior of initially smooth solutions of the deterministic Landau-Lifshitz-Gilbert equation on bounded domains O ⊂ R2 is motivated by computational studies in [17]. In Chapter 2, an implementable finite element based space-time discretization is proposed for bounded Lipschitz domains O ⊂ Rn (n ≤ 3), where iterates construct a weak martingale solution of the SLLG equation (4)–(5) for vanishing discretization parameters. This discretization is structure-preserving, i.e. solutions satisfy a (pointwise) sphere-property, as well as an energy estimate. Computational studies motivate possible pathwise blow-up of solutions, but a smooth evolution of related expectations in the presence of space-time white noise. (iii)Effective macro-spin magnetization dynamics in a heat bath. A challenging goal is to derive macroscopic equations to properly describe macro-spin magnetization dynamics for a broad range of temperatures. Macro-spin m] are first moments of solutions of the infinitemagnetizations m = E[m dimensional SLLG equation (4)–(5), which in the case of the single spin model is approximated in [59] by the solution of the phenomenological LLB model. An independent, simple description of magnetization dynamics leads to (7), where changes of the magnetizations are described in terms of the current magnetization, the torque, and the damping term, together with the Landau power law to account for temperature effects on the saturation magnetization. Comparative computational studies for both, the stochastic mesoscopic system (4)–(5), and the macroscopic model (7) are provided, which motivate increased dissipativity for system (7) at elevated temperatures.

6

Introduction

The main goal in this work is to use constructive methods to verify mathematical results — for instance, to construct an invariant measure in part (i) by a structure-preserving time discretization, and a weak martingale solution in (ii) by finite element based space-time discretizations. This approach then provides a theoretical foundation for computational simulations with such schemes to study phenomena which so far lack a rigorous analytical understanding — such as e.g. the (long-time) dynamics of the stochastic partial differential equation (4)–(5) with space-time white noise. The following three chapters address items (i) to (iii). Chapters 1 and 2 each start with a preliminary section which provides relevant background material. Numerical schemes are proposed in the main parts of the different chapters in order to construct strong SDE-solutions and corresponding invariant measures (Chapter 1), a weak martingale solution for the SPDE (4)–(5) (Chapter 2), and a weak resp. strong solution of (7) (Chapter 3). These schemes are implemented, and corresponding simulations are discussed in each chapter to complement our theoretical results. Most of the work was done when the third author (M. N.) was affiliated with the Universität Tübingen and supported by the DFG-project: ‘Long-time dynamics of the Landau-Lifshitz-Gilbert equation’ (2011–2013); partial support by the ARC Discovery grant DP120101886 is also gratefully acknowledged. Chapters 1 resp. 2 base on ideas from [95] resp. [12], which are considerably extended here. The analysis of the model in Chapter 3 uses concepts from [14].

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles We study the effect of noise on a ferromagnetic chain consisting of N spins, where the magnetization process X : R+ × Ω → (S2 )N evolves according to     ∂X = X × Heff + Hthm − α X × X × Heff . (1.1) ∂t Thermal fluctuations are usually taken into account by augmenting the effective field Heff ≡ Heff (X) = −∇E(X) (see (2)) in the Landau-Lifshitz-Gilbert equation with an isotropic Gaussian white noise field to represent different processes involving magnon, phonon and electron interactions; cf. (1)–(3). This model has been suggested by Brown [24] to study transition states for thermally activated magnetization reversal of a single spin. Empirical studies indicate a different magnetization dynamics at elevated temperatures, such as enhanced damping, increased relaxation rates, and a shrinking saturation magnetization for increasing temperatures [38, 62]. However, starting with [24], most works study only a single spin and the anisotropic energy E = Eani in (2), see e.g. [82, 62], and the references therein. In particular, it has been shown formally in [62] that the Gibbs distribution (with N = 1) 2α

μ[dx] =



e− ν 2 E(x) dx 2α

e− ν 2 E(x) dx

(1.2)

(S2 )N

is the stationary distribution of the stochastic Landau-Lifschitz-Gilbert equa˙ where W denotes an (R3 )N -valued tion (SLLG) (1.1). Here Hthm = ν W, Wiener process, and the Stratonovich form of the stochastic integrals is used in (1.1). On the other hand, according to statistical mechanics, a system in thermal equilibrium is described by the Maxwell-Boltzmann statistics, and consequently the stationary distribution has the form

e

− k 1 τ E(x) B

e

dx

− k 1 τ E(y) B

, dy

(S2 )N

where kB is the Boltzmann constant, and τ ≥ 0 denotes the temperature of the system. Thus we can deduce the following fluctuation–dissipation relation 2α 1 , = ν2 kB τ

(1.3)

8

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

which determines the constant ν > 0 in terms of the temperature in (3). We recall that the basis of this relation to hold is a separation of time scales, where the relaxation time of the heat bath is assumed to be much faster than that of the spin system. Next to thermodynamically consistent equilibria, a physically relevant quantity in the modelling of non-uniform magnetization reversal is the relaxation time, which is the characteristic time for the N -spin system to reach an equilibrium. In Theorem 1.7, we state exponentially fast relaxation of (1.1) to its unique equilibrium for finite ensembles of nanomagnetic particles, i.e. we prove ergodicity of the Gibbs distribution (1.2), with exponential rate of convergence ρ > 0. For simplicity, we consider only the exchange energy Eexch , but the other two energies Eani , Eext may easily be added and do not alter the result; cf. Remark 1.26 for the general case. The rate ρ is related to the Néel-Brown relaxation time τN B = ρ1 of the system, which is the subject of a vast number of physical papers, see e.g. [1, 24, 42], and others. The technical difficulty of the result stems from the fact that the noise is degenerate if we consider the evolution of the system in (R3 )N . Consequently, we need to incorporate the ‘sphere-property’ of each single spin into the configuration space of the system, and hence study the evolution of the system on the compact Riemannian manifold (S2 )N . Another difficulty, when compared to results in [88], lies in the fact that the noise is multiplicative and, consequently, control-type arguments as in [88, Lemma 3.4] to establish the irreducibility of the system are not applicable. Indeed, it is well–known that in general the solution of a SDE is not a continuous function of the driving process in the topology of the space of continuous functions. To circumvent these issues, we apply instead the Girsanov theorem to find a proper representation of transition semigroups, which allows to conclude its irreducibility by the one of the corresponding Wiener process. Furthermore, we show that the energy E is a Lyapunov function of (1.1), and that the transition semigroup satisfies certain regularity properties. The Lyapunov property also proves to be important in Chapter 2 about the corresponding stochastic PDE (4): it appears as an energy inequality in the infinite dimensional situation and allows to show convergence of the numerical scheme which is discussed in this chapter. The result in Theorem 1.7 does not provide a precise rate of convergence towards the equilibrium, which motivates Theorem 1.8 where we show that the exponential rate of convergence in the weaker L2 (S2 )N ; μ -topology is estimated from below by β = ν 2 N κe

−

2 osc(E) kB τ

,

(1.4)

where N is the number of spins, κ is the spectral gap of the Laplace-Beltrami operator on the sphere S2 , and osc(E) = sup E − inf E. Notice that β is an

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

9

increasing function of the temperature τ , and hence a decreasing function with respect to the damping parameter α by (1.3), which contradicts the intuition that the more damping we put on the system the faster becomes the convergence to equilibrium (see Figure 1.1 for numerical simulations of the rate of convergence). We develop several strategies to approximate problem (1.1). We start in Subsection 1.2.4 with an ‘outer approximation’ in (R3 )N with the help of the Ginzburg-Landau penalization term. This approach is motivated by numerical demands where the restriction to schemes with sphere-valued solutions requires to construct non-standard discretizations, and rules out many wellknown (high-order) discretizations; another motivation is to study the impact of stochastic forcing onto approximately sphere-valued solutions, including the asymptotic regime t → ∞. A main observation is that a relaxation of the sphere property of solutions has to go together with a modification of the noise in order to ensure proper (approximate) long-time dynamics. For this purpose, we compare two approximate problems (cases δ = 0 and δ > 0 in system (1.55)). In the first one the noise of the penalized system is the same as for the limiting system. The second one has an additional additive noise. Our results show that the system (1.55) with conservative noise (δ = 0) behaves better on the finite time interval. The problem inherits a natural energy inequality (Proposition 1.37), the solution stays in the unit ball, and converges on each finite time interval to the solution of the non-penalized system if initial data are sphere-valued (Theorem 1.10). The system allows for several natural choices of a configuration space but, as discussed in the Remark 1.33, neither of them justifies the strong irreducibility property; thus, ergodicity of the system is not clear. The modification of the noise (case δ > 0) improves control over the long-time dynamics of the system (1.55): we are then able to show the geometric ergodicity of the system, i.e. the system exponentially converges to the unique invariant measure (Theorem 1.9). Furthermore, if the additive noise is sufficiently small, the solution converges to the solution of the non-penalized system (Corollary 1.39). Thus, we see that the behavior of the system is very sensitive with respect to the type of the used noise, and convergence to the limiting system for finite times is guaranteed only for sufficiently small δ > 0. These issues motivate a second approximation strategy, which uses discretization in time of the system (1.1) where the geometric constraint is preserved at each step of the simulation. In Section 1.3, we present two numerical schemes to simulate system (1.1); as it is well-known, naive time discretizations of SDEs may easily loose not only the geometric rate of convergence, but overall asymptotic convergence properties; see Subsection 1.4.2 for computational evidence. The first scheme (Scheme 1.11) is nonlinear implicit and yields an (S2 )N -valued discrete Markov chain, which inherits the Lyapunov function property from the limiting system. As a consequence, we may show geometric exponential ergod-

10

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

icity of the system with the same method which is used to verify Theorem 1.7. Furthermore, we show local in time strong rates of convergence towards the continuous process for corresponding iterates. This result, together with the geometric ergodicity property for the limiting equation (1.1) from Theorem 1.7 implies convergence of invariant measures from the numerical scheme to the Gibbs measure (1.2), as a consequence of the general results of Shardlow & Stuart in [101]. The second scheme (Scheme 1.16) is linear implicit, and hence computationally more efficient. Iterates of this discrete Markov chain are again (S2 )N -valued, but the discrete Lyapunov condition is not available any more. As a consequence, tools for the first scheme do not apply to verify geometric ergodicity. However, we are able to show convergence of invariant measures to the unique time-asymptotic Gibbs distribution (1.2) of (1.1) for a vanishing discretization parameter by the perturbation result of Shardlow & Stuart in [101]. We also show an optimal rate of weak convergence for finite times. These results are complemented by computational studies in Section 1.4, where evidence is provided that numerical schemes may fail to approximate proper long-time dynamics if the sphere-property of iterates is not accounted for; another series of experiments studies the effect of penalization. Furthermore, computational studies with different projection methods are reported which are related to penalization concepts, and are often used to solve the related deterministic problem (LLG). The chapter is organized as follows: in Section 1.1, we collect background material on ergodic properties of Markov chains, which is used in Section 1.2 to verify exponential ergodicity of the invariant Gibbs measure (1.2) for (1.1). Time discretization schemes, and penalization methods to approximate (1.1) are studied in Section 1.3. Computational studies are reported in Section 1.4. 1

α=0 α = 0.5 α=2 α=5

0.8

0.6

0.4

0.2

0 0

1

2

3 time

4

5

6

Figure 1.1. Scheme 1.11: Speed of convergence to the stationary distribution of (1) for different values of the parameter α ∈ {0, 0.5, 2, 5}.

Section 1.1

1.1

11

Preliminaries

Preliminaries

We collect some results on geometric ergodicity of Markov chains in Subsection 1.1.1. In Subsection 1.1.2 we recall strategies to conclude ergodicity with rates for solutions of SDEs. Subsection 1.1.3 surveys different convergent discretizations of the deterministic LLG equation.

1.1.1

Geometric ergodicity of Markov chains

Here we recall the Meyn-Tweedie theory [89]. We follow the presentation from [88]. Let X ⊂ Rd be a smooth Riemannian manifold, and T be either R+ or Z+ . Let X:= {X(t);  t ∈ T} be a Markov process (or a Markov chain) on a state space X, B(X) , where B(X) is the σ-field of Borel subsets of X. Let   P (t, x, A ) := P {X(t) ∈ A ; X(0) = x} ∀ t ∈ T ∀ x ∈ X ∀ A ∈ B(X) be the transition kernel of the process X. Let Bb (X) denote the set of Borel measurable bounded real-valued functions. Define the semigroup Pt : Bb (X) → Bb (X) for t ∈ T, which is associated with the process X by its values on the indicator function of Borel subsets of X: Pt 1A (x) := P (t, x, A )

∀ t ∈ T ∀ x ∈ X ∀ A ∈ B(X) .

If T := R+ then we denote by L the infinitesimal generator of the semigroup {Pt ; t ∈ T}. Let Bδ (x) ⊂ Rd denote a closed ball around x of radius δ > 0. Definition 1.1. A Markov process (or chain) X with transition probability P (t, ·, ·) is weakly irreducible iff there exists a compact set C ⊂ X with non∗ empty interior such that for  some y ∗ ∈ Int(C ), for any δ > 0, there exists ∗ t ≡ t(δ, y ) ∈ T such that P t, x, Bδ (y ) > 0 for all x ∈ C . A Markov process (or chain) X with transition probability P (t, ·, ·) is strongly irreducible iff for any y ∈ X, t > 0 and any open set A ⊂ B(X) we have P (t, y, A ) > 0. Definition 1.2. A Markov process (or chain) X with transition probability

P (t, ·, ·) is regular  iff the transition kernel has a nonnegative density p(t, x, y); t ∈ T, x, y ∈ X , such that P (t, x, A ) = p(t, x, y) dy ∀ t ∈ T ∀ x ∈ X ∀ A ∈ B(X) , A

where p(t, ·, ·) ∈ C(X2 ) for any t ∈ T.

12

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Definition 1.3. A Markov process (or chain) X satisfies the minorization condition iff there exist s ∈T, an η > 0, a compact set C ⊂ X, and a probability measure ν on C , B(C ) such that P (s, x, A ) ≥ η ν[A ]

∀ A ∈ B(C ) ∀ x ∈ C .

Lemma 1.4. If a Markov process X is weakly irreducible and regular then it satisfies the minorization condition. Proof. Step 1. Local discussion. By the irreducibility assumption, there exist a compact set C ⊂ X, and y∗∈ Int(C ) such  that for any δ > 0 there exists a ∗ ∗ time t = t(δ) > 0 such that P t, y , Bδ (y ) > 0. Then we can  find a possibly  smaller neighborhood Bδ1 (y∗ ) ⊂ C , and t1 > 0 such that P t1 , y∗ , Bδ1 (y∗ ) > 0. Indeed, since y∗ is in the interior of C , there exists a γ > 0 such that Bγ (y∗ ) ⊂ C and we may take δ1 := γ2 and t1 := t( γ2 ). The existence of a density implies that there exists z∗ ∈ Bδ1 (y∗ ) ⊂ C , and some  > 0 such that p(t1 , y∗ , z∗ ) ≥ 2 > 0 . By the regularity assumption, there exist neighborhoods Br (y∗ ), Br (z∗ ) ⊂ C such that ∀ y ∈ Br (y∗ ) ∀ z ∈ Br (z∗ ) . p(t1 , y, z) ≥  Hence we have that



P (t1 , y, A ) =

p(t1 , y, z) dz ≥

A

p(t1 , y, z) dz A ∩Br

  ≥  Leb A ∩ Br (z∗ )

(1.5)

(z∗ )

∀ y ∈ Br (y∗ ) ∀ A ∈ B(X) ,

where Leb : B(X) → R+ is the Riemannian volume measure on X. Step 2. Global discussion in C . By the irreducibility assumption, there exists a time t2 > 0 such that   P t2 , x, Br (y∗ ) > 0 ∀x ∈ C .   Furthermore, by the regularity assumption, the function P t2 , ·, Br (y∗ ) is continuous on the compact set C . Thus,   min P t2 , x, Br (y∗ ) ≥ γ1 > 0 . x∈C

Section 1.1

13

Preliminaries












Figure 1.2. Illustration of the application of the Kolmogorov-Chapman equation in formula (1.6)

Consequently, by the Chapman-Kolmogorov equation and (1.5) we find (see Figure 1.2)   P t1 + t2 , x, A ≥ p(t2 , x, w)P (t1 , w, A ) dw Br (y∗ )



∗





≥  Leb A ∩ Br (z ) 

p(t2 , x, w) dw (y∗ )

Br  ∗

≥ γ1  Leb A ∩ Br (z )   ∗   ∗ Leb A ∩ Br (z ) . = γ1  Leb Br (z ) Leb[Br (z∗ )]   ∩Br (z∗ )] We may now put η := γ1  Leb Br (z∗ ) , and ν := Leb[· Leb[Br (z∗ )] .

(1.6)

Definition 1.5. The mapping V : X → [1, ∞) is a Lyapunov function for the Markov chain {Xj }∞ j=0 if there exist numbers α ∈ (0, 1), and β ∈ [0, ∞) such that     E V (Xj+1 ) σ {X0 , X1 , . . . , Xj } ≤ αV (Xj ) + β , and V is unbounded if the set X is unbounded, i.e., lim

dist(y,x)→∞

V (y) = ∞

∀x ∈ X.

Furthermore, we assume that level sets {y ∈ X; V (y) ≤ a}, a > 1 are either compact or contain compact subsets such that their union (over a) is X. If T = R+ , we can reformulate Definition 1.5 in terms of the infinitesimal generator L of the semigroup {Pt ; t ∈ T} associated with X.

14

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Definition 1.6. A mapping V : X → [1, ∞) is a Lyapunov function for the Markov process X with generator L if there exist constants 0 < c, d < ∞, such that LV ≤ −cV + d , (1.7) and V is unbounded if the set X is unbounded, i.e., lim

dist(y,x)→∞

V (y) = ∞

∀x ∈ X.

Furthermore, we assume that level sets {y ∈ X; V (y) ≤ a}, a > 1 are either compact or contain compact subsets such that their union (over a) is X. Below, we collect a series of propositions which describe the behavior of a Markov process X under the assumption that there exists a Lyapunov function. First we show that the constant β in Definition 1.5 can be replaced by zero outside of a certain compact subset of X at the expense of an increased constant α. Let 1C : X → {0, 1} denote the characteristic function of C . Proposition 1.7. Assume that {Xj }∞ j=0 is a Markov chain with Lyapunov function V : X → [1, ∞). Let γ ∈ (α, 1), and s ≥ 1, and denote  sβ  C (s, γ) := x ∈ X; V (x) ≤ . γ−α Then

     E V (Xj+1 )σ {X0 , X1 , . . . , Xj } ≤ γV (Xj ) + sβ 1C (s,γ) (Xj .

Proof. Fix j ≥ 0. The result is evident if Xj ∈ C (s, γ). Otherwise, V (Xj ) > sβ j j j γ−α . Consequently γV (X ) > αV (X ) + sβ ≥ αV (X ) + β, and the result follows. The next results asserts polynomial convergence to 0 of a Lyapunov function as time converges to infinity. Let a ∧ b := min{a, b}. Proposition 1.8. Let {Xj }∞ Lyapunov function j=0 be a Markov process with   V : X → [1, ∞) (with parameters α and β), that Fj := σ {X0 , X1 , . . . , Xj } , j ≥ 0, J is a stopping time, γ ∈ (α, 1), and C := C (2, γ). Then there exists some C > 0 such that    E V (Xj )1{J>j} ] ≤ E V (Xj )1{J≥j} j∧J      0 ≤ Cγ E V (X ) + E γ −l 1C (Xl−1 ) j

l=1

 γj    ≤ C E V (X0 ) + 1 1−γ

(j ≥ 1) .

Section 1.1

15

Preliminaries

Proof. The first inequality is trivial. The third inequality immediately follows from the following elementary estimate γ

j

j∧J 

γ

−l

1C (X

l−1

l=1

)≤

j 

γ j−l ≤

l=1

1 . 1−γ

To show the second inequality we consider a finite differences representation for the function F : X × T → R+ , defined by F (X, j) := γ −j V (X). We have J∧j           F XJ∧j , J ∧ j = F X0 , 0 + F Xl , l − F Xl−1 , l − 1) l=1

  = F X0 , 0 +

j 

 







1{J>l−1} F Xl , l − F Xl−1 , l − 1) .

l=1

Taking the expectation and applying the tower property for conditional expectation leads to       E F XJ∧j , J ∧ j = E F X0 , 0 + +

j           E E 1{J>l−1} F Xl , l − F Xl−1 , l − 1 Fl−1 . l=1

  Notice that the event {J > l − 1} is Fl−1 –measurable, and F X0 , 0 = V (X0 ). Hence,      (1.8) E F XJ∧j , J ∧ j = E V (X0 ) + j      + E 1{J>l−1} E F (Xl , l) − F (Xl−1 , l − 1)Fl−1 . l=1

We apply Proposition 1.7 with s = 2 to conclude that      E F (Xl , l)|Fl−1 = γ −l E V (Xl )Fl−1     ≤ γ −l γV Xl−1 + 2β 1C (Xl−1 )   = F Xl−1 , l − 1 + 2γ −l β 1C (Xl−1 ) .

(1.9)

We may combine identity (1.8) with inequality (1.9) to deduce that 

E F (X

J∧j

j      0 , J ∧ j) ≤ E V (X ) + 2β E γ −l 1{J>l−1} 1C (Xl−1 ) . l=1

The result then follows from the estimate         E F (XJ∧j , J ∧ j) ≥ E F XJ∧j , J ∧ j 1J≥j = γ −j E V (Xj )1{J≥j} .

16

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

We obtain the following estimates on the first return time τC := min{j > 0; Xj ∈ C } to the set C . Corollary 1.9. Let {Xj }∞ j=0 be a Markov process with Lyapunov function V : X → [1, ∞), and C := C (2, γ) ⊂ X for γ ∈ (α, 1). Then there exists C > 0 such that       (j > 0) , (i) P {τC > j} ≤ C E V (X0 ) + 1 γ j     −τ   (ii) E γ C ≤ C E V (X0 ) + 1 . Proof. We apply the second inequality of Proposition 1.8 with stopping time J = τC . The definition of τC implies that j∧τ C

γ j−l 1C (Xl−1 ) = γ j−1 1C (X0 ) .

l=1

Furthermore,   E V (Xj )1τC >j ≥

 2β 2β  E[1τC >j ] = P τC > j . γ−α γ−α

Hence assertion (i) follows. To show assertion (ii), we observe that 

E γ

−τC



=

∞ 

γ

−l

P



τC = l

l=1



≤

∞ 



γ −l P

 τC > l − 1 .

l=1

Since γ ∈ (α, 1) we can apply (i) with γ ∈ (α, γ) to conclude that there exists κ1 > 0 such that ∞  l       γ E γ −τC ≤ κ1 E V (X0 ) + 1 . γ l=1

The previous Corollary can be generalized to estimate the time τr (C ) := τ[r] (C ), r ≥ 0 of the [r]-th visit to the set C (put τ0 := 0). Corollary 1.10. Assume that {Xj }∞ j=0 is a Markov process with Lyapunov function V : X → [1, ∞), and C := C (2, γ) ⊂ X for γ ∈ (α, 1). There exists a positive constant C ≡ C(C , V ) such that r−1        E γ −τr (C ) ≤ C r sup V + 1 E V (X0 ) + 1 . C

Section 1.1

17

Preliminaries

  Proof. By definition, we can assume that r ∈ N. Denote A(r) := E γ −τr (C ) for r ∈ N. We have by elementary properties of the conditional expectation that for r > 1  r  A(r) = E γ − l=1 [τl (C )−τl−1 (C )]   r  = E E γ − l=1 [τl (C )−τl−1 (C )] |Fτr−1 (C )  r−1   = E γ − l=1 [τl (C )−τl−1 (C )] E γ −[τr (C )−τr−1 (C )] |Fτr−1 (C )  r−1    = E γ − l=1 [τl (C )−τl−1 (C )] E γ −τ1 (C ) Xτr−1 (C )    = A(r − 1) E γ −τ1 (C ) Xτr−1 (C ) , and the result follows from Corollary 1.9, (ii). Corollary 1.11. Assume that {Xj }∞ j=0 is a Markov process with Lyapunov function V : X → [1, ∞). Then there exists an invariant probability measure. Proof. We apply Proposition 1.8 with a constant stopping time J = j to conclude that   sup E V (Xj ) < ∞ . j≥1

Now the existence of invariant measure follows from a standard argument. Indeed, by the Chebyshev inequality, and compactness of the level sets of function V , the sequence of measures n  1  l μn := P {X ∈ ·} (n ∈ N) , n l=1

is tight (Chapter 2, Definition 2.16). Therefore, by the Prohorov Theorem (Chapter 2, Theorem 2.2) there exists a convergent subsequence to the measure μ. Consequently, the measure μ is finite and invariant. Normalizing it if necessary, we obtain an invariant probability measure. Theorem 1.1. Let X be a Markov process (or chain) with transition kernel j P . Fix T > 0. Let {Xj }∞ j=0 , with X := X(jT ) be an embedded Markov chain with transition kernel P (T ). Assume that the Markov chain {Xj }∞ j=0 has a Lyapunov function V : X → [1, ∞) (with parameters α and β), and satisfies the minorization condition with the set  2β  C := C (2, γ) = x; V (x) ≤ γ−α √ for some γ ∈ ( α, 1) and parameter η. Then there exist a unique invariant measure μ, and constants r := r(γ) ∈ (0, 1), κ := κ(γ) ∈ (0, ∞), such that   0        X j f (X ) − f dμ ≤ κrj E V (X0 ) + 1 ∀ measurable f : |f | ≤ V . E X

18

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Proof. Step 1: Construction of an equivalent Markov chain with atomic structure. In this step we will construct a Markov chain {Zj }∞ j=0 with the same j ∞ transition kernel as {X }j=0 , which has an atomic structure, i.e. there exists a subset of the configuration space of non-zero probability such that the transition kernel of the Markov chain is the same for all points of the subset. The minorization condition implies that we can define a new transition kernel as follows: ⎧ ⎨ P (x, A ) (x ∈ / C), P(x, A ) := ⎩ P (x,A )−ην[A ] (x ∈ C ) . 1−η

Let

 X  j, ω  j+1 = h( ) X

( ω ∈ Ω)

be the corresponding Markov chain with transition kernel P. Define the new Markov chain Zj+1 := h(Zj , ωj ) (1.10)   j , φj , ξ j are i.i.d. random variables, where ωj := ω        ω  x, ω  ) + (1 − φ)ξξ + 1 − 1C (x) h  , h(x, ω) := 1C (x) φh(x,    , φ, ξ , where φ, ξ are random which and ω1 is distributed as ω := ω  variables  are independent from ω  , such that P φ = 1 = 1 − η, P φ = 0 = η, and ξ is distributed according to ν. Elementary calculations then imply that the transition kernel of the chain {Zj }∞ j=0 is the same as of the initial Markov chain j ∞ {X }j=0 . j ∞ Step 2: A coupling argument. Let {Zj }∞ j=0 and {Z }j=0 be two realizations of ξ j }∞ the Markov chain (1.10) with the same random variables ({φj }∞ j=0 , {ξ j=0 ) and 1 ∞ 2 ∞ l  l ); l ≤ ωj }j=0 . Denote Fj := σ{(Z , Z independent random variables { ωj }j=0 , { j} for j ≥ 0. Our aim is to estimate the difference        j )  , E f (Zj ) − E f (Z for measurable f such that |f | ≤ V . Without loss of generality we can assume that f is a non-negative function; otherwise, we may decompose f as a difference of non-negative functions. We define the coupling time by

  j ) ∈ C × C ; φj = 0 . ψ := inf (Zj , Z j≥0

Notice that

      E f (Zj ) = E f (Zj )1j≥ψ + E f (Zj )1j 0 such that   (1.80) sup E X ,δ (t) − Y(t)2 ≤ C2 . t∈[0,T ]

Proof. The result follows from Theorem 1.10 and Lemma 1.38. A natural idea at this point would be to try to apply the result of Shardlow and Stuart [101, Thm. 3.3] (using results of Theorem 1.7 and Corollary 1.39) to conclude that invariant measures of the SDE (1.56) weakly converge to the Gibbs measure (1.2) as the parameters , δ tend to zero. The problem with this approach is that invariant measures of the SDE (1.56) are supported on (R3 )N while the Gibbs measure (1.2) is supported on (S2 )N . Consequently, the result of Shardlow and Stuart cannot be applied, and the proof of convergence remains an open problem. In the following Section 1.3, different convergent numerical schemes to approximate solutions of the SLLG equation are proposed, whose distinct feature is structure-preservation. In particular, the configuration space of the schemes is (S2 )N . Consequently, invariant measures of schemes are also supported on (S2 )N , and the problem explained above will disappear.

1.3

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation

In this section we will study various discretizations of the stochastic LandauLifshitz-Gilbert equations in the form (1.37). The Scheme 1.11 below yields (S2 )N -valued iterates, a discrete energy bound, and the long-time dynamics is governed by an invariant measure which approximates the one from (1.37); see Subsections 1.3.1 and 1.3.2. A simplified version of this discretization is Scheme 1.16 in Subsection 1.3.3 which yields (S2 )N -valued iterates and reliable long-time dynamics, but which is not known to satisfy a discrete energy identity.

1.3.1

A structure-preserving discretization of (1.36): the geometric exponential ergodicity

We present a discretization of (1.37) whose iterates inherit the ‘unit-length’ property and the Lyapunov structure of (1.37), see Theorem 1.12, and also the geometric exponential ergodicity property which is stated in Theorem 1.13. Strong convergence with rates to solutions of (1.37) at finite times, as well as convergence of the (discrete) ergodic measure towards the one of (1.37) will be shown in Section 1.3.2.

68

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Fix T > 0. Let k > 0 denote the uniform size of a mesh Ik = {tj }Jj=0 which covers [0, T ], and {Δj Wi }j=0,... i=1,...,N denotes i.i.d. Gaussian random variables in R3 , where each Δj Wi := Wi (tj+1 ) − Wi (tj ) ∼ N (0, kIdL (R3 ) ). For simplicity, we assume below that ci = ν = 1 for all i ∈ N and restrict to the exchange energy function E = Eexch ; cf. (1.30).  j  be an (R3 )N -valued Scheme 1.11. For all j ≥ 0, let Mj = M1j , . . . , MN random variable which solves the following iterative scheme,    j+ 1 j+ 1 J Mj+1 )i − αMi 2 × (J J Mj+1 )i k Mij+1 − Mij = Mi 2 × − (J  +Δj Wi (1.81) Mi0 = Qi ,

Qi ∈ S2 , (1.82)   j+ 1 for all i = 1, . . . , N . We denote Mi 2 := 12 Mij + Mij+1 .   For j ≥ 0, let Fj = σ Mir ; i = 1, . . . , N, r = 0, . . . , j denote the natural filtration, and Q = (Q1 , . . . , QN ) ∈ (S2 )N . We show that the energy function E is a Lyapunov function for the Scheme 1.11.

   Theorem 1.12. There exists Mj ; j ≥ 0 ⊂ L∞ Ω, (R3 )N , a unique adapted solution of Scheme 1.11 which satisfies (i) (ii)

where

|Mij | = 1 ∀j ≥ 0 (1 ≤ i ≤ N ) , P-almost surely , N     j+ 1    J Mj+1 )i |2 Fj E E(Mj+1 )Fj + αk E |Mi 2 × (J i=1

  J , α . ≤ (1 − 2k)E(M ) + k C N, J j

Proof. Step 1. Solvability of System 1.11. Fix a set Ω ⊂ Ω, P[Ω ] = 1 such that {Δj Wi }j=0,... us assume i=1,...,N are finite for all ω ∈ Ω . In the following, let  that ω ∈ Ω . We will show the existence of a solution Mj (ω); j ≥ 0 ⊂ (R3 )N of Scheme 1.11 by induction via the Brouwer fixed point theorem. To prove the existence of Mj+1 (ω) ∈ (R3 )N let us suppose that we found Mj (ω) for some j ≥ 0. Define a continuous mapping Gj (ω, ·) : (R3 )N → (R3 )N by the identity         Gj (ω, X) = k −X × J 2X − Mj (ω) + αX × X × J 2X − Mj (ω)   +2 X − Mj (ω) − X × Δj W .  1/2 . For all X ∈ (R3 )N such that X ≥ Mj (ω) we We denote  ·  := ·, · have !! !!   1 Gj (ω, X), X = X2 − X, Mj (ω) ≥ ||X X|| − Mj (ω) ≥ 0 . 2

Section 1.3

69

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation

The Brouwer fixed point theorem, see e.g. [60, Corollary IV.1.1, p. 279] implies the existence of X∗ ∈ (R3 )N such that Gj (ω, X∗ ) = 0. Then Mj+1 (ω) = 2X∗ − Mj (ω) solves Scheme 1.11. We will omit here a proof of the adaptedness of the discrete process {Mj ; j ≥ 0}. This issue will be discussed in full detail later for the case of the corresponding SPDE in Theorem 2.11 (see Step 2 of Lemma 2.22). Step 2. The iterates are (S2 )N -valued. We can deduce from equation (1.81) j+ 1

that the scalar product of Mij+1 − Mij with Mi 2 is zero for all i = 1, . . . , N , and all j ≥ 0. Consequently, |Mij+1 | = |Mij | for all i = 1, . . . , N , and all j ≥ 0. Now, the ‘unit-length’ property of spins as stated in assertion (i) immediately follows from the assumption that Mi0 = Qi ∈ S2 , with i = 1, . . . , N . Step 3. Stability. We take the scalar product in (R3 )N of equation (1.81) with J Mj+1 , i.e. we have N 

N !  ! j+ 1 J Mj+1 )i k J Mj+1 )i , Mij+1 − Mij = J Mj+1 )i , Mi 2 × (J (J −(J

i=1

i=1

− αk

N 

j+ 12

J Mj+1 )i , Mi −(J

i=1

+

N 

j+ 12

J Mj+1 )i , Mi (J

!  j+ 1 J Mj+1 )i × Mi 2 × (J

(1.83)

! × Δj Wi .

i=1

The left-hand side of equality (1.83) can be transformed as follows, N  i=1

! J Mj+1 )i , Mij+1 − Mij = (J

J Mj+1 , Mj+1 − Mj

!!

!! √ √ √ 1 J Mj+1 , J Mj+1 − J Mj = 2 2  √ √ 1 √  J Mj+1 2 −  J Mj 2 +  J (Mj+1 − Mj )2 = 2 ∀j ≥ 0, = E(Mj+1 ) − E(Mj ) + E(Mj+1 − Mj )

(1.84)

where the third equality follows from the standard identity 2a, a − b = |a|2 − |b|2 + |a − b|2 for a, b, c ∈ R3 . Combining equalities (1.83) and (1.84), we deduce

70

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

that E(M

j+1

) + E(M

j+1

− M ) =E(M ) − αk j

j

N 

j+ 12

|Mi

J Mj+1 )i |2 × (J

i=1

+

N 

j+ 12

J Mj+1 )i , Mi (J

! × Δj Wi .

i=1

Hence, N  j+ 1       J Mj+1 )i |2 Fj E |Mi 2 × (J E E(Mj+1 )Fj ≤E(Mj ) − αk i=1

N  !   j+ 1 J Mj+1 )i , Mi 2 × Δj Wi |Fj . E (J +

(1.85)

i=1

In the remainder of the proof we look at the last term in inequality (1.85). Denote N !  j+ 1 J Mj+1 )i , Mi 2 × Δj Wi . (J (A) = By the definition of

conclude

N !  ! 1 J Mj )i , Mij × Δj Wi + J Mj )i , (Mij+1 − Mij ) × Δj Wi (J (J 2

N 

(A) =

i=1 j+1/2 we Mi

i=1 N 

+

i=1

!   j+ 1 J (Mj+1 − Mj ) i , Mi 2 × Δj Wi = (A1) + (A2) + (A3) .

i=1

For every 1 ≤ i ≤ N and every j ≥ 0, we define   j+ 1 j+ 1 j+ 1 J Xj+1 )i + αXi 2 × Xi 2 × (J J Xj+1 )i Fij (X) = −Xi 2 × (J Notice that

  J 1 + α |Fij (X)| ≤ J

∀ X ∈ (S2 )N .

∀ X ∈ (S2 )N . (1.86)

We use (1.81) to rewrite (A2) as follows, !   1 j+ 1 J Mj )i , Fij (M)k + Mi 2 × Δj Wi × Δj Wi (J 2 N

(A2) = =

k 2

i=1 N 

+

i=1

N ! 1 !   J Mj )i , Fij (M) × Δj Wi + J Mj )i , Mij × Δj Wi × Δj Wi (J (J 2

N 1

4



i=1

 ! J Mj )i , (Mij+1 − Mij ) × Δj Wi × Δj Wi (J

i=1

=(B1) + (B2) + (B3) .

Section 1.3

71

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation

Now we estimate conditional expectations with respect to the σ-field Fj of terms (A1) through (A3) introduced above. We have N     !  J Mj )i , Mij × E Δj Wi |Fj = 0 . E (A1)Fj = (J

(1.87)

i=1

Because of (1.86), we conclude N     !     k    j j     J ≤ E |(B1)| F E (B1) F E (J M ) , F (M) × Δ W ≤  j  j i j i Fj i 2 i=1

N   k   J Mj )i | |Fij (M)| |Δj Wi |Fl E |(J ≤ 2 i=1

N N   k      k  2 2 J  (1 + α) J  (1 + α) E |Δj Wi |Fj = J E |Δj Wi | ≤ J 2 2 i=1 i=1 √ √ 3 3 k J 2 (1 + α)N 3k = J 2 (1 + α)k 2 . N J (1.88) ≤ J 2 2 Similarly, we can deduce that           J , α k , (1.89) E (B3)Fj  ≤ E |(B3)|Fj ≤ C1 N, J

and

         J , α)k . E (A3)Fj  ≤ E |(A3)|Fj ≤ C2 N, J

(1.90)

Furthermore, by the Graßmann identity a × (b × c) = ba, c − ca, b for all a, b, c ∈ R3 we have N  !     1  J Mj )i , Δj Wi Mij , Δj Wi  − Mij |Δj Wi |2 Fj E (J E (B2)Fj = 2

=

1 2 −

i=1 N 

    J Mj )i , Δj Wi Mij , Δj Wi Fj E (J

i=1 N 

1 2

    J Mj )i , Mij |Δj Wi |2 Fj = ... E (J

i=1

By elementary calculations we continue as follows,   E (B2)Fj = 

N 3  p  q   1  J Mj )pi Δj Wi (Mij )q Δj Wi Fj E (J 2 i=1

−

1 2

N  i=1

p,q=1

    J Mj )i , Mij |Δj Wi |2 Fj . E (J

72

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Since Mj is Fj -measurable we deduce that    E (B2)Fj =

3 N  q  p  q   1  J Mj )pi Mij E Δj W i Δj W i Fj (J 2 i=1 p,q=1

N !    1 J Mj )i , Mij E |Δj Wi |2 Fj − (J 2 i=1

= −k

N 

! J Mj )i , Mij . (J

(1.91)

i=1

At last, combining inequality (1.85) with estimates (1.87), (1.88), (1.89), (1.90) and (1.91) settles the proof. Now we are in a position to show that there exists a unique invariant measure μk for the Markov chain M = {Mj ; j ≥ 0} in Scheme 1.11. Moreover, this Markov chain is ergodic with an exponential convergence rate. Theorem 1.13. Assume that the size of the mesh k > 0 is small enough. The Markov chain from Scheme 1.11 has a unique invariant measure μk which is supported on (S2 )N . We have geometric exponential ergodicity,   i.e. there exist Ck , ρk > 0 such that for all Q ∈ (S2 )N and all f ∈ Bb (S2 )N satisfying |f | ≤ E + 1,          E f (Mj )M0 = Q − μk [f ] ≤ Ck E(Q) + 1 e−ρk j

∀j ≥ 0.

Remark 1.40. Under general assumptions, the Markov chain continuously depends upon the driving process. Consequently, there is no problem with a control-type argument as is in the continuous case. Proof. We check the conditions of the Theorem 1.1 for the Markov chain {Mj ; j ≥ 0} on the manifold (S2 )N . Step 1. M is a regular chain. Notice that system (1.81)-(1.82) can be written as

 Mj+1 = Φ(Mj+1 , Mj , Δj W , M0 = Q ,  2 N  ∞ 2 N 3 N 2 N where Φ ∈ C (S ) × (S ) × (R ) ; (S ) , and Φ is Lipschitz with respect to the first argument, with a Lipschitz constant less than 1, provided the mesh size k is small enough. Consequently, by the Banach fixed point theorem there  N   exists Hk ∈ C∞ (S2 × R3 )N , (S2 )N such that   Mj+1 = Hk Mj , Δj W ,

M0 = Q .

(1.92)

Section 1.3

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation

73

Furthermore, let us notice that the chain is forward accessible, i.e. starting from some point in the configuration spacewe can reach point  any2other  within a fiN nite number of steps. Indeed, let x = x1 , . . . , xN ∈ (S ) , y = y1 , . . . , yN ∈ (S2 )N , and yi = −xi , for every i = 1, . . . , N . Then we have  2(x + y )   N i i × y − x − kF (x, y) , i i i |xi + yi |2 i=1  x i + yi x + y i  x i + yi J y)i + α i J y)i . Fi (x, y) = − × (J × × (J 2 2 2   y = Hk x, u(x, y) ,

u(x, y) =

Thus starting from a point x ∈ (S2 )N , we can reach every other point y ∈ (S2 )N in no more than two time steps. Indeed, if yi = −xi for all i = 1, . . . , N , then y may be reached within one step from the above procedure. Otherwise, two steps are needed, by first moving to an auxiliary point z such that zi = −yi , −xi , for i = 1, . . . , N . Hence, the Markov chain M is a regular chain. Step 2. M is strongly irreducible. It is enough to show that for any x ∈ (S2 )N and any open set A ⊂ (S2 )N   P (x, A ) = P Hk (x, Δ0 W) ∈ A > 0. Since the set A is open, there exists z ∈ A with zi = xi , for i = 1, . . . N such that Bδ (z) ⊂ A for some δ > 0. Now         P Hk x, Δ0 W ∈ A ≥ P Hk x, Δ0 W ∈ Bδ (z)    = P Δ0 W ∈ (Hk )−1 x, Bδ (z) > 0,   where the last inequality follows from the fact that the set (Hk )−1 x, Bδ (z) is andnonempty (it contains u(x, z)). Thus we have shown that M ≡

open Mj ; j ≥ 0 is a strongly irreducible chain. Step 3. The probability transition function of M has a continuous density. It can be shown by the induction principle from the representation (1.92) when the mesh size k is small enough. Step 4. Existence of a Lyapunov function for the Markov chain M. It is an immediate consequence of the assertion (ii) proved in Theorem 1.12.

Remark 1.41. To show the existence of a solution in the proof of Theorem 1.13 we have used the Banach fixed point theorem instead of the Brouwer fixed point theorem. For this purpose, we need an additional smallness assumption for the

74

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

mesh size in order to establish a contraction principle. The advantage of the Banach fixed point theorem is that it allows to conclude the existence of a smooth map Hk in equation (1.92). That, in its turn, leads to the proof of forward accessability and irreducibility of the chain. Furthermore, the adaptedness of the Markov chain becomes trivial.

1.3.2

Strong Convergence of Scheme 1.11

We show convergence of the Scheme 1.11 towards system (1.37). We fix a Wiener process W = W1 , . . . , WN ) constructed from N i.i.d. 3D Wiener processes Wi , and choose corresponding Gaussian random variables {Δj Wi }j=1,... i=1,...,N in Scheme 1.11. # k in the nodal points tj ∈ Ik as a solution We define the continuous process M of Scheme 1.11, and construct a natural approximation between neighboring nodal points. Scheme 1.14. Let {Mj ; j ≥ 0} be the solution of Scheme 1.11. For each   #k = M #k,1 , . . . , M #k,N is defined according to k > 0, the continuous process M   1 j #k,i (t) = M − M j+ 2 × (J J Mj+1 )i − M i

i

j+ 1 −αMi 2 j+ 12

+Mi

×



j+ 1 Mi 2

JM × (J

j+1

)i

  × Wi (t) − Wi (tj )



 (t − tj )

∀ t ∈ [tj , tj+1 ) .

#k ∈ M = {Mj ; j ≥ 0} satisfies Scheme 1.11, there holds P-a.s. M Since + 3 N C R0 ; (R ) . Theorem 1.15. Let the process X be a strong solution of the system (1.37). Then there exists β > 0 such that   # k (t) − X(t)2 ≤ C(α, J J , N )eβt k E M ∀t > 0. Proof of Theorem 1.15. Fix T > 0 and J = 1 ≤ i ≤ N , we define  J−1  j+ 1 J Mj+1 )i fk,i (s) = −Mi 2 × (J

T  k

+ 1. For all s ∈ [0, T ], and

j=0 j+ 1 +αMi 2

×



j+ 1 Mi 2

JM × (J

j+1



)i −

j+ 1 Mi 2



1[tj ,tj+1 ) (s) ,

Section 1.3

75

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation J−1 

gk,i (s) =

Mij 1[tj ,tj+1 ) (s) .

l=0

Then, we deduce from Definition 1.14 for 1 ≤ i ≤ N that t t #k,i (t) = Qi + fk,i (s) ds + gk,i (s) × dWi (s) + Ak,i (t) , M 0

(1.93)

0

where 1 Ak,i (t) = 2

t J−1 

t J−1 

1

(Mij+1 −Mij ) [tj ,tj+1 ) (s)×dWi (s)+

j=0

0

0 j=0

j+ 12

Mi

1[tj ,tj+1 ) (s) ds .

     +  2 N 1 3 N F : L1 Ω, Cloc R+ → L R , ; (S ) ; (R ) Ω, C loc 0 0     where F = F1 , . . . , FN , and for u = u1 , . . . , uN , and s ∈ [0, T ],       Fi (u)(s) := −ui (s) × J u i (s) + αui (s) × ui (s) × J u i (s) − ui (s) Define

∀s ≥ 0.

We have   # k (t) − X(t)2 E M % $ N t t 2       (Fi (X) − fk,i ) ds + gk,i − Xi × dWi (s) + Ak,i (t) =E  i=1

≤ 3N

N 

0

0

$ t

E

i=1

|Fi (X) − fk,i |2 ds

(1.94)

0

 t  2    + gk,i − Xi × dWi (s) + |Ak,i (t)|2

%

0



N N  t    t   2 E |Fi (X) − fk,i | ds + E |gk,i − Xi |2 ds ≤ 3N i=1

0

N    E |Ak,i (t)|2 +



i=1

0

  =: 3N (A) + (B) + (C) .

i=1

By elementary calculations, we have for the first term that N N  t   t    2 E |Fi (X) − Fi (gk )| ds + 2 E |Fi (gk ) − fk,i |2 ds (A) ≤ 2 i=1

0

= (A1) + (A2) .

i=1

0

76

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

 The Lipschitz property of each of the maps Fi ; i = 1, . . . , N then leads to t    J , N E (A1) ≤ C α, J X − gk 2 ds . 

0

Moreover, by elementary calculations and the ‘unit-length’ property at nodal points of Ik we have 

E |Fi (gk ) − fk,i |

2





J , N ≤ C α, J

N    E |Mij+1 − Mij |2

∀ s ∈ [tj , tj+1 ) .

i=1

√ The difference M·j+1 − M·j , j ≥ 0 is of the order k by the formula (1.81). Consequently, we deduce that   J , N T k . (A2) ≤ C α, J # k and the ‘unit-length’ property at nodal points we have By the definition of M N    #k,i (s)|2 ≤ C(α, J J , N )k . E |gk,i (s) − M i=1

Consequently, we deduce by elementary calculations that N N  t   t    2 # #k,i |2 ds (B) ≤ 2 E |Mk,i − Xi | ds + 2 E |gk,i − M

≤2

i=1 N 

0

E

i=1



i=1

t 0

   # k,i − Xi |2 ds + C α, J J , N T k . |M

It remains to estimate the third term (C) = for i = 1, . . . , N :

0

N    E |Ak,i (t)|2 in (1.94). We have

i=1

Ak,i (t) = J−1   j+ 1 1  t  j+ 12 j+ 1 J Ml+1 )i − αMi 2 × Mi 2 × (J J Mj+1 )i · × (J Mi − 2 0 j=0

· k 1[tj ,tj+1 ) (s) × dWi (s) J−1 J−1   t j+ 1 1  t  j+ 12 + × Δj Wi 1[tj ,tj+1 ) (s) × dWi (s) + Mi 2 1[tj ,tj+1 ) ds Mi 2 0 0 j=0

= (Di ) + (Ei ) ,

j=0

Section 1.3

Discretizations of the stochastic Landau-Lifshitz-Gilbert equation

77

where (Ei ) is comprised of the last two terms. We easily verify     J , N k 2 T . E |(Di )|2 ≤ C α, J The term (Ei ) can be rewritten as follows, t J−1 J−1  j+ 1  1 t  j+ 12 Mi × Δj Wi 1[tj ,tj+1 ) (s) × dWi (s) + Mi 2 1[tj ,tj+1 ) ds (Ei ) = 2 0 0 j=0 j=0   J−1 t J−1 t   j  1 j Mi × Δj Wi 1[tj ,tj+1 ) (s) × dWi (s) + Mi 1[tj ,tj+1 ) (s) ds = 2 0 0 j=0 j=0  J−1   1 t  j+1 + Mi − Mij × Δj Wi 1[tj ,tj+1 ) (s) × dWi (s) 4 0 j=0  t J−1  j+1 1 j Mi − Mi 1[tj ,tj+1 ) (s) ds = (Fi ) + (Gi ) . + 2 0 j=0

 · we have E Mj+1 By elementary calculations and the ‘unit-length’ property of M  −Mj 2 ≤ Ck. Hence, the second term (Gi ) has the following upper bound  1  E |(Gi )|2 ≤ T 2 k 2 . 4 For the term (Fi ) we have the following equality (Fi ) =

1 2

=

1 2

t J−1  0 j=0

 

Mij

× Δj W i



1[tj ,tj+1 ) (s) × dWi (s) +

 Mij × Δj Wi × Δj Wi

t J−1  0 j=0

Mij 1[tj ,tj+1 ) (s) ds

j: tj+1 0, and e ∈ R3 are respectively the exchange parameter, the anisotropy constant, and the uniaxial anisotropy direction vector. In addition to the structure-preserving, convergent Schemes 1.11 and 1.16 to approximate (1.37), we propose different, more efficient discretizations in Subsection 1.4.1 which are explicit in time and use projection strategies to ensure that iterates are (S2 )N -valued at the end of each iteration step; so far, those schemes lack a theoretical understanding, and some of them may even fail to approximate solutions of (1.37), as is evidenced in Subsection 1.4.2. Moreover, Subsection 1.4.1 discusses the crucial role of nonlinear solvers for the efficiency of the above schemes. The second part of this section studies long-time dynamics of iterates from those schemes: according to Section 1.3, invariant measures which are generated by Schemes 1.11 and 1.16 are supported on (S2 )N , are geometrically exponentially ergodic (see Theorems 1.13, and Corollaries 1.42, 1.44), and approximate the invariant measure of problem (1.37) (see Theorem 1.7). Subsection 1.4.2 supports these results, and provides computational evidence for the different schemes with respect to reliability or failure of related long-time simulations. The last part considers temporal discretizations of the penalization strategies (1.55) which are studied in Subsection 1.2.4. We found computational evidence that iterates for δ = 0 stay inside the unit ball, while they leave it for δ > 0 (see Figure 1.13). As is detailed in Remark 1.33 and Subsection 1.2.4, proper longtime dynamics for (1.55) is not clear analytically for δ = 0, while an invariant measure which is supported on (R3 )N is theoretically constructed for every δ > 0. The computations in Subsection 1.4.3 evidence the Gibbs character of the invariant measure of (1.55) for δ > 0, as well as δ = 0. The results also show that the computed invariant measure for sufficiently small ε is close to the invariant measure of (1.36) which is supported on (S2 )N ⊂ (R3 )N .

86

1.4.1

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

Numerical schemes

Consider a system with N spins which starts at x0 = (x1,0 , . . . , xN,0 ) ∈ (S2 )N . The simplest numerical approximation of (1.37) is the explicit Euler (EulerMaruyama) scheme, which for the Itô version of (1.37) reads:   Scheme 1.19. Let Y0 = x0 . For j ≥ 0 compute Yj+1 := Y1j+1 , . . . , YNj+1 from   j j Yij+1 − Yij = Yij × Heff,i − αYij × (Yij × Heff,i )−ν 2 Yij k + νYij × Δj Wi .   j j Here and below we denote Hjeff ≡ Heff (Yj ) = Heff,1 , and by , . . . , Heff,N   W = W1 , . . . , WN an (R3 )N -valued Wiener process, with Δj Wi := Wi (tj+1 )− Wi (tj ). In general, iterates of Scheme 1.19 are not sphere-valued, and it is not clear whether the noise term in combination with the discretized Stratonovich correction term approximates the corresponding Stratonovich integral for k → 0. As will become clear in Subsection 1.4.2, it is of crucial relevance for simulations to accurately preserve the structural properties of (1.37). A straightforward approach to enforce the norm constraint is to employ a projection step at each time level. This results in the following explicit projection scheme:   Scheme 1.20. Let Y0 = x0 . For j ≥ 0 compute Y∗,j+1 := Y1∗,j+1 , . . . , YN∗,j+1   and Yj+1 := Y1j+1 , . . . , YNj+1 from   j j − αYij × (Yij × Heff,i )−ν 2 Yij k + νYij × Δj Wi , Yi∗,j+1 − Yij = Yij × Heff,i Yij+1 =

Yi∗,j+1 |Yi∗,j+1 |

.

Again, it is not clear if the corresponding Stratonovich integral term is recovered in (1.37) for k → 0, but iterates {Yj ; j ≥ 0} are now (S2 )N -valued. — Next, we restate the Scheme 1.11 from Subsection 1.3.1, which properly approximates the Stratonovich integral term in (1.37) and allows for (S2 )N -valued iterates.   Scheme 1.21. Let Y0 = x0 . For j ≥ 0 compute Yj+1 := Y1j+1 , . . . , YNj+1 from   j+1/2 j+1/2 j+1/2 j+1/2 j+1 j+1 × Heff,i − αYi × (Yi × Heff,i ) k + νYi × Δj W i . Yij+1 − Yij = Yi To ask for discretization schemes where iterates are sphere-valued allows for a restricted class of numerical discretizations; a complementatory viewpoint are ‘outer approximations’, where iterates are again (R3 )N -valued, and an approximate sphere property of iterates is accounted for by a penalization term; see also Subsection 1. The penalization Scheme 1.4 is the basis for the following implicit penalization scheme for general effective fields:

Section 1.4

87

Computational studies

  Scheme 1.22. Choose ε > 0 and let Y0 = x0 . Then Yj+1 := Y1j+1 , . . . , YNj+1 for j ≥ 0 solves   j+1 2  j+1/2  j+1/2 j+1 j+1 α |Yi | + |Yij |2 − 2 Yi k × Heff,i + αHeff,i − 2ε Yij+1 − Yij = Yi j+1/2

+νYi

× Δj W i .

This scheme generalizes the deterministic Scheme 1.4; it is again implicit, nonlinear, and properly addresses the noise term by the midpoint formula; the approximation of the drift is also based on the midpoint formula. The projection Scheme 1.20 is a time-explicit scheme which is not supposed to approximate the stochastic forcing term properly; hence, for comparison, we include the semi-implicit projection scheme, which is based on the Scheme 1.21:   Scheme 1.23. Let Y0 = x0 . For j ≥ 0 compute Yj+1 := Y1j+1 , . . . , YNj+1   and Y∗,j+1 := Y1∗,j+1 , . . . , YN∗,j+1 from     ∗,j+1 ∗,j+1 Yi∗,j+1 − Yij = Yi∗,j × Heff,i + αHeff,i k + ν2 Yi∗,j+1 + |Yi∗,j |Yij × Δj Wi , Yij+1 =

Yi∗,j+1 |Yi∗,j+1 |

. (1.105)

The discretization of the diffusion is chosen such that it does not alter the magnitude of the solution Y∗,j+1 ; see also Remark 1.45 below. The scheme can be interpreted as a semi-explicit penalization scheme with ε = k, cf. [98]: substituting Yij = Yi∗,j+1

−

Yi∗,j

Yi∗,j |Yi∗,j |

in (1.105) leads to 

= − 1− +

1 |Yi∗,j |



  ∗,j+1 ∗,j+1 + αHeff,i k Yi∗,j + Yi∗,j+1 × Heff,i

 ν  ∗,j+1 Yi + Yi∗,j × Δj Wi . 2

(1.106)

It is due to the two sub-steps which independently address the stochastic evolution in (R3 )N and its restriction to (S2 )N that we may not expect a (pathwise) energy balance — which is not available for iterates of Schemes 1.19 and 1.20 as well. The Scheme 1.3 in Section 1.1.3 is based on the reformulation (1.19) of (1.18) to allow for its linear character; it has already been pointed out at the end of Section 1.1.3 t that the reformulation (1.19) leads to the additional stochastic integrals α 0 Xi × ◦dXi which makes a corresponding solution theory for the problem more difficult. Nevertheless, we study the implicit variant (i.e., θ =

88

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

1) of the linear projection Scheme 1.3 in Section 1.1.3 in the present context computationally, and refer to it as linear semi-implicit projection scheme. We refer to p. 26 for the definition of Kj+1 . Scheme 1.24. Let Y0 = x0 . For j ≥ 0 compute a Kj+1 -valued Vj+1 , and Yj+1 such that for all η ∈ Kj+1 holds N   j    α Vj+1 , η − Yi × Vij+1 , ηi i=1

!!  ν  J (Yj + kVj+1 ) + Hani (Yj ), η +(1 + α2 ) Δj W, η , = (1 + α2 ) AJ k j j+1 Y + k Vi (1 ≤ i ≤ N ) . and Yij+1 = ij |Yi + k Vij+1 | Recall that Schemes 1.21 and 1.22 require to solve a nonlinear system at each iteration level j ≥ 0. Suitable nonlinear solution strategies are discussed in Subsection 1.4.1 below. The following remark discusses the discretization of the diffusion in (1.37). Remark 1.45. If the noise term in the numerical methods is constructed by the midpoint method, it does not alter the magnitude of iterates, and a deviation of iterates from (S2 )N is then only due to the discretization of the drift term in (1.37). We motivate this claim for the implicit projection scheme (Algorithm 1.23): taking the scalar product of (1.106) with (Yij+1 + Yi∗,j ) then leads to |Yij+1 |2 − |Yi∗,j |2 = Ri (Yj+1 ) , where Ri (Yj+1 ) represents the discretized drift term, which is expected to be of order O(k). Note that a different discretization of the diffusion term may √ cause a further perturbation of magnitude O( k). The following remark discusses the computation of expected values for iterates {Yj ; j ≥ 0}. Remark 1.46. To obtain expected values for quantities of interest in the longtime limit one can integrate any suitable f : (S2 )N → R with respect to the invariant measure μ,    lim E f X(T ) = f (x) μ[dx] . T →∞

(S2 )N

Section 1.4

89

Computational studies

For the time-discrete setting with time step k > 0 we approximate the above integral on a sufficiently large time interval T ∗ = J ∗ k via ∗

J 1  f (Yj ) lim E f (X(T )) ≈ ∗ T →∞ J





P-a.s.,

j=1

where {Yj ; j = 1, . . . , J} is computed from one realization of the Wiener process. The theoretical results regarding the ergodicity of invariant measures for systems of ferromagnetic spins from the first part of this chapter provide a rigorous basis for this approach. In order to verify the ergodicity of numerically computed invariant measures we have compared both approaches. We found that, in most cases, the long-time average computations are more efficient than the Monte-Carlo approach; see also [106].

Solving nonlinear algebraic systems We describe two algorithms to solve the nonlinear systems which arise from the implicit time discretization of the LLG and SLLG equations: a simple fixed point method and the Newton scheme. We exemplify their construction for Scheme 1.21, which can easily be adopted to any other nonlinear scheme from above. For every j ≥ 0, by using the notations Zj+1 = Yj+1/2 and Yj+1 = 2Zj+1 − Yj , we may recast Scheme 1.21 in the form F j (Zj+1 ) = 0, j F j = (F0j , . . . , FN ), where Fij : (R3 )N → R3 , i = 1, . . . , N is given by  Fij (Zj+1 ) = 2Zij+1 − 2Yij + − Zij+1 × Heff,i (2Zj+1 − Yj )   +αZij+1 × Zij+1 × Heff,i (2Zj+1 − Yj ) k − νZij+1 × Δj Wi . The fixed point iterative scheme to approximate the iterate Yj+1 from Scheme 1.21 can now be formulated as follows. Algorithm 1.47. (i) For j ≥ 0, set Zj+1,0 = Yj , and m := 0. (ii) For m ≥ 1, compute Zj+1,m such that the following linear system of equations is valid P-a.s.,  2Zij+1,m − 2Yij = Zij+1,m × Heff,i (2Zij+1,m − Yij )   −αZij+1,m × Zij+1,m−1 × Heff,i (2Zij+1,m−1 − Yij ) k +νZij+1,m × Δj Wi

(1 ≤ i ≤ N ) .

(1.107)

90

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles ∗

∗

∗

(iii) Stop if Zj+1,m − Zj+1,m −1  < T OL , set Yj+1 := 2Zj+1,m − Yj , and go to (iv). Otherwise set m ← m + 1 and continue with (ii). (iv) Stop if j + 1 = J; otherwise set j ← j + 1 and go to (i). For the deterministic case, it is shown in [17, 10] that limm→∞ Zj+1,m − J ); see also Subsection 3.3.1 Yj+1  = 0, provided the time step obeys k = O(J in Chapter 3. We expect an analogical condition for the convergence of the fixed point algorithm in the presence of noise. A property of computed iterates Yij+1,m := 2Zij+1,m − Yij is that their magnitude does not change: for this purpose, we fix 1 ≤ i ≤ N , and m ≥ 0 in (1.107) and take the scalar product with Zij+1,m . This simplifies to Zij+1,m − Yij , Zij+1,m  = 0, and hence |Yij+1,m |2 = |Yij |2 . J ) for the fixed point iterations depends The convergence condition k = O(J crucially on the properties of the exchange operator J and may often become too restrictive for large spin systems. This situation is typical for discretizations of infinite spin ensembles in dimensions 1 ≤ n ≤ 3 which occupy a bounded domain O ⊂ Rn ; see Chapters 2 and 3. There, the condition becomes k = O(h2 ), where h > 0 is the mesh size of a spatial triangulation of O. Hence, the algorithm requires restrictively small time-step parameters k > 0. As will be shown below, the Newton method is a more robust alternative to simulate large spin systems. The Newton scheme for the equation F j (Zj+1 ) = 0 uses the Jacobian   m j+1,m JF ) ∈ L (R3 )N j ≡ J F j (Z   m = Jm N of the mapping F j . The Jacobian consists of 3×3 sub-blocks J F j F j i,l=1 m = such that J F j F il

F ji (Zj+1,m ) ∂F . ∂Zlj+1,m

F il

For Scheme 1.21 we get for 1 ≤ i, l ≤ N

1

  j+1,m j+1,m J mj = 2I3 + I3 × Heff,i + Zij+1,m × ∂Zi Heff,i k F ii  j+1,m j+1,m j+1,m +α I3 × (Zi × Heff,i ) + Zi × (I3 × Heff,i )  j+1,m j+1,m j+1,m × (Zi × ∂Zi Heff,i ) k − νI3 × Δj Wi , +Zi and for i = l we get   m = Z j+1,m × ∂ H j+1,m + αZ j+1,m × (Z j+1,m × ∂ H j+1,m ) k , JF Zl eff,i Zl eff,i j i i i F il

1

j+1,m j+1,m We denote ∂Zi Heff,i ≡ ∂Z j+1,m Heff,i . i

Section 1.4

91

Computational studies

where I3 denotes the 3 × 3 identity matrix, and for v, w ∈ R3 we define (I3 × v) w = (I3 w) × v = w × v (analogical identities hold for the 3 × 3 matrices j+1,m ∂Zl Heff,i ). The Newton scheme to approximate iterates {Yj ; j ≥ 0} from Scheme 1.21 can then be formulated as follows. Algorithm 1.48. (i) For j ≥ 0, set Zj+1,0 = Yj , and m := 0. (ii)For m ≥ 1, compute Zj+1,m such that the following linear system of equations is valid P-a.s., m j+1,m+1 F j (Zj+1,m ) . JF − Zj+1,m ) = −F j (Z ∗

∗

∗

(iii) Stop if Zj+1,m − Zj+1,m −1  < T OL , set Yj+1 := 2Zj+1,m − Yj , and go to (iv). Otherwise set m ← m + 1 and continue with (ii). (iv) Stop if j + 1 = J; otherwise set j ← j + 1 and go to (i).

Comparison of nonlinear solvers for the SLLG equation It has already been discussed in Subsection 1 that the convergence requirement for the nonlinear solvers places limits on the time step size of the numerical scheme. The following computational results again approximate a dynamics similar to the one of X in Subsection 1 on a lattice of mesh size h = O(N −2 ) which covers [−1, 1]2 ⊂ R2 . We can distinguish between two scenarios: (a) for sufficiently small noise the solution process exhibits a large ‘discrete gradient’ after a finite time, where the spin in the center points up and all the remaining spins point in the opposite direction (see Figure 1.3 (left)); once the maximum possible ‘discrete gradient’ is achieved the central spin quickly switches and the magnetization spin becomes almost homogeneous (see Figure 1.3 (middle and right)); (b) for large noise, the evolution is dominated by the noise effects and spins rotate independently in random directions with a speed which is proportional to the intensity of the noise. The aforementioned features of the dynamics make this problem a good candidate to test the properties of the different solvers which are presented in Subsection 1.4.1. The nonlinear systems are solved almost to machine precision, we set T OL = 10−10 . In addition, to eliminate round-off errors, the linear systems in each iteration are solved using a direct solver, i.e. Gaussian elimination. First we set ν = 0, T = 1 and examine the robustness of the nonlinear solvers with respect to the mesh size hp = 2−p (p = 3, 4, 5). Table 1.1 compares the number of average and maximum iterations for the Newton scheme and the

92

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The role of noise in finite ensembles of nanomagnetic particles

fixed point scheme, respectively. In particular, it shows that the fixed point scheme requires to choose k = O(h2 ) in order to guarantee convergence of the whole sequence.2 In contrast, the Newton algorithm requires significantly less iterations to converge; on the finest level, the Newton scheme admits time steps which are up to 20 times larger than those needed for the fixed point scheme to achieve convergence. Indeed, a further examination of the results from Table 1.1 reveals that the time step k > 0 for the Newton algorithm depends linearly on the mesh size h > 0. Further, we observe that the Newton algorithm requires the most iterations for convergence during the time period that corresponds to the fast switching of the central spin (shown in Figure 1.3). This observation leads to the conjecture that the time step for the Newton algorithm is not restricted by the spatial mesh size, but rather by the rate of change in the discrete solution. To check the dependence of both solvers on the convergence criterion, we increase the tolerance for the stopping criterion T OL = 10−3 and repeat the above calculations for h = 2−5 . The differences between the calculations with tolerances 10−10 and 10−3 were only minor. p

k

m

t∗

p

k

m

t∗

3

2.5 × 10−3

3-5

0.088

3

3 × 10−4

16-24

0.088

4

1.25 × 10−3

4-10

0.046

4

8 × 10−5

19-26

0.0375

5

4 × 10−4

4-16

0.05

5

2 × 10−5

17-26

0.044

Table 1.1. Nonlinear solvers (ν = 0): Time step, number of iterations m, and discrete blow-up time t∗ for hp = 2−p (p = 3, 4, 5). Newton’s method (left), and fixed point scheme (right).

Next, we fix the mesh size h = 2−5 , T = 0.1 and investigate the effect of an increasing noise intensity ν on the convergence of the nonlinear solvers. In Table 1.2 we list the number of iterations and maximum admissible time steps on the given time interval for both nonlinear schemes. The Newton scheme requires slightly smaller time steps and slightly more iterations for convergence than in the deterministic case. We also observe that the time steps need to be decreased with increasing noise intensity. This behavior is to be expected as the Newton scheme requires a good initial guess for convergence, and an increasing noise intensity leads to a faster evolution of the solution. Overall, we may conclude that the Newton scheme remains robust with respect to the 2

In general, the fixed point algorithm admits slightly larger time steps than those presented in Table 1.1. However, large time steps require unproportionally many iterations for convergence, and the fixed point scheme becomes significantly less efficient.

Section 1.4

93

Computational studies

noise intensity. It appears that the fixed point scheme is more robust with respect to the noise intensity, however the iteration counts for the schemes still remain significantly higher than for the Newton scheme. ν

Newton k/m

Fixed point k/m

1

2.5 × 10−4 /7-11

2 × 10−5 /28-30

1.4

1.25 × 10−4 /6-7

2 × 10−5 /26-29

2

1.25 × 10−4 /6-8

2 × 10−5 /24-26

2.8

6.125 × 10−5 /6

2 × 10−5 /22-24

Table 1.2. Nonlinear solvers (ν > 0): Effect of noise on the number of iterations.

1.4.2

Long-time dynamics

We study the long-time dynamics of probability distributions for problem (1.37): Subsection 1.4.2 illustrates failure of the explicit Scheme 1.19 to properly simulate related invariant measures. The computational results in the Subsections 1.4.2 to 1.4.2 use Scheme 1.21, and study the interplay of noise with J X + Hani (X); see (2). different energy portions in Heff (X) = −J

Failure of the Euler-Maryuama scheme We present computational evidence for the crucial importance of length conservation of iterates throughout the simulation. We consider a single spin X = (x1 , x2 , x3 ) with uniaxial anisotropy direction e = (0, 0, 1). The corresponding effective field takes the form Heff = KX, ee . We take an anisotropy constant K = 5, a damping constant α = 0.1, and the noise intensity parameter ν = 1. We compute the stationary distribution of the third component of the spin Y = (y1 , y2 , y3 ) from a single long-time simulation for k = 0.01, T = 105 by means of the explicit projection Scheme 1.20, and the midpoint Scheme 1.21. The analytical stationary distribution for this simple case can be expressed as 2α

2

fani (x3 ) = Z −1 e− ν 2 Eani = Z −1 e− ν 2 (1−x3 ) , where Z =

1

−1 e

2 − αK 2 (1−s ) ν

ds.

αK

(1.108)

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The role of noise in finite ensembles of nanomagnetic particles

0.7

0.9

midpoint explicit projection analytic

explicit explicit N=100000 explicit X/|X| explicit projection analytic

0.8

0.65 0.7

0.6

0.6

0.5 0.55 0.4

0.5

0.3

0.2 0.45 0.1

0.4 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 1.14. Stationary distribution of the third spin component: comparison of analytical and computed values from Schemes 1.20 and 1.21 (left). Stationary distriy3 from the explicit Euler Scheme 1.19, bution of third computed spin component y3 , |Y| and of y3 from the explicit projection Scheme 1.20 (right).

The computed values together with the analytical Gibbs probability distribution are depicted in Figure 1.14 (left). We observe that Scheme 1.21 clearly produces a more accurate approximation of the analytical probability density than Scheme 1.20. We observe that Scheme 1.20 underestimates the probability of energetically favorable states. The Euler-Maruyama scheme without projection (i.e. Scheme 1.19) for k = 0.01 becomes unstable after a few time steps, i.e. the modulus of the solution will become unbounded. For a smaller time step k = 10−4 , we are able to compute the solution on a time interval [0, 2000]. We used the computed values to reconstruct the probability distribution of the third component, and the normalized third component as a long-time average. For comparison, we display in Figure 1.14 (right) the distribution computed with the explicit projection Scheme 1.20 with k = 0.001 as a time-average over the time interval [0, 2000], the distribution averaged over 105 realizations of the Wiener process with the explicit projection scheme (k = 10−4 ) for T = 10, and the analytical probability distribution. Apart from the values near ±1, the probability distribution which is computed via Scheme 1.19 as an average over multiple realizations of the Wiener process gives a reasonable approximation of the stationary distribution. However, there is a clear difference between the probability distribution computed as an average over multiple realizations of the Wiener process, and the probability distribution computed as a long-time average. These results in Figure 1.15 confirm that the Euler-Maruyama scheme does not preserve the ergodic properties of the model — which is due to the failure of Scheme 1.19 to yield (approximately) sphere-valued iterates; see also Remark 1.46.

Section 1.4

95

Computational studies

2.4

2.5 |Y|

Y_3 explicit

Y_3 projection

1

2

2.2

1.5

2

0.5

1 1.8 0.5 1.6 0

0

1.4 -0.5 1.2 -1 -0.5

1

-1.5

0.8

-2

0.6

-2.5 0

500

1000

1500

2000

-1 0

500

1000

1500

2000

0

500

1000

1500

2000

Figure 1.15. From left to right: time evolution of a single trajectory of tj → |Y j | from Scheme 1.19. Evolution tj → Y3j from Schemes 1.19 and 1.20, respectively.

Interplay of anisotropy and exchange with noise in the SLLG equation We use Scheme 1.21 to study computationally how the noise interacts with different energy contributions. We start with a simple one-spin system with anisotropy, then study a two-spin system with exchange, and continue with more complex problems for a system with five spins, with anisotropy and exchange energies. Remark 1.49. In order to obtain an approximation of the probability density function for one spin, the unit sphere is divided into segments ωβγ ⊂ S2 which are associated with points   xβγ = sin(βπ/16) cos(γπ/16), sin(βπ/16) sin(γπ/16), cos(βπ/16) , where β = 0, . . . , 16, γ = 0, . . . , 31. These segments are defined via

 ωβγ = x ∈ S2 : xβγ = arg min |x − xαδ | . xαδ

For this partitioning of the sphere, at a fixed time level tj , we construct a piecewise constant empirical probability density function fˆj : S2 → R via f (x) = f (xβγ ) = ˆj

ˆj

j #{m| Y(m) ∈ ωβγ }

|ωβγ |M

∀ x ∈ ωβγ

j for β = 0, . . . , 16, γ = 0, . . . , 31, where Y(m) denotes the solution at time tj = jk computed with the m-th realization of the Wiener process, and M stands for the total number of realizations of the Wiener process. In order to reduce the fluctuations in the empirical probability density function fˆj due to the finite approximation, we compute a time averaged probability density function f over 1 T /k ˆj the last 100 time levels, i.e. we take f (x) = 100 j=T /k−100 f (x).

96

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The role of noise in finite ensembles of nanomagnetic particles

One spin system with anisotropy We start with a study of the long-time behavior of a single spin (i.e. N = 1) for different values of the damping parameter α ≥ 0, and noise intensity ν = 0.5. Throughout the experiments we use the time step k = 5 × 10−3 , and the expected values of the quantities of interest were computed from M = 104 sample paths of the Wiener process. The final time T is taken sufficiently large in order to obtain solutions close to the steady-state. We start with low-dimensional noise for a single spin at x10 = (0, 1, 0), i.e. we choose Wi = (0, 0, 1)W in (1.36), where W is an R-valued Wiener process, and Heff,i = (1, 0, 0) is a given fixed effective field. The probability density f for different values of the damping parameter α is depicted in Figure 1.16. We observe that for α → 0 the probability density approaches the uniform measure on the sphere. For increasing values of α the probability density becomes increasingly concentrated around the point (1, 0, 0), which is the steady state solution of the deterministic problem. Some anisotropy in the shape of the probability density function can be observed in the direction (0, 1, 1); see Figure 1.16. In order to

Figure 1.16. Scheme 1.21 for Heff = −∇Eani , and low-dimensional noise in (1.36): Asymptotic probability density f , for α = 0.01, 0.1, 0.25, 0.5, 1.

study the speed of convergence towards the steady α ≥ 0, we  state for different  depict in Figure 1.17 (left) the evolution of t → E[Yj ] − E[YJ ] (1 ≤ j ≤ J), which indicates exponential rates of convergence for α ≥ 0. The rate of convergence seems to depend linearly on the damping parameter α for large values of the parameter, i.e. the corresponding graphs in Figure 1.17 can be approximated by e−Cα(t−1) for a suitable constant C > 0. For the case α = 0.01, which is graphically indistinguishable from the case α = 0, the convergence still seems to be exponential, and the corresponding graph can be approximated by e−0.11t . Figure 1.17 (middle and right) shows that for growing values of α > 0 the expected value of the magnetization converges to the steady state of the deterministic problem, i.e. E[Yj ] → (1, 0, 0) for tj → ∞. Different initial conditions seem to lead to the same steady state probability density, which suggest that the invariant measure is ergodic despite the low-dimensionality of the considered noise; cf. [8].

Section 1.4

97

Computational studies

alpha=0.01 exp(-0.1 t) alpha=0.25 exp(-0.2(t-1)) alpha=0.5 exp(-0.4(t-1)) alpha=1 exp(-0.8(t-1))

10

1

0.1

0.01

0.001 0.1

1

10

100

Figure 1.17. Scheme 1.21 for Heff = −∇Eani in (1.37) (modified accordingly): Role of  damping parameter α ≥ 0 for evolution of log tj → logE[Yj ]−E[YJ ] (left). Evolution tj → E[Yj ] for α = 0.01, 0.1, 0.25, 0.5, 1 (middle and right).

In the following experiments, we use high-dimensional noise as stated in Section 1.2.2. It is due to the ergodicity of the problem by Theorem 1.7 that stationary quantities of interest may be obtained by a long-time simulation of a single realization sample path, which will be done for all the remaining experiments in this section; see Remark 1.46. Below, we study long-time dynamics for different energies Eani , Eexch , and Eani + Eexch . We start with the evolution of (1.37) (modified accordingly) for one magnetic spin X = (x1 , x2 , x3 ) (N = 1, c1 = 1) with uniaxial anisotropy. We consider the case with the easy axis e = (0, 0, 1). The magnetization dynamics for the spin is as follows: the spin spends some time around the two energy minima (0, 0, ±1), and the noise allows the magnetization to spontaneously switch between the two states; see Figure 1.18 (right). The time-asymptotic probability density for the problem takes the form (1.108). The results in Figure 1.18 are computed with Scheme 1.21 and indicate very good agreement between the analytic and computed probability densities. 3 X_3

analytic, alpha=0.2, nu=0.5 alpha=0.2, nu=0.5 analytic, alpha=0.1, nu=0.5 alpha=0.1, nu=0.5 analytic, alpha=0.05, nu=0.5 alpha=0.05, nu=0.5 analytic, alpha=0.1, nu=1 alpha=0.1, nu=1

2.5

2

1

0.5

1.5 0

1

-0.5 0.5

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 1000

2000

3000

4000

5000

6000

Figure 1.18. Analytic and computed probability density of the third component for different values of ν and α (left). Time evolution of the third component for ν = 0.5, α = 0.2 (right).

98

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

System with five spins with exchange and anisotropy In the following experiment we focus on the long-time dynamics of (1.37) (modified accordingly) for multiple spins (N = 5, c1 , . . . , c5 = 1) in the presence of combined exchange and anisotropy effects. The constants A = 0.25, K = 10, α = 0.05, and the noise intensity ν = 0.5 have been chosen to demonstrate complex interactions between exchange and anisotropy energy, and the noise. Similar to a single spin, the spins will spend some time in the two local energy minima which are aligned with the easy axis, and spontaneously switch from one minimum to the other, which is due to the noise. While the probability distribution for the third component of each individual spin is of Boltzmann type, the probability distribution of the averaged magnetization seems to approach the Gaussian distribution for a growing noise intensity; see Figure 1.19 (left). When compared to the single spin case, the effect of the exchange interactions on the probability distribution for the third component of individual spins can be interpreted as increased damping, or a decreased noise intensity. 3.5

1.8

single spin z-component, A=0.25, K=10, alpha=0.05, nu=0.5 average z-component, A=0.25, K=10, alpha=0.05, nu=0.5 single spin z-component, A=0.25, K=10, alpha=0.1, nu=0.5 average z-component, A=0.25, K=10, alpha=0.1, nu=0.5 single spin z-component, A=0.25, K=10, alpha=0.05, nu=1 average z-component, A=0.25, K=10, alpha=0.05, nu=1

3

2.5

midpoint, single spin, A=0.25, K=10, alpha=0.05, nu=0.5 midpoint, average spin, A=0.25, K=10, alpha=0.05, nu=0.5 Alouges, single spin, A=0.25, K=10, alpha=0.05, nu=0.5 Alouges, average spin, A=0.25, K=10, alpha=0.05, nu=0.5 penalization, single spin, A=0.25, K=10, alpha=0.05, nu=0.5 penalization, average spin, A=0.25, K=10, alpha=0.05, nu=0.5

1.6

1.4

1.2 2

1

0.8

1.5

0.6 1 0.4 0.5 0.2

0

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 1.19. Probability distribution of the averaged third component and the third component of the central spin (left). Comparison of the Schemes 1.21, 1.22, and 1.24 (right).

Finally, for comparison we provide graphs of the stationary distributions from the midpoint Scheme 1.21, the penalization Scheme 1.22 (ε = 0.001), and the semi-implicit projection Scheme 1.24 in Figure 1.19 (right). We observe that there is a good agreement between the results obtained with the first two schemes; the results for Scheme 1.24 are slightly different, the differences can be attributed to the numerical damping in Scheme 1.24.

1.4.3

Interplay of penalization and noise

In this section we study the interplay of noise and penalization in Scheme 1.22, and its slight modification — which is later referred to as Scheme 1.22 —,   j+1/2 j+1/2 × Δj Wi + δΔj Wi is used instead of νYi × Δj Wi , where the noise ν Yi with δ = ε. As already mentioned, the projection Schemes 1.20 and 1.23 may

Section 1.4

99

Computational studies

be interpreted as semi-implicit penalization methods, and hence share common penalization effects.

Two-spin system with exchange energy We consider a system of two  spins  (N = 2) with a symmetric positive definite 3 N exchange operator J ∈ L (R ) which corresponds to the negative discrete J X)i = (−1)i 2(X2 − X1 ) Laplacian with Neumann boundary conditions, i.e. (J for i = 1, 2. The initial condition is X1 (0) = (0, 1, 0) and X2 (0) = (0, 0, −1). If not mentioned otherwise, the time step size is k = 0.01. The expected values have been computed as an average over 200 realizations of the Wiener process or as a time-average simulation. We study the distance to (S2 )N 2over a long-time j  via tj →  E i=1(1 − |Yi |) , and the expected value of the exchange energy tj → E Eexch (Yj ) for various parameters. To examine the differences between the two Schemes 1.22 and 1.22 we first fix α = 0.1, ν = 0.2, and take ε = 0.1, 0.01, 0.001. In Figure 1.20 we display results  for Scheme 1.22; we observe that the error in the magnitude of E 2i=1 (1−|Yij |) depends linearly on the parameter ε, and is larger during periods of a faster decrease of the energy. The results for the Scheme 1.22 are very similar, and differences become negligible for decreasing values of ε. 0.14

2

eps=0.1 eps=0.01 eps=0.001

0.12

eps=0.1 eps=0.01 eps=0.001

1.8 1.6

0.1

1.4 1.2

0.08 1 0.06 0.8 0.6

0.04

0.4 0.02 0.2 0

0 0

2

4

6

8

10

0

2

4

6

8

10

2

   Figure 1.20. Evolution tj → E i=1 (1 − |Yij |) (left) and tj → E Eexch (Yj ) (right) for ε = 0.1, 0.01, 0.001 (Schemes 1.22 and 1.22 ).

Next, we consider ε = 0.01, α = 0.1, and take values ν = 0.2, 0.5, 1. We deduce from Figure 1.21 that both schemes produce comparable results; we display the results for Scheme 1.22 with ν = 1, where the differences between the two schemes are most pronounced. The stationary density for both 2α schemes is close to the Gibbs distribution density f (c) = Z −1 e ν 2 Eexch (c) , where X1 ,X2  Eexch (c) = (1 − c), and c = |X is the cosine of the angle between the two 1 ||X2 | spins. The distributions for various parameters, which are computed from longtime simulations are depicted in Figure 1.22, the time step taken was k = 0.005.

100

Chapter 1

The role of noise in finite ensembles of nanomagnetic particles

0.025

2.5

nu=1 nu=0.5 nu=0.2

nu=1 nu=0.5 nu=0.2

0.02

2

0.015

1.5

0.01

1

0.005

0.5

0

0 0

Figure 1.21. ν = 0.2, 0.5, 1.

2

4

6

8

Evolution of E

10

0

2

4

6

8

10

 2

  j  j i=1 (1 − |Yi |) (left) and E Eexch (Y ) (right) for

3.5

scheme1 A=1, nu=1, epsilon=0.01 scheme2 A=1, nu=1, epsilon=0.01 scheme1 A=2, nu=1, epsilon=0.01 scheme2 A=2, nu=1, epsilon=0.01 scheme2 A=1, nu=2, epsilon=0.05 scheme2 A=1, nu=2, epsilon=0.01, gamma=0.5 scheme2 A=1, nu=2, epsilon=0.05, gamma=0.5 scheme1 A=2, nu=2, epsilon=0.01 scheme1 A=2, nu=2, epsilon=0.05 f_exch A=1, alpha=1, nu=1 f_exch A=2, alpha=1, nu=2

3

2.5

2

1.5

1

0.5

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 1.22. Stationary distribution for different values of A, ν, ε.

We observe that both numerical schemes preserve ergodicity, and that the approximation of the computed invariant measure is robust with respect to the penalization parameter ε. Five-spin system with exchange In this problem we consider five spins with exchange interactions, see Figure 1.23 (left). Most results in this section were computed using Scheme 1.22, and if not mentioned otherwise we set A = 1. A comparison of results for Schemes 1.22 and 1.22 is given at the end of the section. In Figure 1.23 (right) we display the long-time probability distribution of the magnitude error 1 − |Yi | for i = 1, 3, where Y1 describes the spin in the corner, and Y3 is the spin in the center of the domain (−1, 1). The results were computed from one long-time single realization for different values of ν, ε with time step k = 0.01, using Scheme 1.22 . We observe that the ‘peak’ of the probability distribution depends linearly on ε, for ν fixed. We also see that due to stronger exchange interactions, the error in the magnitude for the middle spin is larger for the spins at the boundary of the domain (−1, 1). In Figure 1.24 (left) we display the stationary distribution of the magnitude of the middle spin |Y3 | for a range of values of the parameters ε. While it

Section 1.4

101

Computational studies

40

X3 nu=4.24, epsilon=0.01 X1 nu=4.24, epsilon=0.01 X3 nu=3, epsilon=0.02 X1 nu=3, epsilon=0.02 X3 nu=3, epsilon=0.01 X1 nu=3, epsilon=0.01 X3 nu=9, epsilon=0.02 X1 nu=9, epsilon=0.02

35

30

25

20

15

10

5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 1.23. System of five interacting spins (left). Stationary distribution for 1−|Y1 | and 1 − |Y3 | (right).

appears that the magnitude of the middle spin |Y3 | depends linearly on the penalization parameter ε, the results also indicate a more complex nonlinear relation between the noise intensity ν and the magnitude of the computed solution. In Figure 1.24 (right) we show that there is a close agreement of computed results for both penalization schemes which implies that the parameter δ does not play a significant role in the simulations. 16

16 eps=0.01, nu=4 eps=0.02, nu=4 eps=0.05, nu=4 eps=0.1, nu=4 eps=0.1, nu=2.83 eps=0.1, nu=2

14

scheme1, eps=0.01, nu=4 scheme1, eps=0.02, nu=4 scheme1, eps=0.1, nu=4 scheme2, eps=0.01, nu=4 scheme2, eps=0.02, nu=4 scheme2, eps=0.1, nu=4

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Figure 1.24. Stationary distribution of |Y3 | for different values of ν and ε computed with Scheme 1.22 (left). Stationary distribution of |Y3 | computed with Schemes 1.22 and 1.22 (right).

Five spins with exchange and anisotropy In the final experiment we take a system with five spins (N = 5), and ε = 0.05, ν = 0.5, A = 0.25, k = 0.005. We also include anisotropy interactions with K = 10, e = (0, 0, 1). We study the effects of the additive noise in Scheme 1.22 , with δ = 0 and δ = 0.1. The values of δ and ε have been chosen large in order to better demonstrate their effects on the numerical solution. It can be seen in Figure 1.25 that tj → |Y3j | remains strictly bounded below 1, and its value only occasionally drops below 0.5 for δ = 0; for δ = 0.1 we observe frequent

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The role of noise in finite ensembles of nanomagnetic particles

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Figure 1.26. Trajectory of tj → Y3j for δ = 0 (left), and δ = 0.2 (right).

overshoots above the value of 1 and occasionally drops below 0.2. In addition, in Figure 1.26 we visualize the trajectories of tj → Y3j for both values of the δ parameter. We observe that for δ = 0 it is contained within the unit sphere and stays above the sphere with radius 0.5, while the trajectory for δ = 0.5 occasionally exits the unit sphere. Both trajectories are concentrated around the north and south pole of the unit sphere, which correspond to the two minima of the energy. The described behavior is in agreement with the theory from Section 1.2.4, which suggests extremely small values for the parameter δ; see Lemma 1.31, and Figure 1.13.

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation In Chapter 1, we studied different effects regarding the (long-time) dynamics of ferromagnetic N -spin chains, which are caused by low- and high-dimensional noise, time discretization, and relaxation of the dynamics from (S2 )N to (R3 )N . Asymptotic rates of convergence were obtained in certain cases, which depend on the number of spins N , in particular. The aim in this chapter is to analyze the role of noise in the context of infinitely many spins, where the dynamics is governed by the stochastic partial differential equation (SLLG)     m ∂m = m × Heff + Hthm − α m × m × Heff ∂t m ∂m =0 ∂n m (0, ·) = m 0

on (0, T ) × O × Ω on (0, T ) × ∂O × Ω on O × Ω ,

(2.1)

m) = Δm m where O ⊂ Rn , n = 1, 2, 3, is a bounded Lipschitz domain, and Heff (m ˙ for the ease of presentation. The thermal noise Hthm = W to e.g. enable transitions between different local equilibrium states is modelled by a Wiener process W ≡ W(t, x). A physically relevant question is to clarify the role of noise for an infinite spin ensemble, which sets forth the studies of the thermal fluctuations for finite spin ensembles with general interactions in Chapter 1. There is a strong physical indication that the driving Wiener process should be uncorrelated in space [21, Sec.s 3.2 and 3.3]. This scenario puts challenges on the existing general mathematical theory for stochastic parabolic PDEs as developed in [44], which is not applicable here because of the irregularity of the noise, and the nonlinear constraint; indeed, even the local existence results seem to be out of reach of the current theory in the case of driving space-time white noise. Furthermore, there is an example in the recent works [70, 99] which shows that even the additive space-time white noise can lead to a non-existence of solutions for stochastic PDEs of parabolic type in dimensions higher than one. However, we remark that there is also a physical motivation to use colored noise away from thermodynamic equilibrium where the flucuation-dissipation relation need not be true, and where the weakly coupled microscopic events in the heat bath interact with individual spins; see e.g. [85]. We begin in Section 2.2 with the construction of a weak martingale solution in the case of driving colored (in space) noise by means of an implementable

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Figure 2.1. Discretely sphere-valued finite element functions on O ⊂ R (time fixed)

space-time discretization of problem (2.1), i.e. Scheme 2.9. This scheme is based on the finite element method. Due to the piecewise polynomial character of finite element functions the approximate solution processes are not S2 -valued in general, see Figure 2.1 for an illustration. A second problem which has to be overcome to construct a convergent discretization is the limited regularity of finite element functions such that the following energy bound (which holds for approximate solutions from standard Galerkin methods) is not clear to be true any more, T   2 mW1,2 + α m × Δm m2L2 ds ≤ CT . E sup m m (2.2) 0≤t≤T

0

Finally, it is not immediate to see how to construct a continuous adapted process from the iterates of Scheme 2.9. In Subsection 2.2.3 we will show the convergence of continuous versions of the iterates of Scheme 2.9 to a weak martingale solution of (2.1). In Subsection 2.2.4 we study the convergence behavior of a modification of Scheme 2.9, i.e. Scheme 2.10, where the increments of the Wiener process W are replaced by increments {ξξ j }Jj=0 of a related random walk. If compared to Scheme 2.9, an additional difficulty in the study of the convergence behavior of Scheme 2.10 is to construct an embedding of iterates of the Scheme 2.10 to a continuous process. We do not have a continuous driving process, such as e.g. the Wiener process in Subsection 2.2.3, which would serve as a starting point for the construction of such an embedding. There are several approaches to this problem. The first strategy is to work with the discrete processes directly, which has e.g. been realized in [26] for the stochastic Navier-Stokes equation. The second strategy is to apply a martingale embedding theorem to construct a continuous martingale out of the discrete one. This approach is used in Subsection 2.2.4. The choice of the appropriate process to embed into a continuous one is not straightforward here. A natural idea is to embed the driving random walk into a continuous martingale, but this leads to technical problems which are discussed in detail in Remark 2.44. Instead, we embed the diffusion term into a continuous martingale. The main steps of the

Chapter 2

105

The stochastic Landau-Lifshitz-Gilbert equation

convergence analysis for Schemes 2.9 and 2.10, and a comparison of used tools in the corresponding proofs are put together in Table 2.1 on page 134. Necessary results for the finite element analysis and the approximation of the Wiener process by a random walk, as well as useful tools from stochastic analysis are collected in the preliminary Section 2.1. We remark that the convergence analysis in Section 2.2 for the Schemes 2.9 and 2.10 is based on variational arguments rather than on semigroup methods which are not known to be applicable for problem (2.1), and is motivated by the work [56], where variational arguments for a general Galerkin approximation and a different problem are used. The convergence of the iterates of the schemes is shown for general (regular) meshes, and hence includes adaptive meshing strategies to e.g. resolve singular phenomena. The main motivation to construct solutions of problem (2.1) by

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Figure 2.2. Blow-up scenario for the deterministic LLG: Evolution of strong solutions of the deterministic LLG; from [17].

implementable schemes comes from their potential to supplement theoretical results by reliable quantitative computational studies. For instance, the simulations which are reported in Section 2.3 are based on Scheme 2.10, and motivate a non-trivial behavior of solutions which so far lacks theoretical understanding. As it is evidenced from the computational studies for the deterministic LLGequation in [17], the regularity of solutions may break down in finite time for the space dimension n = 2; see Figure 2.2. Below, we report on computational studies for problem (2.1) which indicate a pathwise blow-up of discrete solutions, while the corresponding expectations remain smooth for space-time white noise. In another series of computational experiments we study properties of

106

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the stationary distributions for space-time white noise in dimensions n = 1 and 2 for varying discretization parameters.

2.1

Preliminaries

We collect some material which is used in Section 2.2 for the purpose of constructing a weak martingale solution to problem (2.1). Our method is based on implementable approximations which are introduced below as Schemes 2.9 and 2.10.

2.1.1

Finite elements and temporal discretization

Standard references for the finite element method used in this section are the monographs [39, 23]. Let O ⊂ Rn , n ≤ 3 be a polygonal or polyhedral bounded Lipschitz domain. We denote by K ⊂ O an n-simplex, which is the convex hull of n + 1 nodal points, and which is specified by its diameter hK , as well as by the diameter ρK of the largest ball contained in K. The positive number K accounts for the shape of the n-simplex K. σK = hρK For each h > 0, let Th be a triangulation of O which consists of n-simplexes K such that 3 (i) O = K. K∈Th

(ii) The intersection of the interior of two different simplexes is empty. (iii) The diameter of each K is bounded by h = max{diam(K); K ∈ Th }. The set of nodes of the triangulation Th will be denoted by Eh := {x ;  ∈ L}. Definition 2.1. Assume that O ⊂ Rn , n ≤ 3 is a bounded polygonal Lipschitz domain in Rn . Let {Th }h>0 be a family of triangulations of O, each of them satisfying the conditions listed above. (i) The family {Th } is said to be regular, iff there exists a number σ > 0, such that for every Th , σK ≤ σ ∀ K ∈ Th . (ii) The family {Th } is said to be quasi-uniform, iff it is regular and there exists a constant γ > 0, such that for every Th , γh ≤ hK

∀ K ∈ Th .

In the following, we will always assume Th to be regular, and that the cardinality of the set of neighboring nodes of each node x is bounded independently of h > 0, i.e.,

 C = # m ∈ L : ∃K ∈ Th such that xm , x ∈ K < ∞ . (2.3)

Section 2.1

107

Preliminaries

Let L2 (O; R3 ) denote the standard Lebesgue space of (equivalence classes of) square integrable Lebesgue measurable functions f : O → R3 . For given two functions f , g ∈ L2 (O; R3 ), the L2 (O; R3 )-scalar product of them will be denoted by   f, g = f (x), g(x) dx , O

where ·, · denotes the inner product in R3 . By W1,2 (O; R3 ) we denote the Banach space of those f ∈ L2 (O; R3 ) whose weak first order partial derivatives belong to L2 (O; R3 ) as well. We associate with each element K ∈ Th the space P 1 (K; R3 ) of R3 -valued functions on K which are polynomials of degree less or equal to one in each component. We define the lowest order finite element space Vh ⊂ W1,2 (O; R3 ) by  

Vh = φ h ∈ C(O; R3 ) : φ h K ∈ P 1 (K; R3 ) ∀ K ∈ Th . (2.4)  h denote the counterpart of Vh for real-valued mappings. For each  ∈ L, Let V  h the nodal basis function associated with the nodal point we denote by ϕ ∈ V x ∈ Eh , i.e. ϕ (x ) = 1, and ϕ (xm ) = 0 for all m ∈ L \ {}. The (affine) nodal interpolation operator I h is a bounded linear map from C(O; R3 ) to Vh such that the following condition is satisfied for every φ ∈ C(O; R3 ) ∀ ∈ L. (2.5) I hφ (x ) = φ (x ) Assume that p > n2 . Then, for all natural numbers m such that 0 ≤ m ≤ 2, the following error estimates are well-known, see e.g. [23, Chapter 4] φ − I hφ Wm,p ≤ Ch2−m |φ φ|W2,p φ φ|Wm,p := with the semi-norm |φ

  |β|=m

∇β φ pLp

∀ φ ∈ W2,p (O, R3 ) ,

(2.6)

1

p

. Note also that the restric-

tion of I h to Vh is an identity on Vh ; moreover, the following inequality is valid, φ L ∞ I hφ L∞ ≤ φ ∀ φ ∈ C(O; R3 ) . (2.7) I   We define a bilinear form ·, · h : C(O; R3 ) × C(O; R3 ) → R by        φ, ψ  (x) dx = φ, ψ h = Ih φ ζ φ (x ), ψ (x ) ∀ φ , ψ ∈ C(O; R3 ) , O

∈L

(2.8) and the induced mapping   φ2h = φ , φ h φ

∀ φ ∈ C(O; R3 ) ,

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where ζ = O ϕ dx. Note that  · h is a norm on Vh . For the next two inequalities we refer to [39, 23], see also [40, Lemma 2.1] for the case n = 2, φh L2 ≤ φ φh h ≤ (n + 2)1/2 φ φh L2 φ    (φ φh , ψ h ) ≤ Ch φ φh L2 ψ ψ h W1,2 φh , ψ h )h − (φ

∀ φ h ∈ Vh , (2.9) ∀ φ h , ψ h ∈ Vh . (2.10)

The inverse estimates relate various norms on a finite element space on furtherly restricted triangulations. The following lemma holds for quasi-uniform triangulations Th of O; see e.g. [23, Sec. 4.5]. Lemma 2.2. Let Th be a quasi-uniform triangulation. Let (l, r) and (m, q) be two pairs of real numbers such that m ≥ l ≥ 0, and r, p ∈ [1, ∞]. Then there exists a constant C ≡ C(l, r, m, p) > 0 independent of h, such that 1  1  p r l−m−n min{0, r1 − p1 } φh pWm,p (K) φh rWl,r (K) φ ≤ Ch φ ∀ φ h ∈ Vh . K∈Th

K∈Th

h : Vh → Vh by the Lax-Milgram Next, we define the discrete Laplacian Δ theorem and the following variational identity hφ h , χ h )h = (∇φ φh , ∇χ χh )L2 −(Δ

∀ φ h , χ h ∈ Vh .

(2.11)

Lemma 2.3. Let Th be a quasi-uniform triangulation of O ⊂ Rn , and φ h ∈ Vh hφ h ∈ Vh satisfying (2.11). Moreover, there be given. There exists a unique Δ exists a constant C > 0 independent of h such that (i) (ii)

hφ h h ≤ Ch−1 ∇φ φ h L 2 , Δ −2 hφ h L∞ ≤ Ch φ φh L∞ . Δ

 h φ h ∈ Vh Proof. Let φ h ∈ Vh be given. The uniqueness and the existence of Δ follows from the Riesz lemma. hφ h in (2.11). Then, by inequality (2.9), and Lemma 2.2 we (i) Insert χ h = Δ have for all φ h ∈ Vh ,  hφ h 2 = − ∇φ hφ h ) ≤ ∇φ hφ h L2 ≤ Ch−1 ∇φ  h φ h h . φ h , ∇Δ φh L2 ∇Δ φh L2 Δ Δ h This proves inequality (i). hφ h (x )ϕ in (2.11). (ii) Fix a nodal point x for some  ∈ L, and insert χ h = Δ Then we get the following inequalities   hφ h (x )|2 = ζ −1 Δ  hφ h , χ h |Δ

h    −1  φ(xm )| (∇ϕm , ∇ϕ ) ≤ |Δhφ h (x )| ζ |φ m∈L:∃K∈Th , xm ,x ∈K

hφ h (x )| ζ −1 φ φh L∞ ∇ϕ 2L2 ≤ C|Δ

hφ h (x )| ζ −1 φ hφ h (x )| h−2 φ φh L∞ h−2 ϕ 2L2 ≤ C|Δ φ h L ∞ , ≤ C|Δ

Section 2.1

109

Preliminaries

where we have used (2.3), Lemma 2.2, as well as the following bounds which easily follow from the quasi-uniformity assumption for Th , ζ −1 ϕm 2L2 ≤ C ,

and

∇ϕm L2 ≤ C∇ϕ L2

∀m ∈ L.

This concludes the proof of (ii). Recall that Ik = {tj }Jj=0 denotes a partition of the interval [0, T ] of constant width k = tj+1 − tj = TJ . For a Banach space E and p ∈ [1, ∞) the space of

J E-valued sequences U j j=0 with finite norm J 1/p   U j pE 0, J ∈ N and Ik = {tj }Jj=0 be a partition of interval [0, T ] of width k = TJ . For t ∈ [tj , tj+1 ) define

(ii)

t − tj j+1 tj+1 − t j U U , + k k − + (t) := U j , Uk,h (t) := U j+1 , Uk,h

(iii)

Uk,h (t) := U j+1/2 .

(i)

Uk,h (t) :=

A useful compactness result for sequences of such E-valued functions whose norm is uniformly bounded with respect to k > 0 is provided in Lemma 2.8 below. Next, we will present an approximation result for Hölder continuous, and piecewise affine E-valued functions. First, we will need the following technical result which shows that the difference between the affine approximation of a Hölder continuous function and the function itself is also Hölder continuous. Lemma 2.5. Assume a, b ∈ R, a < b and γ : [a, b] → E, where E is a normed vector space, and γ is a Hölder continuous function with exponent α ∈ (0, 12 ), i.e. there exists some finite C > 0 such that γγ (t2 ) − γ (t1 )E ≤ C|t2 − t1 |α

∀ t1 , t2 ∈ [a, b] .

Define a function Z : [a, b] → E by the following formula,  t − a  b−t γ (t) − γ (a) − γ (b) − γ (t) Z (t) := b−a b−a

∀ t ∈ [a, b] .

(2.12)

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Then Z (a) = Z (b) = 0 , and Z is a Hölder continuous function with exponent α, i.e. (b − a)α 2α ≤ 2C|t − s|α

Z (t)E ≤ C Z Z (t) − Z (s)E Z

∀ t ∈ [a, b] , ∀ s, t ∈ [a, b] .

We remark that Z is the difference between the function γ and its affine approximation. This observation will be used later on. Proof. The first claim is obvious. The proof of the second part is based on direct calculations and of the following two easy identities, (t − a)(b − t) b−a t∈[a,b]

=

sup

b−a , 4

sup (1 − x)1−α + x1−α = 2α . x∈[0,1]

To prove the third claim we observe that     Z (t) := (b − t) γ (t) − γ (a) − (t − a) γ (b) − γ (t) (b − a)Z   = (b − a)γγ (t) + t γ (a) − γ (b) + aγγ (b) − bγγ (a)

∀ t ∈ [a, b] .

Therefore for s, t ∈ [a, b], + +    +  + +(b − a) Z (t) − Z (s) + = +(b − a) γ (t) − γ (s) + (t − s) γ (a) − γ (b) + E E ≤ |b − a| γγ (t) − γ (s)E + |t − s| γγ (a) − γ (b)E . Hence, by the assumptions on γ , Z (t) − Z (s)E ≤ 2C|t − s|α Z

∀ s, t ∈ [a, b]

as claimed. The above lemma can be used to prove the following approximation result. Proposition 2.6. Suppose that γ : [0, T ] → E, where E is a normed vector space, is a Hölder continuous function with exponent α ∈ (0, 12 ), i.e. satisfies inequality (2.12). Let Ik := {tj }Jj=0 be an equi-distant partition of [0, T ] of mesh size k = TJ > 0. Let γ k : [0, T ] → R be a piecewise affine approximation of the function γ , i.e. γ k (tj ) = γ (tj ) for j = 0, . . . , J, and γ k is affine on every segment [tj−1 , tj ], j = 1, . . . , J. Then γ k is a Hölder continuous function with exponent α and sup γγ k (t) − γ (t)E ≤

t∈[0,T ]

C α k , 2α

(2.13)

Section 2.1

111

Preliminaries

and in case E is an algebra, sup γγ 2k (t) − γ 2 (t)E ≤ Ck α 21−α γγ L∞ (0,T ;E) .

t∈[0,T ]

Proof. It follows from Lemma 2.5 that the restriction of γ k to each interval [tj , tj+1 ] is α-Hölder continuous. Therefore, by applying for instance [27, Corollary 3.3] we infer that γ k is α-Hölder continuous on the whole interval [0, T ]. However, the C α -norm of γ k is bounded from above by the sum over j = 0, . . . , J − 1 times the C α -norm of the restriction of γ k to each interval [tj , tj+1 ]. Hence, by Lemma 2.5 again, γγ k (t) − γ k (s)E ≤ 2CJ|t − s|α

∀ s, t ∈ [0, T ] .

In order to prove inequality (2.13) we notice that sup γγ k (t) − γ (t)E = max

sup

1≤j≤J t∈[tj−1 ,tj )

t∈[0,T ]

γγ k (t) − γ (t)E ,

so that inequality (2.13) is a direct consequence of the second inequality in Lemma 2.5. Since γγ k L∞ (0,T ;E) ≤ γγ L∞ (0,T ;E) , the second result is then a direct consequence of the first one.

2.1.2

Fractional Sobolev spaces and related compact embeddings

Solutions of problem (2.1) are not expected to be differentiable with respect to time, which is due to the limited temporal regularity properties of the driving Wiener process W. As a consequence, the classical compactness lemma of Aubin-Lions is not applicable, and a corresponding result for fractional Sobolev spaces will be needed below to show the convergence of the numerical schemes to a solution of equation (2.1). Definition 2.7. Let E be a Banach space, T > 0, and 0 < s < 1, 1 ≤ p < ∞. (i) The fractional Sobolev spaces are defined by      W s,p 0, T ; E = f ∈ Lp 0, T ; E ; f W s,p (0,T ;E) < ∞ , where

 f W s,p (0,T ;E) =

T 0



T 0

f (r) − f (t)E p drdt |t − r|s |r − t|

1

p

.

(ii) The Hölder spaces are defined by      Lips [0, T ]; E = f ∈ L∞ 0, T ; E ; f Lips (0,T ;E) < ∞ ,

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where f Lips ([0,T ];E) = esssupr,t∈[0,T ]

f (r) − f (t)E . |r − t|s

(iii) The Nikolskii spaces are defined by 

   N s,p 0, T ; E) = f ∈ Lp 0, T ; E : f N s,p (0,T ;E) < ∞ , where

f N s,p (0,T ;E) = sup k −s f (· + k) − f (·)Lp (−k,T −k;E) . k>0

  We use here a standard convention that for f ∈ Lp 0, T ; E we set f = 0 outside of the interval (0, T ). Assume that s, r ∈ (0, 1), and p, q ∈ [1, ∞). Then the following properties are valid, cf. [103]: (i) W s,p ⊂ N s,p , (ii) W s,p ⊂ W r,p and N s,p ⊂ N r,p , if s ≥ r, (iii) W s,p and N s,p are both embedded in W r,p , and N r,p , provided that s > r, s− 1

(iv) if s > p1 , then both, W s,p and N s,p are embedded in Lip p , and hence are embedded in the set C([0, T ]; E) of E-valued continuous functions, ≥ r − 1q and r ≤ s, p ≤ q, then W s,p ⊂ W r,q and N s,p ⊂ N r,q .  Let G ∈ C [0, T ]; E) be piecewise affine on subintervals [tj , tj+1 ) of a partition Ik of [0, T ]. The following criterion is useful to derive uniform bounds in Nikolskii spaces for space-time functions which are piecewise affine in time.

(v) if s −

1 p

Lemma 2.8. Assume that k > 0 and that Ik = {tj }Jj=0 is an equi-distant mesh  of size k > 0 covering [0, T ]. Assume that G ∈ C [0, T ]; E) is such that for every j ∈ {0, . . . , J − 1} the function [tj , tj+1 )  t → G (t) ∈ E

is affine.

Assume that for some p ≥ 1, α ∈ (0, 1), and C > 0, and every 0 ≤  ≤ J, k

J−  + + +G (tj+ ) − G (tj )+p ≤ C p tαp .

E

(2.14)

j=0

Then G ∈ N α,p (0, T ; E), and there exists a constant C = C(T ) > 0 which does not depend on k > 0, such that G N α,p (0,T ;E) ≤ C . G

Section 2.1

113

Preliminaries

Proof. Note that T = tJ = kJ. We have to show that for some constant C > 0 independent of k we have T −δ + + +G (t + δ) − G (t)+p dt ≤ Cδ αp Xp (δ) := ∀ δ ∈ (0, T ) . (2.15) E 0

For this purpose, we distinguish three cases. (i) δ ∈ (0, k). Then Xp (δ) =

J−2   j=0

tj+1 −δ

+

tj

tj+1 tj+1 −δ



+

tJ −δ  tJ−1

G (t + δ) − G (t)pE dt . G

Since for each j ∈ {0, . . . , J − 1} the function G |[t

j ,tj+1 ]

+ + + + +G (t) − G (s)+ ≤ |t − s| +G (tj+1 ) − G (tj )+ E E k

(2.16)

is affine, we infer that ∀ t, s ∈ [tj , tj+1 ] .

Let us fix j ∈ {0, . . . , J − 1} and take t ∈ [tj+1 − δ, tj+1 ] ⊂ [tj , tj+1 ]. Then |tj+1 −t| ≤ δ. Moreover, t+δ ∈ [tj+1 , tj+1 +δ] ⊂ [tj+1 , tj+2 ], and |(t+δ)−tj+1 | ≤ δ. Hence, + + + + + + +G (t + δ) − G (t)+ ≤ +G (t + δ) − G (tj+1 )+ + +G (tj+1 ) − G (t)+ E E E + + +  δ + +G (tj+2 ) − G (tj+1 )+ + +G (tj+1 ) − G (tj )+ . ≤ E E k Accordingly, let now t ∈ [tj , tj+1 − δ] ⊂ [tj , tj+1 ]. Then also t + δ ∈ [tj , tj+1 ], and hence + + + + +G (t + δ) − G (t)+ ≤ δ +G (tj+1 ) − G (tj )+ . E E k Consequently, by using equality (2.16), assumption (2.14) with  = 1 (so that t = t1 = k) leads to  δ p J−1 + + +G (tj+1 ) − G (tj )+p k E k j=0  δ p  δ (1−α)p ≤ 2p C p k αp ≤ (2C)p δ αp , k k

Xp (δ) ≤ 2p

(2.17)

which concludes the proof of (2.15). (ii) The argument from (i) straightforwardly generalizes to the case δ = k. (iii) δ > k. In this case we can find 1 ≤  ≤ J − 1, and η ∈ (0, 1) such that δ = k( + η). Let t ∈ [tj , tj+1 ] ∩ [0, T − δ]. By the triangle inequality we have + + + + + +G (t + δ) − G (t)+ ≤ +G (t + δ) − G (t + t )+E + +G (t + t ) − G (tj+ )E E + + + + ++G (tj+ ) − G (tj )+ + +G (tj ) − G (t)+ =: I + II + III + IV .

E

E

114

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

We may proceed as in the first step to control the terms I, II, and IV ; on using (2.14), we arrive at Xp (δ) ≤ 34p−1 k 

≤ C 4 k p p

J−1 

G (tj+1 ) − G (tj )pE + 4p−1 k G

j=0 αp

+

 = tαp

αp

≤ (4C) k (1 + ) p αp

J− 

G (tj+ ) − G (tj )pE G

j=0

(4C) k (1 +  ) p αp

αp

≤ (4C)p k αp 2αp αp ≤ (22+α C)p δ αp .

This concludes the proof. The following compactness results will be useful to identify limits in the numerical schemes for (2.1); see e.g. [56] for related proofs. Lemma 2.9. Assume that E0 ⊂ E ⊂ E1 are Banach spaces, with E0 and E1 being reflexive. Assume that the embedding E0 ⊂ E is compact, q ∈ (1, ∞) and α ∈ (0, 1). Then the embedding       Lq 0, T ; E0 ∩ W α,q 0, T ; E1 → Lq 0, T ; E is compact. Lemma 2.10. Assume that E0 , E are Banach spaces such that the embedding E0 ⊂ E is compact. Let q ∈ (1, ∞), and 0 < α < β < 1. Then the embedding     W β,q 0, T ; E0 → W α,q 0, T ; E is compact. Lemma 2.11. Assume that E0 , E are Banach spaces such that the embedding E0 ⊂ E is compact. Let q ∈ (1, ∞), and α > 1q . Then the embedding       W α,q 0, T ; E ∩ C [0, T ]; E0 → C [0, T ]; E is compact.

2.1.3

Young integral

We present the Love-Young inequality from [110]; see also [53] for a modern introduction into this topic. Definition 2.12. Let Ik = {tj }Jj=0 be a regular partition of the interval [a, b] of size k = b−a J , and p ∈ [1, ∞). Define the p-variation vp (f ) of a function f : [a, b] → R by J  |f (tj+1 ) − f (tj )|p . vp (f ) := sup k>0 j=0

Section 2.1

115

Preliminaries

Let Wp ([a, b]) denote the set of functions such that vp (f ) < ∞. Define the p-variation semi-norm  · (p) on Wp ([a, b]) as follows, 1

f (p) := vp (f ) p , and the p-variation norm via f [p] := f (p) + sup |f (t)| . t∈[a,b]

The next theorem is known as Love-Young inequality and has been shown in [110]. Theorem 2.1. Let p, q ∈ (1, ∞) be such that p1 + 1q > 1. Then there exists Cp,q > 0 such that for all f ∈ Wp ([a, b]), g ∈ Wq ([a, b]),  b      ≤ Cp,q f [p] g(q) , f dg (2.18)   a

where the integral is understood in the Young-Stieltjes sense; see [53]. We refer the interested reader to the book [53] for more details. The following result is immediate. Corollary 2.13. Assume that p, q ∈ (1, ∞) satisfy p1 + 1q > 1. Then there exists C > 0 such that for all f, f˜ ∈ Wp ([a, b]), and g, g˜ ∈ Wq ([a, b]),  b  b      ˜ f dg˜ − (2.19) f dg  ≤ Cp,q f˜ − f [p] g ˜ (q) + f [q] g˜ − g(p) .  a

2.1.4

a

Wiener process and the approximating random walk

  Let P := Ω, F, F, P be a complete probability space with a filtration F :=

 Ft ; t ∈ [0, T ] . For a separable Hilbert space K, let W = W(t); t ∈ [0, T ] be a K-valued Wiener process on P. We denote by Q ∈ T1 (K) the symmetric, non-negative definite, trace-class covariance operator

of W. Hence,  there exists a sequence of i.i.d. R-valued Brownian motions β (t); t ∈ [0, T ] ∈N on P, such that W(t) =

∞  √

q β (t)e

∀ t ∈ [0, T ] ,

(2.20)

=1

where {e } ≥1 is an orthonormal basis of K consisting of eigenfunctions of Q, and {q } ≥1 ⊂ R+ are corresponding eigenvalues. We refer to e.g. [44, Chapter 4] for further details.

116

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

  For p ≥ 1, and another separable Hilbert space H, let M p [0, T ], F; H (:=   M p Ω × [0, T ], F; H ) be the space of equivalence classes of F-progressively measurable processes u : [0, T ] × Ω → H, such that * ) T p u(s)H ds < ∞ . E 0

Let us denote by L (K, H) the space of linear bounded operators from K to H, and by T2 (K, H) the space of linear Hilbert-Schmidt operators from

t K to H. We define the stochastic integral ϕ (s) dW(s); t ∈ [0, T ] for 0  any ϕ ∈ M 2 [0, T ], F; T2 (K, H) as a continuous H-valued F-martingale, such J  that if ϕ is a step process of the form ϕ = ϕ (tj−1 )1[tj−1 ,tj ) belonging to j=1   M 2 [0, T ], F; T2 (K, H) , then

t 0

ϕ (s) dW(s) =

J 

  ϕ (tj−1 ) W(t ∧ tj ) − W(t ∧ tj−1 )

∀ t ∈ [0, T ] ;

j=1

see e.g. [44, Chapter 4]. The stochastic integral  satisfies the Itô isometry (see e.g. [44, Proposition 4.5]), i.e. for each ϕ ∈ M 2 [0, T ], F; T2 (K, H) , +2    t + t + + ϕ(s)Q1/2 2T2 (K,H) ds ϕ E + ϕ (s) dW(s)+ = E H

0

0

∀ t ∈ [0, T ] . (2.21)

It also satisfies the Burkholder-Davis-Gundy inequality, see [44], i.e. for every 0 < r < ∞ there exists a constant Cr > 0 such that $ $ %  r2 % +r + t T 1 + + 2 ϕ(s)Q 2 T2 (K,H) ds . (2.22) ϕ (s) dW(s)+ ≤ Cr E ϕ E sup + 0≤t≤T

0

H

0

We also recall that the Q-Wiener process W satisfies the following inequality   n n ∀n ∈ N, (2.23) E W(t) − W(s)2n K ≤ Cn (t − s) (Tr Q) where for n = 1 we have equality with C1 = 1; see for instance [72, Corollary 1.1]. Next, we approximate the K-valued Q-Wiener process W on P by a Kvalued Q-random walk. To choose discrete, bounded random variables instead of (unbounded) Gaussians per iteration step is a common strategy when discretizations of stochastic differential equations are implemented on a computer. For this purpose, recall Ik := {tj }Jj=0 , consider the (time-discrete) filtration   Fk := {Ftj ; tj ∈ Ik } ⊂ F, and set Pk := Ω, F, Fk , P .

Section 2.1

117

Preliminaries

Definition 2.14. Let Q ∈ T1 (K) be symmetric and non-negative definite, and Ik := {tj }Jj=0 . A Q-random walk on Pk is a sequence {ξξ j }Jj=0 of K-valued i.i.d. random variables such that for each 0 ≤ j ≤ J the conditions below are satisfied: (SI1 ) ξ j is Ftj -measurable and independent of {Ft ; 1 ≤  ≤ j − 1}.       (SI2 ) E ξ j = 0, and E (ξξ j , x)K (ξξ j , y)K = k Qx, y K for all x, y ∈ K.   p (SI3 ) For every integer p ≥ 1 there exists Cp > 0 such that E ξξ j 2p K ≤ Cp k . Definition 2.14 generalizes the definition of an R-valued random walk. In the finite-dimensional case, the weak limit of a random walk is a Wiener process by the Donsker’s invariance principle, see [77, p. 70]. We now provide examples of R-, and more general K-valued sequences of random variables {ξξ j }Jj=0 . Example 2.15. 1. For K = R, let {ξj }Jj=0 be an i.i.d. sequence of R-valued √ random variables, and put ξ j = k ξj . We give two admissible choices for {ξj }Jj=0 such that the conditions (SI1 )–(SI3 ) are satisfied by {ξ j }.   (i) P ξj = ±1 = 12 , or √     (ii) P ξj = ± 3 = 16 , and P ξj = 0 = 23 . We refer to [90, Sec. 6.4] and [80, Sec. 9.7] for various other approximations of an R-valued Wiener process. 2. Let K be a Hilbert space with orthonomal basis {e }, and {q } be eigenvalues of Q ∈ T1 (K). Then the conditions (SI1 )–(SI3 ) are satisfied for the following system of random variables: ξj =

∞  √

q ξ ,j e

j ≥ 1,

=1

where {ξ ,j } ,j∈N are i.i.d. R-valued random variables from part 1 of this example.

2.1.5

Convergence of random variables and representation theorems

We will present mathematical tools which are necessary to show the convergence of the discretization schemes below, and to identify the limit of iterates of the schemes with a solution of the corresponding SPDE. We start with the Prohorov theorem about weak convergence of probability measures. This theorem is used later to show the weak convergence of laws of solutions of the various

118

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

discretization schemes when the mesh size tends to zero. Next we will present a version of the Skorokhod theorem to strengthen the convergence, and finish the section with a martingale representation theorem and a criterion of invariance of the martingale property with respect to taking the limit of a parameter. In the following, E will be a separable and complete metric space with the corresponding Borel σ-field denoted by B(E). We denote by Br (x), for x ∈ E, and r > 0 the open ball in E of radius r with center x.   Definition 2.16. A family Λ of probability measures on E, B(E) is tight iff for an arbitrary ε > 0 there exists a compact set Kε ∈ B(E) such that μ [Kε ] ≥ 1 − ε

∀μ ∈ Λ.

A family consisting of only one probability measure is always tight. This follows from the following classical result.   Lemma 2.17. Let μ be a probability measure on E, B(E) . Then for arbitrary  > 0 there exists a compact set K ∈ B(E) such that μ [K ] ≥ 1 −  . Proof. Fix  > 0. Let {yi }i∈N be a dense set in E, i.e. for an arbitrary δ > 0, ∞ 3 E= B δ (yi ). We infer that there exists a sequence {nk }∞ k=1 ⊂ N such that i=1

μ

nk 4 i=1

Denote Fk =

n 3k i=1

  B 1 (yi ) ≥ 1 − k . k 2

B 1 (yi ), and F = k

∞ 5 k=1

Fk . Then F is compact, and

∞ 4    μ [F ] = 1 − μ E \ F = 1 − μ E \ Fk ≥ 1 −  . k=1

  Definition 2.18. A family Λ of probability measures on E, B(E) is relatively compact iff an arbitrary sequence {μn } of elements from Λ contains a subsequence   {μnk } which is weakly convergent to a probability measure μ on E, B(E) . The following theorem is known as the Prohorov theorem.   Theorem 2.2. A family Λ of probability measures on E, B(E) is relatively compact iff it is tight.

Section 2.1

119

Preliminaries

Proof. Part 1: Relative compactness of Λ implies its tightness. Assume the ∞ opposite. Then, there exist δ > 0, a sequence of measures {μ n }n=1 ⊂ Λ, and a 3∞ ∞ sequence of compacts {Kn }n=1 with Kn ⊂ Kn+1 and E = n=1 Kn , such that μn [Kn ] < 1 − δ

∀n ∈ N.

By the relative compactness, there exists a weakly convergent subsequence of {μn }, and we deduce the following contradiction: 1 = μ [E] = lim μnk [Knk ] < 1 − δ . k→∞

Part 2: Tightness of Λ implies its relative compactness. We consider two cases. a) Assume first that E is compact. the Banach-Alaoglu theorem, an  By ∗ arbitrary closed ball of the space C(E) is compact in the weak-∗ topology.  ∗ Since by the Riesz representation theorem C(E) may be identified with the  space of all finite signed measures on E, B(E)), the weak-∗ convergence on  ∗  ∗ C(E) coincides with the weak convergence on C(E) , and the result follows. b) In the general case we can construct a sequence of compact sets {Kn } such that μ [Kn ] > 1− n1 , for all μ ∈ Λ and all n ∈ N. By a diagonalization procedure and the argument from part a) we can construct a sequence of measures {μn } ⊂ m 3 Λ such that for each m ∈ N, their restrictions to Kj converge weakly to a j=1 3  m m 3 1 Kj such that μ˜ m Kj ≥ 1 − m . Furthermore, measure μ˜ m on the set j=1

j=1

we can assume that the sequence is self-consistent, i.e. for all k > m we have  m μ˜ m = μ˜ k   . Hence, we can conclude that the measure μ via j=1

Kj

k   4 μ[A ] := lim μ˜ k A ∩ Kj k→∞

A ∈ B(E) ,

j=1

is the weak limit of the sequence {μn }. We will need the following generalization of the Skorokhod theorem (cf. Theorem 1.10.4 and the Addendum 1.10.5 in [107]). It allows to deduce from the weak convergence of a sequence of probability measures almost sure convergence of corresponding random variables on a new probability space. Theorem 2.3. Assume that (Ωk , Fk , Pk ) for k ∈ N are probability spaces and Uk : Ωk → E, k ∈ N are (Fk , B(E))-measurable mappings such that Uk → U∞ weakly (i.e. the law of Uk converges weakly to the law of U∞ ). Then there

120

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

 and perfect1 , measurable maps: φk : Ω  P),  F,  → Ωk exist a probability space (Ω, such that  ◦ φ−1 , (i) Pk = P k

 k := Uk ◦ φk almost surely converges to U  ∞, (ii) U  k and Uk are equal. (iii) For each k ∈ N, the laws of U The martingale representation theorem is another result which will be useful in Section 2.2 to identify the stochastic integral in (2.1). Fix T > 0, and let M2T (H) denote the Banach space of all H-valued continuous, square integrable F-martingales N , which is endowed with the norm, cf. [44], 1    2 2 N M2 (H) := E sup N N (t)H N . T

t∈[0,T ]

We recall the notion of a quadratic variation process. Definition 2.19. Let N ∈ M2T (H). We call a T1 (H)-valued, continuous, Fadapted and increasing process V the quadratic variation of the process N if and only if V (0) = 0 , and the process N (t) ⊗ N (t) − V (t)

∀ t ∈ [0, T ]   is a continuous F-martingale. We denote V (t) := N (t). We may now state the martingale representation theorem in the form that we will use it in Section 2.2; cf. [44, Theorem 8.2, p. 220]. Theorem 2.4. Let Q ∈ T1 (K) be a symmetric, non-negative operator on K. 1 Define a Hilbert space K0 = Q 2 K, which is endowed with its natural norm. Assume that N ∈ M2T (H), and   N (t) =

t

1

1

Φ(s)Q 2 )(Φ Φ(s)Q 2 )∗ ds (Φ

∀ t ∈ [0, T ] ,

0

where Φ is a F-progressively measurable T2 (K0 , H)-valued process. Then there  P),  and a K-valued Q-Wiener process  F,  F, exist a filtered probability space (Ω,  P × P),  such that  F × F,  F × F, W, defined on (Ω × Ω, t Φ (s, ω) dW(s, ω, ω) ˜

N (t, ω) =

∀ t ∈ [0, T ]

. ∀ (ω, ω) ˜ ∈Ω×Ω

0 1

A perfect map is a continuous, closed, surjective map, such that the inverse image of any set consisting of one point is compact.

Section 2.1

121

Preliminaries

Next, we provide criteria when the martingale property of a family of processes is transferred to its limit; the following theorem builds upon material from [44, p. 232]. Theorem 2.5. Let H be a separable Hilbert space and (E,  · E ) be a separable Banach space. Assume that P = (Ω, F, P) is a probability space, and T > 0. For each s > 0, let Ms : [0, T ] × Ω → H ,

Xs : [0, T ] × Ω → E

be stochastic processes. Assume also that M : [0, T ] × Ω → H ,

X : [0, T ] × Ω → E

are stochastic processes such that P-a.s. for every t ∈ [0, T ], Ms (t) → M(t)

(2.24)

Xs (t)  X(t) and that for some q > 1,

weakly

(s → 0) ,

  sup E Xs (t)qE < ∞ .

(2.25)

(2.26)

s>0

For each s > 0, let Fs denote the natural filtration on the probability space P which is generated by the process Ms . Similarly, we denote by F the natural filtration on the probability space P generated by the process M. Finally, by F we denote the augmentation of the filtration F. If Xs is an Fs -martingale for each s > 0, and X is F-adapted, then X is an F-martingale. Proof. In view of a deep result from the monograph [51, p. 75] it is enough to show that X is an F-martingale. Let us fix t, r ∈ [0, T ] such that r ≤ t. We have to show that for any choice of times 0 ≤ r1 < r2 < · · · rK ≤ r, where K ∈ N, any bounded and continuous functions hi : H → R, i = 1, · · · , K, and any φ ∈ E∗ the following equality holds K  ! '   E φ , X(t) − X(r) ∗ hi M(ri ) = 0 . (2.27) E ,E

i=1

By assumption, Xs is a Fs -martingale for each s. Hence, for each K ∈ N, E



φ , Xs (t) − Xs (r)

!

K ' E∗ ,E

i=1

  hi Ms (ri ) = 0 .

(2.28)

    & In view of Assumptions (2.24) and (2.25), φ , Xs (t)−Xs (r) E∗ ,E K i=1 hi Ms (ri )     & converges P-a.s. to φ , X(t) − X(r) E∗ ,E K i=1 hi M(ri ) . Moreover, in view of

122

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

the assumption (2.26), since each hi is bounded,  that the family of

  we infer &K random variables φ , Xs (t)− Xs (r) E∗ ,E i=1 hi Ms (ri ) ; s > 0 is uniformly integrable. Hence by a well-known result, see e.g. [97, Appendix C], we infer that lim E

s→0



!

φ , Xs (t) − Xs (r) =E



K ' E∗ ,E

  hi Ms (ri )

i=1

! φ , X(t) − X(r)

K ' E∗ ,E

(2.29)

  hi M(ri ) .

i=1

This equality in view of equality (2.28) concludes the proof of equality (2.27). The next result identifies the quadratic variation of the martingale X introduced in the previous theorem. Theorem 2.6. Let us assume that the objects P, H, E, M, X, Ms , Xs satisfy the assumptions of Theorem 2.5. Furthermore let us assume that E is a Hilbert space. Let Q s = Xs  be the quadratic variation process of the Fs -martingale Xs for each s > 0. Moreover, let Q be a T1 (E)-valued process such that Pa.s. for every t ∈ [0, T ], Q s (t)x, yE → Q (t)x, yE

∀x, y ∈ E .

Assume also that for some r > 1, and for every t ∈ [0, T ],   sup E Xs (t)2r E < ∞, s>0   sup E Q s (t)rL (E) < ∞ .

(2.30)

(2.31) (2.32)

s>0

Then Q is F-adapted and is equal to X, the quadratic variation process of the F-martingale X. Proof. The F-adaptedness of Q is clear. In view of the Doob-Meyer theorem [76, Theorem 22.5, p. 412] in order to prove that Q is the quadratic variation process of the F-martingale X, it is enough to prove that for all x, y ∈ E the process N = {N (t); t ∈ [0, T ]} defined by       N (t) := X(t), x E X(t), y E − Q (t)x, y E ∀ t ∈ [0, T ] is an F-martingale. Let us fix x, y ∈ E, and two values t, s ∈ [0, T ] such that s ≤ t. We have to show that for any choice of times 0 ≤ r1 < r2 < · · · rK ≤ s, where K ∈ N,

Section 2.1

123

Preliminaries

and any bounded and continuous functions hi : H → R (i = 1, · · · , K), the following equality holds K  '   hi M(ri ) = 0 . (2.33) E N (t) − N (s) i=1

For each s > 0, we define a process Ns = {Ns (t); t ∈ [0, T ]} by the following formula       ∀ t ∈ [0, T ] . Ns (t) := Xs (t), x E Xs (t), y E − Q s (t)x, y E

Let us fix s > 0. Since by the assumptions Q s is the quadratic variation process of the Fs -martingale Xs , in view of the Doob-Meyer theorem we infer that Ns is an Fs -martingale. Note that the assumptions (2.30) and (2.25) imply that P-a.s. for all t ∈ [0, T ] Ns (t) → N (t) . (2.34) Moreover, by assumptions (2.31) and (2.32) we deduce that the martingales Ns satisfy   sup E |Ns (t)|r < ∞ . (2.35) s>0

Hence, the family {Ns ; s > 0} satisfies the assumptions of Theorem 2.5 and therefore we infer that the process N is an F-martingale as required. This concludes the proof.

2.1.6

Stability of solutions of the Landau-Lifshitz-Gilbert equation

Let O ⊂ Rn be a bounded smooth domain. We recall certain aspects of finitetime finite energy blow-up behavior of solutions u : OT → S2 of the deterministic Landau-Lifshitz Gilbert equation (LLG) (α, β ≥ 0)   ∂u = β u × Δu − α u × u × Δu on OT := (0, T ) × O , (2.36) ∂t u(0, ·) = u0 on O , which is supplemented with Dirichlet or homogeneous Neumann boundary conditions; see e.g. [81]. By putting β = 0 we recover the harmonic map heat flow, for which possible blow-up behavior of corresponding gradients caused by the geometry of the target manifold is known (n ≥ 2). Subsection 2.1.6 discusses this phenomenon, also when β is changed to positive values. Subsection 2.1.6 studies the complementary case α = 0, β = 0 where (2.36) becomes conservative and describes the Schrödinger map. In a certain case, the problem may be restated as the cubic Schrödinger equation (2.44), for which recent works study the effect of acting noise on the stable formation of singularities of function values in the deterministic problem.

124

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Singular behavior of solutions of LLG Let B1 (0) ⊂ R2 be the unit ball. We start with the harmonic map heat flow (n = 2), where β = 0 in (2.36). Consider radially symmetric functions u : [0, T ] × B1 (0) → S2 given in terms of polar coordinates by ⎛

cos ϕ sin θ(t, r)

⎞

⎟ ⎜ ⎟ u(t, r, ϕ) = ⎜ ⎝ sin ϕ sin θ(t, r) ⎠ . cos θ(t, r)

(2.37)

It can be easily seen that functions of this type solve (2.36) provided θ : (0, T )× (0, 1) → R satisfies 1 sin 2θ θt = θrr + θr − r 2r2

on (0, T ) × (0, 1) .

(2.38)

If we add boundary conditions θ(t, 0) = 0 and θ(t, 1) = θ∗ > π, the solution u : OT → S2 blows up at a finite time T ∗ > 0, i.e. limt↑T ∗ θr (t, 0) = ∞; see the survey [105]. The blow-up rate and additional structural details of the singularity formation are known through a matched asymptotic analysis in [74]; in fact, small temporal (for t < T ∗ ) and spatial length scales (L(t) > 0 for t < T ∗ ) are interwoven in this phenomenon according to [74, 33] via θr (t, r) → 

2L(t) 2 L(t) + r2

where

L(t) ↓ 0

(t ↑ T ∗ ) .

(2.39)

Moreover, solutions may be continued beyond the blow-up time T ∗ where the strong solution terminates by an instantaneous rotation over an angle π; see e.g. [105]. Finite time blow-up behavior of solutions for the harmonic map heat flow is so far only known for the special case of radial symmetry of the form (2.37). The results in [75] suggest that these singularities are not generic, i.e. they vanish for small (non-symmetric) perturbations of (initial) data. Solutions of problem (2.36) where β = 0 are not invariant any more under radial symmetry, which is due to the additional precessional motion. As a consequence, ansatz (2.37) may not be used to construct explicit blow-up examples; however, there is computational evidence for a corresponding blow-up behavior [18, 17]. According to [75], those singularities are again not generic. This observation will be a strong motivation in Section 2.2 below to study solutions of a modification of (2.36) with driving (spatially non-uniform) noise. Different numerical schemes for (2.36) have been constructed in the last two decades, which e.g. motivate singularity formations of solutions; besides efficient discretizations, including (explicit) ‘splitting-projection’ methods, preference is

Section 2.1

125

Preliminaries

given to structure-preserving ones to reliably resolve the singular behavior of solutions of (2.36); see Sections 1.1.3 and 1.4.1 in Chapter 1, and [81]. A relevant early work on the numerical analysis of a finite difference discretization of the deterministic LLG equation is [6]; we further mention the more recent works [3, 5] which employ a finite element discretization, and either use an explicit time stepping in combination with a projection step (i.e. θ = 0) to ensure sphere-valued iterates at the end of each iteration step, or amount to solving linear problems in this setting (i.e. θ ∈ (0, 1]); see Scheme 1.3 on p. 26. In both cases, iterates satisfy an energy inequality, and (un-)conditional convergence2 to a weak solution of (2.36) may be shown for existing subsequences of (continuified in time) iterates for vanishing discretization parameters. The construction of all these schemes starts with an equivalent reformulation of problem (2.36) by formally applying u × · and some elementary calculations: β ut + α u × ut = (β 2 + α2 ) u × Δu

on OT ;

(2.40)

see also Subsection 1 on p. 25. A different approach has been chosen in [17], where iterates take values in S2 at every nodal point x ∈ Eh of the mesh Th which covers O ⊂ Rn . This scheme is presented on p. 130 (Scheme 2.8), and it is related to the Scheme 1.5 on p. 28: reduced integration in space is used to compromise between the pointwise requirement for iterates to take values in the target manifold S2 , and the variational formulation of the problem. The discrete Laplacian is used to allow for a discrete energy identity for iterates. The disadvantage of Scheme 2.8 as compared to the schemes in [3, 5] lies in its nonlinear structure; cf. p. 89. However, unconditional overall subsequence convergence for (again discretely sphere-valued) iterates of a (linear) simple fixed point method with stopping criterion to solve Scheme 2.8 can be shown; see Subsection 3.3.1 on p. 216 for further details in this direction. We remark that according to (2.39), adaptive space-time meshes are needed in practical studies to resolve relevant space-time scales during the formation of singularities. It has been pointed out in [33] that a fixed (uniform) mesh could prevent inherent scaling structures in the discrete evolution to develop, and thus to reliably approximate blow-up dynamics. This requirement on used meshes is compatible with the convergence analysis for the above results which only assumes regular meshes {Th } and {Ik }. It has already been pointed out in Section 1.1.3 of Chapter 1 that reformulation (2.40), in the case of the stochastic Landau-Lifshitz Gilbert equation (2.47) on p. 129, would in particular lead to the additional stochastic integral T α 0 u × ◦du, which causes severe additional analytical difficulties to properly set up a solution concept for problem (2.47). For instance, the stochastic analog of an energy inequality for (2.40) is not well-defined; it would include the term 2

The quasi-uniform mesh Th has to satisfy a (mild) angle condition. For θ ≤ cretization parameters k, h > 0 need to be coupled.

1 , 2

the dis-

126

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The stochastic Landau-Lifshitz-Gilbert equation

T

ut 2L2 ds which is not defined, due to the limited time regularity properties of the K-valued Q-Wiener process W. This is why we use the stochastic analog of (2.36), i.e. equation (2.47), and the numerical Scheme 2.8 to construct corresponding weak martingale solutions in Section 2.2. 0

Stochastic nonlinear Schrödinger equations The conservative Heisenberg model for the ferromagnetic spin is a special case of equation (2.1) when we set α = 0 and Hthm = 0, ut = u × Δu ,

u(0) = u0 .

(2.41)

Let us assume that n = 1. A well-known strategy to study the stability properties of solutions to equation (2.41) is based on the following reformulation, where with the unit-length vector u ≡ u(t, ·) we associate the orthonormal x basis (u, |uuxx | , u×u |ux | ). A standard calculation shows that ⎛ d ⎜ ⎜ dx ⎝

⎞

u ux |ux | u×ux |ux |

⎛

0

|ux |

⎟ ⎜ ⎟ = ⎜ −|ux | 0 ⎠ ⎝ (u,ux ×uxx ) 0 − |ux |2

0 (u,ux ×uxx ) |ux |2

0

⎞⎛ ⎟⎜ ⎟⎜ ⎠⎝

u ux |ux | u×ux |ux |

⎞ ⎟ ⎟ . (2.42) ⎠

xx ) The quantities κ = |ux |, resp. η = (u,u|uxx×u are the curvature resp. the torsion |2 of the introduced orthonormal basis. These quantities uniquely define a function u(t, ·) : R → S2 since we can consider the identity (2.42) as an equation with respect to u. The Hashimoto transform is a map u → q, where  x  q(·, x) = κ(·, x) exp i η(·, y) dy ∀x ∈ R. (2.43)

−∞

It is possible to show [43, 36, 94] that if u : (0, T ) × R → S2 satisfies equation (2.41) then q given by formula (2.43) satisfies the focusing nonlinear Schrödinger equation (NLS) for complex-valued solutions (n = 1, σ = 1), 1 i∂t q − Δq − |q|2σ q = 0 , 2

q(0) = q0 ∈ W1,2 (Rn ) .

(2.44)

Many results are available which discuss the stability of solutions to the Cauchy problem (2.44) in n ≥ 1 dimensions: for n = 1, no singularities appear, whereas singularities may form in the critical and supercritical cases n ≥ 2 for negative initial energies (σ = 1)   1 1 2 Eσ q0 := |∇q0 | dx − |q0 |2(σ+1) dx < 0 . 2 Rn 2(σ + 1) Rn

Section 2.1

127

Preliminaries

These studies of the deterministic problem (2.44) benefit heavily from the in 2 variant quantities being the L -norm, i.e. t → Rn |q(t)|2 dx is constant — which may be obtained by multiplying (2.44) with q, integration, and taking the real part —, and the energy, i.e. t → Eσ q(t) is constant — where we multiply (2.44) by q t , integrate, and afterwards take the imaginary part. The following stochastic version of (2.44) is e.g. considered in the works [49, 50, 19, 47] to study the effect of noise on the stability of solutions,   i dq(t, x) + Δq(t, x) + |q(t, x)|2σ q(t, x) dt =  q(t, x) ◦ dW (t, x) ∀ (t, x) ∈ R+ × Rn , q(0, ·) = q0 ,

(2.45)

where  > 0 and W ≡ W (t, x) denotes a W1,2 (Rn )-valued Q-Wiener process on  a filtered probability space F with Q ∈ T1 L2 (Rn ) ; cf. Subsection 2.1.4. Realvalued noise of Stratonovich type is used in order to inherit P-a.s. conservation of the L2 -norm, while conservation of the energy is violated. The following result from [48] asserts existence and uniqueness of stochastically strong solutions. We denote by an ‘admissible pair’  (r, p) a couple of real-valued positive  numbers, with r > 2 and 2r = n 12 − p1 . Theorem 2.7. Let us assume that W W1,2 (Rn )-valued F0 -measurable random ⎧ ⎨ σ>0 ⎩ 0 0 . L2k Ω, C [0, T ]; W1,2 (Rn ) ∩ L1 Ω, Lr 0, T ; W1,p (Rn )

128

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

A main motivation for (2.45) is to study the influence of noise on the stability of the system, ranging from the formation of new space-time singularities to their delay, or even disappearance. It turns out that the kind of noise and the strength of the nonlinearity σ ≥ 0 are crucial for the stability of solutions of (2.45):   (i) (critical case σ = n2 ) Colored noise in space (i.e. Q1/2 ∈ T2 L2 , W1,α ) for the critical case σ = n2 causes initial data with sufficiently negative energy to form singularities in finite time [49, Thm. 4.1], i.e.,   ∃t > 0 P {τ ∗ (q0 ) < t} > 0 . This occurrence is well-known for the deterministic model ( = 0). (ii) (supercritical case σ > n2 ) Colored noise in space causes any initial data to blow up in finite time, i.e.   ∀t > 0 P {τ ∗ (q0 ) < t} > 0 ; cf. [49, Thm. 5.1]. This is in contrast to the deterministic case where there are initial data which do not form singularities. The techniques used to show claim (ii) are not applicable for the critical case σ = n2 , which motivates computational blow-up studies. Corresponding 1Dstudies in [50] indicate that colored noise in (2.45) does not initiate singularity formation for solutions which are stable in the deterministic case ( = 0). Moreover, further 1D- and 2D-experiments in [50, 19] for (2.45) with white noise (i.e., Q = Id) indicate its potential to stop the deterministic blow-up formation for any initial data: the computational studies in [19, p. 836 f.] evidence regular trajectories which otherwise blow up in the deterministic case ( = 0). We remark that these numerical studies so far lack solid mathematical understanding since even local solvability for (2.45) for driving white noise is not clear. The numerical scheme which is proposed, analyzed, and used for simulations in [47, 50, 19] uses finite differences in space, and is implicit in time. It is constructed in such a way that the conservation of the L2 -norm and the energy of iterates is respected in the deterministic case ( = 0); the drift part in (2.45) is discretized by a Crank-Nicolson type ansatz, and the Stratonovich product on the right-hand side of the equation is approximated by the product of the increment of the noise with the value of the solution at the midpoint. Let Ik = {tj }Jj=0 be an equi-distant mesh of size k > 0, then the W1,2 (Rn )-valued random variable q j is an approximation of q(tj , ·), where 0 ≤ j ≤ J − 1,       i q j+1 − q j + k Δq j+1/2 + f |q j |, |q j+1 | q j+1/2 =  q j+1/2 Δj W , (2.46)

Section 2.2

where

129

Convergent discretization of SLLG

⎧ ⎨ f (a, b) = 1 a2(σ+1) −b2(σ+1) σ+1 a2 −b2 2σ ⎩ f (a, a) = a .

if a = b ,

The scheme has the advantage that the L2 -norm is still a conserved quantity P-a.s. in (2.46), but the inherent noise has to be truncated in a corresponding analytical study to obtain a bound for the energy, and thus a relevant uniform a priori estimate. To a certain extent, this strategy is needed to circumvent the lack of the Itô formula in a discrete setting. The existence of an W1,2 (Rn )valued discrete process {q j } then follows from a fixed point argument, and its adaptedness can be established as well. Convergence in probability of the whole sequence of (extended) solutions of the semi-discretization (2.46) for k → 0 to the unique solution of (2.45) according to Theorem 2.7 may then be deduced by those uniform priori bounds, and a compactness argument; cf. [47, Theorem 2.2].

2.2

Convergent finite element based discretization of the SLLG equation

The stochastic LLG equation (2.1) we are going to approximate numerically may be re-written in the form   m(t, x) = −α m (t, x) × m (t, x) × Δm m(t, x) dt + m (t, x) × Δm m(t, x)dt dm m(t, x) × ◦dW(t) +m ∂nm (t, x) = 0 m (0, x) = m 0 (x)

∀ (t, x) ∈ OT := (0, T ) × O , ∀ (t, x) ∈ ∂OT := (0, T ) × ∂O , ∀x ∈ O,

(2.47)

where W = {W(t); t ≥ 0} is a K-valued Q-Wiener process defined on the filtered probability space P, and O ⊂ Rn is a polyhedral bounded domain. Here, K is a separable Hilbert space and Q ∈ T1 (K) a trace-class covariance m| = 1, operator. In order to accommodate for the pathwise sphere property |m the stochastic term is again understood in Stratonovich sense; see also [22]. In [30], the existence of a weak martingale solution for (2.47) and K = R by a general abstract Faedo-Galerkin method is shown,where the corresponding  mn }n ⊂ L2 Ω; C([0, T ]; L2 ) satisfy for every t ≥ 0 approximate solutions {m m0 L2 P-almost surely. By a mn (t, ·)L2 = m and every n ≥ 1 the identity m compactness argument, corresponding limits are then shown to satisfy (2.47) in a proper sense; see Definition 2.20 below. Our aim here is to provide an alternative proof, by constructing a weak martingale solution of equation (2.47) as proper limit of iterates which solve an implementable finite element based space-time discretization; see Schemes 2.9 and 2.10 below. The corresponding

130

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

iterates are S2 -valued at all nodal points of a regular triangulation Th — a (pointwise) property which is not valid for the general Faedo-Galerkin method used in [30]. The construction of a convergent discretizations of equation (2.47) requires to address the properties that solutions of (2.47) are S2 -valued and have bounded energies. The first pointwise property of solutions is hard to mimic in the variational framework of a finite element method, where functions are piecewise polynomial; see Figure 2.1. A possibility to circumvent this first problematic issue is to use a penalization strategy to approximate the sphere S2 . This approach is reliable for smooth solutions, but makes finding proper choices of the penalization parameters in the neighborhood of singularities a nontrivial endeavor; see also Subsection 1 in Chapter 1. Another numerical strategy to compute discretely S2 -valued iterates is via a projection step, where the intermediate R3 -valued vectors at nodal points x ∈ Eh are projected to S2 at the end of every iteration in time. However, it is due to the numerical splitting at every time level that the iterates usually miss a (discrete) energy law or inherit it only under restrictive mesh conditions; see also Subsection 1.1.3 in Chapter 1. These drawbacks are avoided for the space-time discretization which is proposed in [17]. Here, reduced integration is used to adjust the variational formulation to the requirement of S2 -valued iterates at nodal points x ∈ Eh of the mesh Th , and the midpoint rule in time is used to avoid numerical damping; see also Subsection 1.1.3. Moreover, the (lumped) discrete Laplacian (see (2.11)) is employed to compensate for the limited regularity of the Lagrangean finite element method, and allows for a discrete energy identity. The following scheme to approximate solutions of the deterministic LLG (2.36) (β = 1) has been proposed and studied in [17]; see Subsection 2.1.1 for the used notation. M 0 (x )| = 1 for all x ∈ Eh . For Scheme 2.8. Let M 0 ∈ Vh be such that |M j+1 every j ≥ 0, determine M ∈ Vh such that for all Φ ∈ Vh   hM j+1/2 ], Φ M j+1/2 × Δ (dtM j+1 , Φ )h + α M j+1/2 × [M h  j+1/2  j+1/2  − M × Δh M ,Φ h = 0 . In order to construct a convergent structure-preserving discretization of (2.47), we have to account for the stochastic effects as well. In fact, an additional issue here is to properly arrange the iterates along Ik to approximate the Stratonovich integral, and to simultaneously allow for discretely S2 -valued random variables, M j } which take values in S2 at all nodal points x ∈ Eh . It turns i.e. iterates {M out from the analysis below that keeping the averages M j+1/2 of subsequent iterates in the leading position of the two nonlinear terms in Algorithm 2.8 is hM j+1/2 essential to allow for discretely S2 -valued iterates, while changing Δ j+1  to ΔhM in those terms is needed to allow for relevant a priori bounds; see Lemma 2.22.

Convergent discretization of SLLG

131

Scheme 2.9. Let W be a K-valued Q-Wiener process on a filtered probability M 0 (x )| = 1 for all x ∈ Eh . For every space P, and M 0 ∈ Vh be such that |M j ≥ 0, and increments Δj W := W(tj+1 ) − W(tj ) ∼ N (0, kQ), determine a Vh -valued Ftj+1 -measurable random variable M j+1 such that P-almost surely for all Φ ∈ Vh   hM j+1 ], Φ M j+1 − M j , Φ )h + αk M j+1/2 × [M M j+1/2 × Δ (M (2.48) h    j+1/2  hM j+1 , Φ = M j+1/2 × Δj W, Φ . ×Δ −k M h h The analysis below studies the interplay between the space-time discretization and the stochastic effects, and shows unconditional convergence for k, h → 0 towards a process which is a weak martingale solution of (2.47). Computational studies which use this scheme, and Scheme 2.10 below are reported in Section 2.3, where also the (regularizing) role of noise in the context of possible finite time blow-up behavior and long-time dynamics is discussed. We remark that this scheme has already been studied in Chapter 1 for fixed h > 0 with respect to its convergence behavior for k → 0 at finite and infinite times. There is a rich literature on approximations of various linear and nonlinear stochastic PDEs, see e.g. [25, 65, 64, 63, 66, 47, 86, 67] and the references cited therein. The closest of these works to the following convergence analysis is [47]. In the following, we detail specific characteristics of our approach. (i) A numerical analysis of general (semi-)linear stochastic partial differential equations with Lipschitz-continuous drift in the above cited works is usually accomplished by employing a semigroup method, or with the help of a Green’s function. Furthermore, even in the case of non-Lipschitz drift the notion of a mild solution plays an important role. For instance, the notion of a mild solution for the numerical scheme (2.46) is used in the work [47] to deduce relevant a priori estimates. This approach from [47] benefits from the gradient-type form of the nonlinearity. The latter property does not hold for problem (2.47), and also the concept of a mild solution in this context is not obvious. (ii) Problem (2.47) is a nonlinear SPDE, with a (nonconvex) pointwise sphereconstraint. It is the interplay of geometric aspects and (multiplicative) stochastic forcing which requires specific numerical discretizations to verify the convergence. Iterates of Scheme 2.9 take values in S2 for all x ∈ Eh of a regular triangulation Th and satisfy a discrete version of (2.2), which are key stability properties of the scheme. (iii) In one space dimension, the deterministic LLG equation without the dissipative term is equivalent to the Schrödinger equation; see Subsection 2.1.6. The time discretization (2.46) of the stochastic nonlinear Schrödinger equation is considered in [47]. The multiplicative noise considered in [47] con-

132

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

serves the L2 -norm of the solution which corresponds to the conservation of energy for the stochastic Landau-Lifshitz equation. Thus the equation considered in [47] conserves energy, while the system we consider dissipates energy. (iv) The construction of a weak martingale solution to problem (2.47) uses the implementable Scheme 2.9, which is a space-time discretization based on a finite element method. Unconditional convergence of (subsequences of) iterates to a solution of (2.47) is shown, see Theorem 2.11. It is due to the possible (pathwise) finite-time blow-up behavior of solutions (see [10]) that no regularity properties superior to the basic ones are expected.





















 
























 





















 




Figure 2.3. (a) Analytic affine interpolation M k,h vs. (b) stochastic interpolation # k,h . (c) Backwards shift in time. M

The convergence analysis addresses the interplay of the discretization in both, space and time variables. The key properties next to the stability properties in (ii) are a uniform control of increments of numerical solutions in different spatial norms (see Lemma 2.24), a compactness argument involving fractional Sobolev spaces (Definition 2.7) used to overcome difficulties stemming from the limited temporal regularity properties of solutions due to the driving Wiener process, and some tools from stochastic analysis. In M k,h )} which are related to comparticular, we prove tightness of laws {L(M M j } on some chosen probability space P and for a chosen puted iterates {M K-valued Q-Wiener process, and use the Skorokhod almost sure representation theorem to identify the limits of corresponding deterministic integrals (see Lemma 2.32). In order to identify the limit of the stochastic part in (2.47), we slightly # }, cf. Figure 2.3. These M k,h } to {M perturb the family of processes {M k,h # processes are close (Lemma 2.33) for each (k, h), while each process M k,h




Convergent discretization of SLLG

133

is adapted to some filtration F k,h , with respect to which the Vh -valued approximate Itô process in (2.98) is a square-integrable martingale. A proper limit (k, h → 0) of it is a square-integrable F -martingale, which by the martingale representation theorem then allows to identify the limit of the stochastic part in (2.47) on another probability space P . This approach independently addresses analytically (see Figure 2.3, (a)), and stochastically (see Figure 2.3, (b)) relevant approximation properties in the convergence analysis. It is hence different from [47], where a backwards M k,h } is used to obtain an adapted process; in time shift of the processes {M see Figure 2.3, (c). (v) Problem 2.47 is intrinsically a Stratonovich equation, what makes the analysis more difficult. This is related to the well-known Wong-Zakai approximation of the Wiener process by a sequence of processes, usually not adapted to the original filtration, which have smoother trajectories. Note that the Wong-Zakai approximation for the SLLG equation is an open problem. Our approach encounters a similar difficulty as the proof of the various versions of the Wong-Zakai theorem and overcomes it by a splitting off of the noise terms, see formula (2.69) and the following analysis. Another aspect of the Stratonovich formulation is the necessity to take into account the Itô correction term in the analysis. This term causes serious difficulties, in particular in the presence of irregular, e.g. space-time white noise, where solvability of (2.47) seems to be open, including the case n = 1. The following convergence analysis for Scheme 2.9 is performed for a colored in space noise. It will become clear below that these arguments differ significantly from the most existing works which address the numerics and analysis of nonlinear SPDEs, and it is expected that the elaboration of the steps (ii)-(v) will be useful for the implementable construction of a weak martingale solution for a broad range of problems. While (ii), and the first part of (iv) mainly use tools from nonlinear numerical analysis, those used in the second part of (iv) draw heavily from concepts of stochastic analysis; see Subsections 2.1.4 and 2.1.5. Scheme 2.9 uses increments of a given Wiener process W to update magnetization iterates. However, it is computationally advantageous to replace the Gaussian increments {Δj W}Jj=0 of a K-valued Q-Wiener process in Scheme 2.9 by a simpler (e.g., bounded and discrete) i.i.d. K-valued Q-random walk {ξξ j }Jj=0 , which satisfies appropriate moment conditions (SI1 )–(SI3 ) as stated in Definition 2.14. The following scheme is then a simple modification of Scheme 2.9. M 0 (x )| = 1 for all x ∈ Eh . For Scheme 2.10. Let M 0 ∈ Vh be such that |M every j ≥ 0, and a K-valued Q-random walk {ξξ j }Jj=0 which satisfies (SI1 )–(SI3 ),

134

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Scheme 2.9

Scheme 2.10

M j } and its increments: Lemmata 2.22 and 1. A priori bounds for {M 2.24. 2. (a) Construct the piecewise affine interpolation M k,h . (b) Show the convergence in P-a.s. sense to some m i.

Apply the Prohorov Theorem: Lemma 2.26.

ii.

Apply the Skorokhod Theorem: Lemma 2.29 & Corollary 2.30.

(c) Identification of the limit of the drift term: Lemma 2.32

3. (a) Construct the stochastic in# terpolation process M k,h # (b) Show closedness of M k,h and M k,h (Lemma 2.33) and deduce convergence (Corollary 2.35). (c) Identification of the limit of the stochastic term (Lemmata 2.38 and 2.39) with the help of the martingale convergence Theorem 2.6.

3. (a) Construct an approximation of the diffusion term

with the help of the X k,h martingale embedding Theorem 2.13 (p. 180). Then, define the stochastic inter8 polation process M k,h in (2.133). 8 (b) Show closedness of M k,h and M k,h (pp. 180-182). (c) Identification of the limit of the stochastic term (pp. 182-184).

Table 2.1. Convergence proof for Schemes 2.9 and 2.10: same and different tools

135

Convergent discretization of SLLG

determine a Vh -valued Ftj+1 -measurable random variable M j+1 such that Palmost surely for all Φ ∈ Vh   hM j+1 ], Φ M j+1 − M j , Φ )h + αk M j+1/2 × [M M j+1/2 × Δ (M (2.49) h     j+1/2 hM j+1 , Φ = M j+1/2 × ξ j , Φ . ×Δ −k M h h M j }Jj=0 of The existence and stability of sequences of numerical solutions {M Scheme 2.10 can be shown in the same way as for Scheme 2.9. Then the tightness of laws of the corresponding piecewise affine interpolation of the resulting numerical solutions follows exactly as for iterates from Scheme 2.9 from arguments which are listed above in (ii), (iv). The main issue is then to construct the stochastic integral; cf. (2.100). The right-hand side term of formula (2.49)

k,h which is defined by the formula gives the discrete Fk -martingale X    l

k,h (tj ) := I h M− X k,h (tl ) × ξ , l 0. By assumptions (SI1 )–(SI3 ), the K-valued Q-random walk {ξξ j }Jj=0 has proper measurability and independence properties with respect to the time-discrete filtration Fk , and its expectation and covariance coincides with those of a corresponding Wiener process. Moreover, higher moments show the right scalings with respect to the time step k > 0. Owing to these assumptions, it is possible to show that the

k,h } (in which the discrete diffusion term sequence of continuous martingales {X is embedded) approximates the stochastic integral term in (2.47) as the mesh sizes k, h converge to zero. This convergence result can be compared with the Donsker invariance principle. In fact, any sequence of i.i.d. random variables which have the same expectation and covariance as the Wiener increments may be used to construct the stochastic integral in (2.47), since the convergence is independent of the specific distribution of the driving random walk {ξξ j }Jj=0 in Scheme 2.10. We refer to Table 2.1 which surveys relevant steps in the convergence analysis of Schemes 2.9 and 2.10. We begin with the definition of a weak martingale solution of problem (2.47). Throughout the remainder of this section, we make the following assumptions for the driving K-valued Q-Wiener process W in (2.47): (S1 ) K ⊂ W1,∞ (O, R3 ) ∩ W2,2 (O, R3 ), and the embedding is continuous, 1

(S2 ) Q is a symmetric, non-negative operator, and Q 2 ∈ T1 (K). Notice that property (S1 ) implies f W1,∞ ∩W2,2 ≤ Cf K

∀f ∈ K.

(2.50)

136

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

  Definition 2.20. Let T > 0. A weak martingale solution P, W, m of problem (2.47) consists of (i) a filtered probability space P, where the filtration F satisfies the usual conditions, (ii) a K-valued Q-Wiener process W = {W(t); t ∈ [0, T ]}, and (iii) an F-adapted process m : [0, T ] × Ω → L2 (O, R3 ) such that (a) for P-almost every ω ∈ Ω,

  m (·, ω) ∈ C [0, T ]; L2 .

  m(·, ·)| = 1 is satisfied m(t, ·)2L2 < ∞, and the equality |m (b) E sup ∇m t∈[0,T ]

Lebesgue almost everywhere in OT , P-almost surely. (c) for every ϕ ∈ C∞ (O, R3 ) and every t ≥ 0 the following equation is satisfied P-almost surely,     m (t, ·), ϕ − m 0 , ϕ t n ! m ϕ ∂m ∂ϕ (s, x), (x) × m (s, x) dxds =− ∂xp 0 p=1 O ∂xp t n ! m m × ϕ) ∂m ∂(m (s, x), (s, x) × m (s, x) dxds −α ∂xp 0 p=1 O ∂xp  t  + m (s, x) × ◦dW(s, x), ϕ (x) . (2.51) 0

Here we understand the Stratonovich integral as a sum of the L2 -valued Itô integral  t  m (s, x) × dW(s, x), ϕ (x) , 0

and the corresponding Itô correction term which in this case and by (2.20) is equal to t ∞  ! 1 q m (s, x), e (x) × e (x) × ϕ (x) dxds , 2 O 0

=1

where convergence of the series follows from assumptions (S1 ), (S2 ), and part (iii), (b) of the definition of the weak martingale solution. Indeed, t ∞  ! 1 q m (s, x), e (x) × e (x) × ϕ (x) dxds 2 O 0

=1

≤

∞

t ϕL1 ≤ Ct ϕ ϕL1 Tr Q . q e (x)2L∞ ϕ 2

=1

Convergent discretization of SLLG

137

In the following, we mostly drop the explicit dependence of processes on x ∈ O. The first main result of this chapter is (subsequence) convergence of iterates of Scheme 2.9 to a weak martingale solution of (2.47), which is made precise in the following theorem. Theorem 2.11. Let O ⊂ Rn , n ≤ 3 be a polyhedral bounded domain, and T = tJ > 0. Let W be a K-valued Q-Wiener process on a filtered probability space P, such that Q ∈ T1 (K) satisfies (S1 ) and (S2 ). For every finite (k, h) > 0, let Th be a regular triangulation of O, and Ik be an equi-distant m0 | = 1 Lebesgue partition of [0, T ]. Assume that m 0 ∈ W1,2 (O, R3 ) with |m M 0 (x )| = 1 for all  ∈ L, as almost everywhere, and M 0 ∈ Vh such that |M well as M 0 → m 0 in W1,2 (O,R3 ) for h → 0. Then there exists a solution M j }Jj=0 ⊂ L2 Ω; W1,2 (O, R3 ) of Scheme 2.9 which satisfies {M (i) (ii)

M j (x )| = 1 |M for all  ∈ L , and all 1 ≤ j ≤ J , P-almost surely , J    hM j+1 2 ≤ CT , M j 2L2 + k M j+1/2 × Δ E sup ∇M M h

(iii) E

1≤j≤J

j=1

J−1  j=0

1 M j+1 − M j 4h + M j+1 − M j 2h + M M k  M j+1 − M j ]2L2 ≤ CT , +∇[M

(iv) for each 0 ≤ j ≤ J, the map M j : Ω → W1,2 (O, R3 ) is Ftj -measurable. # : [0, T ] × O × Ω → R3 for (k, h) > 0 be the continuous process Let M k,h M j }j≥0 by formulas (2.100) and (2.108). Then, obtained from the iterates {M there exist a filtered probability space P , a K-valued Q-Wiener  process W, 2 3 and an F -adapted process m : Ω → L (OT ; R ) such that P , W , m is a # }k,h such weak martingale solution of problem (2.47), and a subsequence {M k,h 1 that for any κ ∈ [1, ∞), and all α ∈ (0, 2 ), r ∈ (1, 4), P -almost surely # → m M k,h

in

      Lκ 0, T ; Lκ ∩W α,r 0, T ; (W1,2 )∗ ∩C [0, T ]; L2

(k, h → 0) .

Remark 2.21. 1. For n = 1, the weak martingale solution m of (2.47) may be shown to be stochastically strong, i.e. m : [0, T ] × O × Ω → S2 is a measurable function of a given Wiener process W on a given filtered probability space P. This result is shown in [31, Theorem 12] for an R-valued Wiener process, where pathwise uniqueness of weak martingale solutions is proven, and the result then follows from the Yamada and Watanabe theorem. 2. Let n = 1, and m : [0, T ] × O × Ω → S2 be a stochastically strong solution of (2.47) with sufficiently regular noise W in space, and regular initial data m 0 .

138

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

In [54, Theorem 4.1], local rates of convergence are shown for iterates {mj } of a semi-discretization in time according to Scheme 2.9: for every  > 0 there  k ] ≥ 1 + C , and  k ⊂ Ω such that P[Ω exists a set Ω log(k) E





m(tj , ·) − mj 2L2 ≤ Ck 1− log(k −ε ) 1Ω k sup m 1≤j≤J

( > 0) .

(2.52)

By the Chebyshev inequality, this estimate implies convergence in probability with rate 0 < ν < 12 (1 − ) for every  > 0, i.e.,    ν = 0. m(tj , ·) − mj L2 ≥ Ck lim lim P sup m

k→0  C→∞

1≤j≤J

Note that statement (2.52) may be sharpened in the finite dimensional case (Scheme 1.11); see the related result in Theorem 1.15 on p. 74. The proof of this result consists of several steps. In Subsection 2.2.1, we will derive stability properties for iterates from Scheme 2.9; see Lemma 2.22 and Theorem 2.12. Subsections 2.2.2 and 2.2.3 are concerned with the proof of (subsequence) convergence of iterates from Scheme 2.9, and the identification of the limit as a weak martingale solution. In particular, the existence and the regularity properties of the process m are addressed in Lemma 2.29 (which is part (a) of the definition of a weak martingale solution), and Corollary 2.30, (i), and (2.89) (which is part (b) in the definition of a weak martingale solution). Furthermore, the convergence and the identification of the limit of the deterministic integrals (drift terms) for k, h → 0 are provided by the results (2.93), (2.94), and Lemma 2.32. Subsection 2.2.3 identifies a filtered probability space, a Wiener process, and the limit of the stochastic (Stratonovich) integral for the vanishing discretization parameters. Hence we are able to show part (c) of the definition of weak martingale solution and thus conclude the proof of Theorem 2.11. Subsection 2.2.4 shows convergence for Scheme 2.10 where a Q-Wiener process is replaced by a Q-random walk; the analysis focuses on the identification of the stochastic integral in (2.51), since iterates may easily be shown to verify the same properties as iterates from Scheme 2.9. The second main result of this chapter is then Theorem 2.14.

2.2.1

Unconditional Stability of Scheme 2.9

Fix h > 0, J ∈ N, and put k = TJ . In the following lemma we verify relevant M j }Jj=0 . properties of Vh -valued iterates {M Lemma 2.22. Let Th be a regular triangulation of the polyhedral bounded doM 0 (x )| = 1 main O ⊂ Rn . Suppose that the initial data M 0 ∈ Vh is such that |M

139

Convergent discretization of SLLG

M 0 L2 ≤ C uniformly in h > 0, and that a K-valued Q-Wiener for all  ∈ L, ∇M process W on a filtered probability space P satisfies (S1 ) and (S2 ). Then there M j }Jj=0 which solves Scheme 2.9 and satisfies properties (i)exists a sequence {M (iv) as stated in Theorem 2.11. Proof of Lemma 2.22. The proof is divided into five steps. Step 1: Solvability. Fix a set Ω ⊂ Ω, P[Ω ] = 1 such that W(t, ω) ∈ K for all t ∈ [0, T ], and ω ∈ Ω . In the following, let us assume that ω ∈ Ω . M j (ω)}Jj=0 ⊂ Vh solving Scheme 2.9 follows by The existence of a sequence {M M r (ω)}jr=0 ⊂ Vh has induction. Therefore, let us suppose that a sequence {M already been found for some j ∈ {0, . . . , J − 1}. Then we consider a continuous map F ωj : Vh → Vh which is defined for every Φ ∈ Vh as follows, ∞      √ Φ) = 2 Φ − M j (ω) − I h Φ × e F ωj (Φ q Δj β (ω)I

(2.53)

=1

    h [2Φ h [2Φ Φ − M j (ω)] − Φ × Δ Φ − M j (ω)] Φ× Φ×Δ +k I h αΦ By using the classical formula a × b, a = 0

∀ a, b ∈ R3 ,

Φh ≥ M M j (ω)h that we infer for every Φ ∈ Vh with Φ      1 ω Φh −M Φ), Φ h = Φ Φ2h − M j (ω), Φ h ≥ Φ Φh Φ M j (ω)h ≥ 0 . (2.54) F j (Φ 2 Hence the Brouwer theorem, see for instance [60, Corollary IV.1.1], implies  Φ∗ h ≤ the existence of Φ ∗ ≡ Φ ∗ (ω) ∈ Vh , such that F ωj Φ ∗ = 0 and Φ M j (ω)h . Then obviously M j+1 (ω) := 2Φ Φ∗ − M j (ω) solves Scheme 2.9. M Step 2: Fk -adaptedness of the solution. We have shown that for each M j (ω) ∈ ∞  M j+1 (ω)h ≤ Vh and every z = z e , there exists M j+1 (ω) ∈ Vh such that M M j (ω)h and M

=1

  G j M j+1 (ω), M j (ω), z = 0 ,

where G j : Vh × Vh × K → Vh is the function defined by the right-hand side of (2.53), i.e., ∞        G j Φ (ω), M j (ω), z := 2 Φ (ω) − M j (ω) − z I h Φ (ω) × e

+ kIh



=1

   h [2Φ h [2Φ Φ(ω) − M j (ω)] − Φ (ω) × Δ Φ(ω) − M j (ω)] Φ × Φ (ω) × Δ αΦ ∀ Φ (ω) ∈ Vh .

.

140

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Φ, M , z) = 0 for a cerMoreover, by inequality (2.54) we infer that if G j (Φ  Φh ≤ M M h . Since tain Φ , M , z ∈ Vh × Vh × K, then Φ  also the  function

F ωj : Vh → Vh is continuous, we deduce that the set Λ M j (ω), z := Φ ∈      Vh ; G j Φ , M j (ω), z = 0 is closed for all M j (ω), z ∈ Vh × K. Hence we proved that the map     Λ : Vh × K  M j (ω), z → Λ M j (ω), z ∈ P(Vh ), where P(Vh ) denotes the set of all subsets of Vh , is well-defined and the values of Λ lie in closed and bounded subsets of Vh . Moreover, since the function G j is continuous, the set      graph(Λ) = Φ (ω), M j (ω), z ∈ Vh × Vh × K; G j Φ (ω), M j (ω), z = 0 is closed in Vh × Vh × K. Hence, by applying [20, Theorem 3.1] we infer that there exists a universally and Borel measurable map κj : Vh × K → Vh such that κj (s1 , s2 ) ∈ Λ(s1 , s2 ) for all (s1 , s2 ) ∈ Vh × K. M j (ω); j = 0, . . . , J} by the following inductive We define the sequence {M formula   M j+1 (ω) = 2κj M j (ω), Δj W(ω) − M j (ω) (ω ∈ Ω, j ∈ {0, . . . , J − 1}) . (2.55) Now, in order to prove property (iv) we proceed by induction: Fix j ∈ {0, . . . , J− 1} and assume that the map M j : Ω → W1,2 (O, R3 ) is Ftj -measurable. Since the function Δj W : Ω → K is Ftj+1 -measurable, the claim follows from (2.55), and the measurability properties of κj .

Figure 2.4. Ftj+1 -measurability of the solution M j+1 .

Step 3: Proof of assertion (i). By choosing Φ = M j+1/2 (ω, x )ϕ ∈ Vh for  h is the nodal basis function attached to x ,  ∈ L in Scheme 2.9, where ϕ ∈ V the properties of the vector product yield the assertion (i).

141

Convergent discretization of SLLG

Step 4: Proof of assertion (ii). First we assume that ω ∈ Ω is fixed. Then hM j+1 (ω) ∈ Vh as a test function in (2.48) to find that we choose Φ (ω) = −Δ   j+1 hM j+1 M − M j , −Δ h   hM j+1 ], −Δ hM j+1 M j+1/2 × Δ +αk M j+1/2 × [M h   j+1/2 j+1 j+1   −k M × Δh M , −ΔhM (2.56) h ∞ √   hM j+1 . = q Δj β (ω) M j+1/2 × e , −Δ h

=1

By the binomial formula 2a − b, a = |a|2 − |b|2 + |a − b|2 , h , the first term and the definition (2.11) of the discrete Laplace operator −Δ on the left-hand side of the equality (2.56) becomes  j+1  hM j+1 − M j , −Δ M h  1 j+1 2 M j 2L2 + ∇(M M j+1 − M j )2L2 . M L2 − ∇M = ∇M 2 By the definition (2.8) of the scalar product (·, ·)h , and the formula a × b, c = a, b × c for a, b, c ∈ R3 , we get   hM j+1 ], −Δ hM j+1 = M hM j+1 2 . M j+1/2 × Δ M j+1/2 × Δ M j+1/2 × [M h h

Since a × b, b = 0 for a, b ∈ R3 , the third term in (2.56) is equal to zero. Putting these identities together then yields  1 M j+1 2L2 − ∇M M j 2L2 + ∇(M M j+1 − M j )2L2 ∇M (2.57) 2 ∞    √ hM j+1 2 = − hM j+1 . M j+1/2 × Δ q Δj β (ω) M j+1/2 × e , Δ +αk M h h

=1

We proceed independently with the last term. When we want to apply hM j+1 we encounter a difficulty since in general definition (2.11) of −Δ j+1/2 random variable M × e is not Vh -valued. As a consequence, we property (2.5) of the Lagrange interpolation, and the definition (2.8) of inner product (·, ·)h to get    j+1/2  j+1/2 hM j+1 hM j+1 M M × e , Δ = I [M × e ], Δ h

h h     M j+1/2 × e ] , ∇M M j+1 . = − ∇ I h [M

the the use the

142

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

In the next step, we consider the integral on each element K ∈ Th , and therefore continue as follows,    M j+1/2 × e ] , ∇M M j+1 ∇ (Id − I h )[M (2.58) = 2 K∈Th n  

−

q=1

L (K)

M j+1   j+1/2 ∂e ∂M M j+1  Mj ∂M 1  ∂M × e , × , + M , 2 ∂xq ∂xq ∂xq ∂xq

 M j+1  M j+1 ∂M is equal to where in the last equality we use the fact that 12 ∂M ∂xq × e , ∂xq 0 (q = 1, 2, 3).     Next, we use the algebraic identity 12 a + b = b + 12 a − b to rewrite the first term in the sum on the right-hand side of (2.58) as    M j+1/2 × e ] , ∇[M M j+1 − M j ] ∇ (Id − I h )[M L2 (K)    M j × e ] , ∇M Mj + ∇ (Id − I h )[M (2.59) L2 (K)    j+1 1 Mj M + ∇ (Id − I h ) (M − M j ) × e , ∇M 2 L2 (K) j j j =: IK + IIK + IIIK .

Let us denote hK = diam K. For the first term above, we use the approximation property (2.6) for the Lagrange interpolation on each K ∈ Th , and  ∇2M j+1/2 K = 0 to conclude that j M j+1/2 × e ]L2 (K) ∇[M M j+1 − M j ]L2 (K) |IK | ≤ ChK ∇2 [M n    M j+1/2 ∂ 2 e ∂M ∂e M j+1/2 × ≤ ChK L2 (K) +  × L2 (K) M ∂xq ∂xr ∂xq ∂xr q,r=1 j+1

− M j ]L2 (K) .

M ×∇[M

M j+1 (ω)L∞ = 1, and since By the already proven property (i) we have M e W1,∞ ∩W2,2 ≤ C by assumption (2.50), we obtain the following upper bound for the sum in (2.58),  j    M j+1/2 L2 (K) ∇[M M j+1 − M j ]L2 (K) |IK | ≤ C hK 1 + ∇M K∈Th

K∈Th

≤ C



 2 1/2 M j+1/2 L2 (K) h2K 1 + ∇M ×

K∈Th

×



M j+1 − M j ]2L2 (K) ∇[M

1/2

K∈Th



 M j+1/2 L2 ∇[M M j+1 − M j ]L2 , ≤ C 1 + M

(2.60)

143

Convergent discretization of SLLG

M j+1 (ω)L∞ = 1 by an inverse estimate on each K ∈ Th (Lemma 2.2). Since M for 0 ≤ j ≤ J − 1, by using inequality (2.60) we get, for some constant C > 0, ∞ ∞  2  1 √  j √   M j+1 − M j ]2L2 + C  q |IK ||Δj β | ≤ ∇[M q |Δj β | . 16 K∈Th

=1

=1

j is Ftj -measurable for every j = 0, . . . , J. Notice that the random variable IIK Hence, we may use the Burkholder-Davis-Gundy inequality (2.22), and the Lagrange interpolation inequality (2.6) to estimate the second term in (2.59) as follows,



E sup

∞ m−1 

m≤J j=0 =1

√



q

j IIK Δj β



K∈Th

$

∞  2  1  J−1      2 M j × e ] , ∇M M j L2 (K)  ∇ (Id − I h )[M q  ≤ CE k

$ ≤ CE

 k

j=0 =1

K∈Th

∞ J−1 



q

j=0 =1

M j × e ]2L2 (K) h2K ∇2 [M

K∈Th

 K1 ∈Th

%

M j 2L2 (K1 ) ∇M

1 2

% .

 M j × e ]L2 (K) is estimated as follows, Since ∇2M j K = 0, the term ∇2 [M M j × e ]L2 (K) ≤ C(1 + ∇M M j L2 (K) ) . ∇2 [M Therefore, ∞ m−1    √  j q IIK Δj β E sup m≤J j=0 =1

≤ CE

(2.61)

K∈Th

∞ 1   J−1    2  2 M j L2 (K) M j 2L2 (K1 ) q h2K 1 + ∇M ∇M k j=0 =1

≤ ...

K1 ∈Th

K∈Th

M j L∞ = 1 P-a.s., By an inverse estimate on each K ∈ Th (Lemma 2.2), and M we conclude that % $ J−1 ∞ 1     2  2 j j 2 M L2 (K) M L2 (K1 ) . . . ≤ CE k 1 + M q ∇M $

j=0 =1

K∈Th

∞ 1  J−1   2 M j 2L2 (K) ≤ CE k q ∇M j=0 =1

≤ kE

J−1  j=0

K∈Th

 M j 2L2 + C Tr Q . ∇M

%

K1 ∈Th

  J−1  1  2 M j 2L2 ≤ C Tr Q E k ∇M j=0

144

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

The third term in equality (2.59) may be similarly controlled as the first one, +  j+1 + j M j L2 (K) M |IIIK | ≤ ChK +∇2 (M − M j ) × e +L2 (K) ∇M n +  ∂ 2 e + + M j+1 − M j ) × ≤ ChK +(M (2.62) ∂xq ∂xr L2 (K) q,r=1

n +  M j+1 − M j ] ∂e + ∂[M + + M j L2 (K) . × ++ ∇M + 2 ∂xq ∂xr L (K) q,r=1

M j+1 − M j 2L2 . For this purpose, let In order to continue, we need to control M j+1 j (ω) − M (ω) in (2.48). After absorbing some terms, we us choose Φ = M arrive at   1 hM j+1 2 M j+1 − M j 2h ≤ Ck 2 α2 M M j+1/2 2L∞ + 1 Mj+1/2 × Δ M h 2 ∞ 2 1 √ M j+1/2 × e h |Δj β | + q M . (2.63) 2

=1

We may now come back to inequality (2.62) and sum it over all K ∈ Th . By the inverse estimate on each K ∈ Th (Lemma 2.2), and the pathwise L∞ (O, R3 )M j }Jj=0 , by assertion (i) we find boundedness of {M ∞  √

=1

q



j |IIIK ||Δj β | ≤ C

K∈Th

× 



∞ 1/2  √  2 M j 2L2 (K) q hK ∇M |Δj β |

=1

K∈Th

M j+1 − M j 2L2 (K) + ∇[M M j+1 − M j ]2L2 (K) M

K∈Th

M j+1 − M j L2 + ∇[M M j+1 − M j ]L2 ≤ C M

∞  √

1/2

q |Δj β | .

=1

Because of inequalities (2.8) and (2.63), we may resume that ≤

∞  2   1 √

 hM j+1 2 +  M j+1 − M j ]2L2 + C k 2 M M j+1/2 × Δ ∇[M q |Δ β |  . j

h 16

=1

Summing the last inequality over j = 0, . . . , m − 1 and taking the supremum

145

Convergent discretization of SLLG

over m ≤ J − 1 we conclude from inequalities (2.60), (2.61) that E

 sup

m−1 ∞ 

√

m≤J−1 j=0 =1

q Δj β



j j j IK + IIK + IIIK



K∈Th

J−1  J−1   1  M j+1 − M j )2L2 + k E M j 2L2 ∇(M ∇M ≤ E 8 j=0

2

+ Ck E

J−1 

(2.64)

j=0

  2 hM j+1 2 2 + CT Tr Q 12 . M j+1/2 × Δ M L

j=0

Concerning the second term on the right-hand side of inequality (2.58), by using the fact that a × b, a = 0, we obtain   ∞ n     Mj Mj M j+1 − M j ]   ∂M M j  ∂M ∂[M ∂M √

1 q |Δj β |  × e , × e , +  2 ∂xq ∂xr ∂xq ∂xr 

=1

q,r=1

∞ 2   1 √   j+1 j 2 j 2 M M M − ]L2 + C∇M L2  q |Δj β | + 1 . (2.65) ≤ ∇[M 16

=1

Next, we employ the algebraic identity 12 (a + b) = 12 (a − b) + b to restate the last term in equality (2.58) in the way (0 ≤ j ≤ J − 1,  ∈ N) IV j := = =

n   M j+1  ∂e ∂M , M j+1/2 × ∂xq ∂xr q,r=1 % $ n  M j+1   j ∂e ∂M M j+1  1  j+1 ∂e ∂M j M −M ]× , , + M × [M 2 ∂xq ∂xr ∂xq ∂xr q,r=1 $ n  M j+1 − M j ]  1  j+1 ∂e ∂[M M [M − M j] × , + 2 ∂xq ∂xr q,r=1

Mj  ∂e ∂M 1  j+1 M − M j] × , [M 2 ∂xq ∂xr %  M j+1 − M j ]   j ∂e ∂M Mj  ∂e ∂[M j , , + M × + M × . ∂xq ∂xr ∂xq ∂xr

+

Then, by (2.9), ∞  √

=1

q IV j Δj β ≤

+2 1+ 1+ +∇[M M j+1 − M j ]+L2 + +M j+1 − M j 2h 16 16

146

Chapter 2

+

The stochastic Landau-Lifshitz-Gilbert equation

n ∞   Mj  √  j ∂e ∂M q M × , Δj β ∂xq ∂xr

(2.66)

=1 q,r=1 ∞ 

√

 +C 

2    M j 2L2 + ∇M M j 2L2 . M j+1 − M j 2L2 + M q |Δj β | M

=1

+ 1 + j+1 M j 2h . Summing up over j = 0, . . . , m− We employ (2.63) to control 16 −M M 1, for a fixed m ≤ J in (2.66), we use Lemma 2.22 (i) to find m−1 ∞ 

√

q IV j Δj β ≤

j=0 =1

m−1 +2  1 + +∇[M M j+1 − M j ]+L2 + 16 j=0

m−1  Ck   hM j+1 2 M j+1/2 × Δ M k h 16

+

j=0

+C

∞ m−1  

 

√

2    M j 2L2 q |Δj β | 6 + ∇M

j=0 l=1 ∞  n m−1  √

+

j=0 =1 q,r=1

 M j  ∂e ∂M q Δj β M j × , . ∂xq ∂xr

Taking the supremum over m ≤ J, and the expectation we deduce that 

E sup

m−1 ∞ 

√

m≤J j=0 =1

q IV Δj β j



+2  1 + +∇[M M j+1 − M j ]+L2 ≤ E 16 J−1 j=0

J−1  Ck   hM j+1 2 + M j+1/2 × Δ E k M + h 16 j=0

+C E

∞ J−1  

 

√

2    M j 2L2 ) q |Δj β | 6 + ∇M

j=0 =1



+E sup

∞ m−1 

n  √

m≤J j=0 =1 q,r=1

 M j  ∂e ∂M q Δj β M j × , . ∂xq ∂xr

Next, we use the Cauchy-Schwarz and Burkholder-Davis-Gundy inequalities to estimate the last term as follows, ⎡ ⎤ m−1 n ∞    j  M ∂e ∂M √ E ⎣ sup (2.67) q M j × , Δj β ⎦ ∂x ∂x q r m≤J j=0 =1 q,r=1

147

Convergent discretization of SLLG

≤ CE

n ∞    J−1  M j 2  12  ∂e ∂M  k q  M j × ,  ∂xq ∂xr j=0 =1 q,r=1

 1    J−1 J−1   2 M j 2L2 M j 2L2 + C Tr Q . ∇M ≤ kE ∇M ≤ C Tr Q E k j=0

j=0

By summing over iteration steps j = 0, . . . , m − 1 in equality (2.57), then taking the supremum over m ≤ J and the expectation, using (2.63), (2.64), (2.65), (2.67), and absorbing terms for k > 0 sufficiently small yields  1   1  M m 2L2 − ∇M M 0 2L2 + E M j+1 − M j ]2L2 E sup ∇M ∇[M 2 m≤J 4 J−1 j=0

J−1  α   hM j+1 2 M j+1/2 × Δ M + E k h 2 j=0

≤ CE

J−1 

M j 2L2 + 1 ∇M

∞ 2   √  q |Δj β | + 2k . 

j=0

=1

Because ∞ ∞ 2  2       √ √     M j 2L2  M j 2L2  E ∇M q |Δj β | q |Δj β | |Ftj = E E ∇M

=1

=1

∞ 2   √  j 2 M L 2 E  = E ∇M q |Δj β | |Ftj





≤ Ck Tr Q

1 2

2

=1

 M j 2L2 , E ∇M 

the discrete Gronwall inequality then leads to   1  1  M m 2L2 + E M j+1 − M j ]2L2 + E sup ∇M ∇[M 2 m≤J 4 J−1 j=0

J−1  α   hM j+1 2 M j+1/2 × Δ M (2.68) + E k h 2 j=0      *   )    1 2 1 2 0 2 2 2 M L2 + C Tr Q ≤ E ∇M exp C 1 + Tr Q T ,

which is assertion (ii). This completes the proof in Step 4. Step 5: Proof of assertion (iii). Assertion (iii) consists of three inequalities. The first one follows from inequalities (2.63), (2.68). The third one is a consequence of (2.68). To show the second inequality, we multiply inequality (2.63)

148

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

M j+1 − M j 2h , and get by M 1 M j+1 − M j 4L2 M 2

  M j+1 − M j 2L2 Ck 2 α2 M M j+1/2 2L∞ + 1 × ≤ M hM j+1 2 M j+1/2 × Δ ×M h ∞  2  1 √ M j+1 − M j 2L2 M j+1/2 × e h . + M q |Δj β | M 2

=1

M j+1 − M j 2L2 ≤ 4. Hence in Because of part (i) in Lemma 2.22, we obtain M view of the Young inequality we find that for a fixed δ < 12 there exists C > 0 such that   1 hM j+1 2 M j+1 − M j 4L2 ≤ Ck 2 α2 M M j+1/2 2L∞ + 1 M M j+1/2 × Δ M h 2 ∞   4 √ M j+1 − M j 4L2 + C M j+1/2 × e h . +δM q |Δj β | M 

=1

 4

Thanks to an inequality E |Δj β | ≤ Ck 2 , assumption (S2 ), and inequality (2.68), summation over all indices j = 0, . . . , J − 1 then implies (iii)2 . This completes the proof of Lemma 2.22. In what follows, we put Φ = fp ϕr (1 ≤ p ≤ 3, r ∈ L) in (2.48), where {fp }3p=1  h. is the standard orthonormal basis in R3 , and {ϕr }r∈L is the nodal basis of V 1 1 By using 2 (a + b) = b + 2 (a − b), we conclude that equality (2.48) takes the following form for all j = 0, . . . , J − 1,   1 1 hM j+1 ) − M j+ 12 × Δ hM j+1 M j+ 2 × Δ M j+1 − M j + k I h α M j+ 2 × (M =

∞  √

=1

= =

∞  √

=1 ∞ 

√

=1

  1 q Δj β I h M j+ 2 × e    1 M j+1 − M j ) × e q Δj β I h M j + (M 2

(2.69)

∞   k    1 q Δ j β I h M j × e + q I h M j+ 2 × e × e + A j , 2

=1

where, in view of (2.48), we have ∞

Aj

:=

   j+ 1 1  M 2 × e ) × e l q |Δj β |2 − k I h (M 2 −

=1 ∞ 

k 2

=1

√

q Δj β I h



1

1

hM j+1 ) M j+ 2 × Δ M j+ 2 × (M αM

(2.70)

149

Convergent discretization of SLLG

  1 hM j+1 × e M j+ 2 × Δ −M    j+ 1  1  √ M 2 × em 1 ) × e m 2 + qm1 qm2 Δj β m1 Δj β m2 I h (M 2 m1 =m2

=:

1 j 1 A 1 − k A j2 + A j4 2 2

(j = 0, . . . , J − 1) .

The first term in the representation on the right-hand side of formula (2.69) is an approximation of the Itô integral, the second one represents the Itô correction term, and the last one, given by the formula (2.70), is the remainder term. We will show smallness of the remainder as the size of the mesh converges to zero in the following auxiliary lemma. Lemma 2.23. In the framework above the following limit is valid,  J−1 + J−1  j+  j+  + j +  + J−1   j +A + 2 = 0 . A 1 L ∞ + E k lim E + A 1 + L 2 ++ A 4 +L2 + max E A 2 L

k→0

j=0

1≤j≤J

j=0

j=0

j J−1 Aj1 }J−1 Proof. Step 1. The sequence {A j=0 . Consider the sequence A 3 j=0 , defined by A j3

∞

   j 1  M × e ) × e = q |Δj β |2 − k I h (M 2

(j = 0, . . . , J − 1) .

=1

We will show that

+ J−1  j+  A 3 +L2 = 0 . lim E +

k→0

j=0

+  "    + + J−1 j + j 2 A  Since E + j=0 A 3 + 2 ≤ E  J−1 3 L2 , it is sufficient to show that j=0 L

+ J−1  j +2  A 3 +L2 = 0. lim E +

k→0

j=0

   Since A l3 is Ftj -measurable for l < j, and satisfies E A j3 Ftj = 0, we infer that 2 Aj3 }J−1 the finite sequence {A j=0 is an L -valued martingale difference with respect to filtration F, and therefore  J−1  J−1   j   j 2 A A3 2L2 ≤ Ck 2 J = CT k → 0 E  E A 3 L 2 = j=0

j=0

(k → 0) .

150

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Thus, in order to prove the first claim, it is sufficient to prove that E

J−1 + j +  +A − A j + 2 → 0 1 3 L

(k → 0) .

(2.71)

j=0

Note that A j1 − A j3 =

∞  2    j+ 1 1   M 2 − M j ] × e × e q Δj β  − k I h [M 2

=1

=

∞ 1

4

 2     j+1 M q Δj β  − k I h [M − M j ] × e × e .

=1

Because of the Cauchy-Schwarz inequality, and the standard identity  2  = 2k 2 , E |Δj β |2 − k

(2.72)

we deduce that E

J−1 + j +  +A − A j + 2 1 3 L j=0

≤C

J−1  j=0

≤C

J−1  j=0

≤C

J−1  j=0

∞  |Δ β 2 + j+1 +   k j M − 1+ [M q E  − M j ] × e × e +L2 4 k

=1

  ∞ 1/2  + +2 1/2 |Δj β |2 k 2 − 1| q E | E +M j+1 − M j +L2 4 k

=1

∞ +2 1/2 k   + q E +M j+1 − M j +L2 ≤ ... 2

=1

Next, we use Lemma 2.22, (iii)1 and inequality (2.9) to further conclude that 0 2 1 J−1   1  + +2 1/2  k . . . ≤ C Tr Q2 E +M j+1 − M j +L2    2 j=0 0 1 J−1  1  + +2  k ≤ C Tr Q2J E +M j+1 − M j +L2 2 j=0 √ = kT CT k . ≤ C Tr Q JCT = CTr Q 2 2 Since the right-hand side of the last inequality converges to 0 when k → 0, the proof of (2.71) is complete.

151

Convergent discretization of SLLG

We will now prove the third claim, by using Lemma 2.22, (i), and identity (2.72). Indeed, by the Cauchy-Schwarz inequality, ∞ +   +    j 1 + + Mj+ 2 × e ) × el + A1 L∞ = E + q |Δj β |2 − k I h (M E A

=1

≤

 L∞

∞    1    2  12   1 2 Mj+ 2 × e ) × e 2L∞ I h (M q E |Δj β |2 − k E I

=1

≤ Ck Tr Q .

(2.73)

Aj2 }J−1 Step 2: The sequence {A j=0 . 0, . . . , J − 1,

Similarly to the above, we have for j =

∞ +    j  +   1 1 √ + hM j+1 ]+ A 2 L 2 ≤ α M j+ 2 × M j+ 2 × Δ E A q E |Δj β | +I h [M + 2

+

∞  √

=1

≤ (α + 1) ≤ (α + 1)

L

=1

+ +   1 + hM j+1 ]+ M j+ 2 × Δ q E |Δj β | +I h [M + 2 L

∞ 

=1 ∞ 

√

+ +   1 + hM j+1 ]+ M j+ 2 × Δ q E |Δj β |+I h [M + 2 L

1/2   1/2 1 √   hM j+1 ]2 2 M j+ 2 × Δ I h [M q E |Δj β |2 E I L

=1

 + + 1/2 1 hM j+1 +2 ≤ C (α + 1) Tr Q 2 k E +M j+1/2 × Δ , h where we used (2.9) to deduce the last inequality. Hence, J−1  J−1   1  j   1 hM j+1 2 2 A2 L2 ≤ C Tr Q 2 k(α + 1) M j+1/2 × Δ E k A k E M h j=0

j=0

 J−1    1 1 hM j+1 2 2 M j+1/2 × Δ ≤ C Tr Q 2 k(α + 1)J 1/2 k E M h j=0 1

= C Tr Q 2

 J−1   √ √  1 hM j+1 2 2 . M j+1/2 × Δ T k (α + 1) k E M h j=0

Therefore, thanks to assertion (ii) of Lemma 2.22, this implies  J−1   j A 2 L 2 = 0 . A lim E k

k→0

j=0

(2.74)

152

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Aj4 }J−1 Step 3: The sequence {A j=0 . The proof is similar to Step 1 and is left to the reader. As a consequence of the three steps the assertion of Lemma 2.23 is proved. The following result is a sharpened version of Lemma 2.22, (iii). Its proof uses in particular the reformulation (2.69) of (2.48), and Lemma 2.22, (iii). Lemma 2.24. For every δ > 0, there exists a constant Cδ > 0 such that for every K ∈ {1, . . . , J}, the following inequality holds  J−K   M j+K − M j 2h + tδK M M j+K − M j 4h ≤ Cδ t2K . E k tK M j=0

Proof. Step 1: First inequality. Let us fix K ∈ {1, . . . , J}, and ω ∈ Ω. We sum in (2.69), then take the (·, ·)h inner product with Φ = M j+K (ω) − M j (ω). The Young inequality, and assertion (i) of Lemma 2.22 lead to  K−1  + 2 + 1 hM j+r+1 + M j+K − M j 2h ≤ k M α + 1 +M j+r+1/2 × Δ h 2 r=1

∞  K−1 2  M j+r+1/2 × e ) × e h + I + II , + k q (M

(2.75)

r=1 =1

where ∞ +j+K−1 +2 +  √ + M i × e ] Δ i β + , I = C+ q I h [M i=j

II = C

K−1 

h

=1

Aj+r A 1 h

2

r=1

+C

K−1 

Aj+r A 4 h

2

+ Ck

r=1

K−1 

Aj+r A 2 h .

r=1

We will consider the terms I and II separately. Since M j is Ftj -measurable and Δj W is Ftj -independent, by using Lemma 2.22, (i), and properties of the conditional expectation, we infer that ∞ +2  +j+K−1 +  √ + M i × e ]Δi β + E[I] = E + q I h [M i=j

=

j+K−1 ∞   i=j

=1

=1

h

+  +2 M i × e ]+h q |Δi β |2 ≤ CtK Tr Q . E +I h [M

153

Convergent discretization of SLLG

By (2.72), (2.73), and Lemma 2.22, (i), the first part of II is bounded as follows, E

K−1 

Aj+r A 1 h

r=1

≤KE

K−1 

2 

+E

K−1  

Aj+r A 4 h

r=1

2 Aj+r A 1 h + K E

K−1 

r=1

2 

 2 2 Aj+r A  h ≤ CtK . 4

(2.76)

r=1

Similarly to inequality (2.74), the remaining part of term II can be controlled by  K−1   √ √ Aj+r E k A  tK k . ≤ C h 2 r=1

We may use these bounds in (2.75): summing up over j = 1, . . . , J − K, multiplying by k, and finally taking the expectation then yields to  J−K     M j+K − M j 2h ≤ CtJ (α2 + 1)tK + 1 tK , M E k

(2.77)

j=1

where we use assertion (ii) in Lemma 2.22. Step 2: Second inequality. Fix σ ∈ [1, 2). Let us multiply inequality (2.75) by M j+K − M j 2h and then use Lemma 2.22, (i). This together with the Young M inequality allows us to find that  K−1  + 2 + j+r+1/2 1 j+K j 4 hM j+r+1 + +M M α + 1 M − M  ≤ 4 k × Δ h h 25 r=1

∞  K−1 2  M j+r+1/2 × e ) × e h +4 k ql (M r=1 =1

+I + II + III , where ∞ +j+K−1 +4  √ +  + I = + I h[ q M i × e ] Δi β + , i=j

II



= C

K 

Aj+r A 1 h

r=1

III

h

=1

2

+C

K   σ Aj+r = Cσ k A . 2 h r=1

K  r=1

2 Aj+r A 4 h ,

(2.78)

154

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Again, we separately consider the terms I , II and III . By Lemma 2.22, (i) there exists a constant C > 0 such that for all i = 0, . . . , J − 1, P-a.s. M i × e ]L∞ ≤ C . I h [M I By using the Burkholder-Davis-Gundy inequality (2.22), we infer that ∞ +4  +j+K−1   +  √ + M i × e ] Δi β + q I h [M E I = E + i=j

≤ CE

h

=1

∞ j+K−1 2     2 I h [M M i × e ]2h k q I ≤ C Tr Q t2K . i=j

=1

Because of inequality (2.76), it only remains to find a bound on the following term, E

∞ K    + + σ  1 √ hM j+1 + q |Δj β |+M j+ 2 × Δ k h r=1 =1

$

≤

k E σ

K  ∞  √ r=1

=: k

2−σ 2

 K % σ + σ−1 + σ  1 j+ j+1  hM + q |Δj β | +M 2 × Δ 1 h

r=1

=1

σ−1 σ2 tK k A.

By using the Young inequality, inequality (2.23) and Lemma 2.22, (ii)2 we can σ estimate k 2 A from above in the following way:  σ 2

k A ≤

E

K  ∞  √ r=1



q |Δj β |

=1

≤ Cσ Tr Q Kk

σ 2−σ

 2−σ 2



2σ 2−σ



 2−σ  2

K   + j+ 1 +  hM j+1 +2 +M 2 × Δ E k h

σ

r=1 2−σ 2

≤ Cσ Tr Q tK k

σ−1

. σ

σ

2 ≤ tσK then Putting things together in (2.78) and using the inequality k 2 tK leads to

 J−K    2 M j+K − M j 4h ≤ C Tr Q (α2 + 1)t2K + Cσ Tr Q tσK E k M

(1 ≤ σ < 2) .

j=0

(2.79) Inequalities (2.77) and (2.79) yield the assertion of the lemma. We may now use the bounds from Lemmas 2.22 and 2.24 for (increments M j }Jj=1 that solve Scheme 2.9 to construct proper of) the Vh -valued iterates {M limiting functions as k, h → 0, which are possible candidates for a weak martingale solution of (2.47). For this purpose, we define M k,h to be the piecewise

2

155

Convergent discretization of SLLG

M j }. Moreover, we define M − affine interpolation in time of those iterates {M k,h , M+ M k,h to be the piecewise constant interpolations; see Definition 2.4 k,h and for details. Using this notation, we may rewrite equality (2.48) in the form    +  hM + ), Φ Mk,h × Δ M k,h − M − k,h k,h , Φ h + αk M k,h × (M h     +  −k M k,h × ΔhM k,h , Φ h = M k,h × Δj W, Φ h ∀ Φ ∈ Vh . (2.80) Theorem 2.12. Assume that T > 0, α ∈ (0, 12 ) and δ ∈ (0, 3]. Then there exists a constant C = CT,α,δ > 0 such that every solution M k,h : [0, T ] × O × Ω → R3 of (2.80) satisfies the following inequalities   2 M Mk,h 4−δ + M  (i) E M 1,2 k,h α,4−δ 2 C([0,T ];W ) ≤ CT,α,δ , (0,T ;L ) W (ii)

Mk,h L∞ (0,T ;L∞ ) ≤ 1, M

P − a.s.

Proof. Let us fix T > 0, α ∈ (0, 12 ) and δ ∈ (0, 3]. Choose next s1 ∈ (α, 12 ). Since N s1 ,r (0, T ) ⊂ W α,r (0, T ) (i.e. property (iii) in Section 2.1.2), by applying Lemmata 2.8 and 2.24 we infer the first part of assertion (i). The second part of assertion (i) follows immediately from part (ii) of Lemma 2.22. Finally, the Mk,h } and part inequality in (ii) follows from the definition of the sequence {M (i) of Lemma 2.22. Remark 2.25. Since each process M k,h is affine on intervals [tj , tj+1 ), for j = 0, . . . , J − 1, with the nodal values belonging to L∞ ∩ W1,2 , it follows that P-almost surely   M k,h (ω, ·) ∈ W 1,2 0, T ; L∞ ∩ W1,2 . However, we do not expect that the estimate 2.12  in assertion (i) of Theorem 1,2 ∞ 1,2 1,2 is true with the space W 0, T ; L ∩ W instead of C([0, T ]; W ).

2.2.2

Convergence of iterates from Scheme 2.9

The estimates from Lemma 2.22 and Theorem 2.12 allow for the following tightness result. Lemma 2.26. Let 2 ≤ κ < ∞, α ∈ (0, 12 ) and  1 ≤ r< 4. Then  the sequence  Mk,h )}k,h is tight on the space Lκ 0, T ; Lκ ∩ W α,r 0, T ; (W1,2 )∗ ∩ of laws {L(M C([0, T ]; L2 ). We need the following auxiliary Lemma to prove Lemma 2.26. Lemma 2.27. Let 2 ≤ κ < ∞, γ ∈ (0, 1), r0 > 1. Then the embedding         (2.81) L2 0, T ; W1,2 ∩ W γ,r0 0, T ; L2 ∩ L∞ 0, T ; L∞ → Lκ 0, T ; Lκ is compact.

156

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Proof. We first prove the assertion for κ = 2. Take a bounded sequence {un }∞ n=1 in the space on the left-hand side of (2.81). By Lemma 2.9 with E0 = W1,2 (O, R3 ) (which is compactly embedded into E = E1 = L2 (O, R3 ) we infer that the embedding       L2 0, T ; W1,2 ∩ W γ,r0 0, T ; L2 → L2 0, T ; L2 is compact. Hence, we can find a subsequence, denoted again by {un }∞ n=1 , and     u ∈ L2 0, T ; L2 such that un → u in L2 0, T ; L2 . This completes the proof for κ = 2. If κ > 2, then by the Hölder inequality 2

1− 2

f Lκ (0,T ;Lκ ) ≤ f Lκ2 (0,T ;L2 ) f L∞κ(0,T ;L∞ )

  ∀ f ∈ L2 0, T ; L2 ∩L∞ (0, T ; L∞ ) ,

  ∞ 0, T ; L∞ we infer that u → u in and the boundedness of {un }∞ n n=1 in L  Lκ 0, T ; Lκ . Proof of Lemma Let us fix κ ∈ [2, ∞), α ∈ (0, 12 ) and r ∈ [1, 4). Choose

1  2.26. 1 β ∈ (max α, 4 , 2 ) and q ∈ ( β1 , 4). We use Lemma 2.10 with E0 = L2 (O, R3 ), which is compactly embedded into E = W1,2 (O, R3 ))∗ , to conclude that the following embedding is compact,     W β,r 0, T ; L2 → W α,r 0, T ; (W1,2 )∗ .

(2.82)

By Lemma 2.27 with γ = β and r0 = r, we have that the following embedding is compact,         L2 0, T ; W1,2 ∩ W β,r 0, T ; L2 ∩ L∞ 0, T ; L∞ → Lκ 0, T ; Lκ (2.83) By Lemma 2.11 with E0 = W1,2 (O, R3 ) we infer that the embedding,       C [0, T ]; W1,2 ∩ W β,q 0, T ; L2 → C [0, T ]; L2 (2.84) is compact. Therefore, using compactness of embeddings (2.82)-(2.84), we deduce that the embedding,         C [0, T ]; W1,2 ∩ W β,q 0, T ; L2 ∩ W β,r 0, T ; L2 ∩ L∞ 0, T ; L∞     → Lκ 0, T ; Lκ ∩ W α,r 0, T ; (W1,2 )∗ ∩ C([0, T ]; L2 ) , is compact as well. Now the result follows from Theorem 2.12.   Remark 2.28. In the corresponding result in [30] the space Lκ 0, T ; Lκ is replaced by L2 0, T ; Lκ with κ < 6. Our result is stronger, since contrary to [30] we have pointwise estimates for M k,h ; see assertion (i) in Lemma 2.22.

157

Convergent discretization of SLLG

From now on we fix numbers κ ∈ [2, ∞), α ∈ (0, 12 ) and r ∈ [1, 4). By Mk,h }k,h , Lemma 2.26 and the Prohorov theorem there exists a subsequence {M denoted in the same way as the full sequence, such that the sequence of laws   κ 0, T ; Lκ ∩ M )} converges weakly to a probability measure μ on L {L(M k,h k,h   W α,r 0, T ; (W1,2 )∗ ∩ C([0, T ]; L2 ). The following result follows then from the Skorokhod embedding Theorem 2.3; see also [107, Theorem 1.10.4 and Addendum 1.10.5].   Lemma 2.29. There exist a filtered probability space P = Ω , F , F , P and perfect, measurable maps φk,h : Ω → Ω, such that P = P ◦¯ φk,h

∀ (k, h) > 0 ,

(2.85)

the maps M k,h := M k,h ◦ φk,h , where     M k,h : Ω → Lκ 0, T ; Lκ ∩ W α,r 0, T ; (W1,2 )∗ ∩ C([0, T ]; L2 )

(2.86)

are measurable, and     Lκ 0, T ; Lκ ∩W α,r 0, T ; (W1,2 )∗ ∩C([0, T ]; L2 ) . (2.87)     κ κ α,r 1,2 ∗ 0, T ; (W ) ∩ C([0, T ]; L2 )Moreover, there exists an L 0, T ; L ∩ W valued random variable m defined on the new probability space P , such that     m) = μ on Lκ 0, T ; Lκ ∩ W α,r 0, T ; (W1,2 )∗ ∩ C([0, T ]; L2 ) , L(m

M k,h ) Mk,h ) = L(M L(M

on

and, P -almost surely for k, h → 0     M k,h → m in Lκ 0, T ; Lκ ∩ W α,r 0, T ; (W1,2 )∗ ∩ C([0, T ]; L2 ) . (2.88) Define next C(R+ 0 ; K)-valued random variables Wk,h by the formula := W ◦ φk,h Wk,h

(k, h > 0) .

It follows from equalities (2.85) that each process Wk,h is a Wiener process on P.

M k,h }k,h from Corollary 2.30. There exists a subsequence of the sequence {M Lemma 2.29, denoted in the same way as the original one, such that as (k, h) → 0 the following convergences hold:   ∗ (i) M k,h  m in L2 Ω ; L∞ (0, T ; W1,2 ) ,   (2 ≤ κ < ∞) , (ii) M k,h → m in L2 Ω , Lκ (0, T ; Lκ ) (iii) M k,h → m almost everywhere in [0, T ] × O , P − almost surely.

158

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

M k,h }k,h satisfies the same inequalProof. By (2.85) and (2.86) the sequence {M Mk,h }k,h . Therefore, conity as in (i) of Theorem 2.12 the original sequence {M vergence in assertion (i) follows from the Banach-Alaoglu theorem. FurtherM k,h }k,h more, by the uniform estimate (ii) of Theorem 2.12 the sequence {M ∞ ∞ P -a.s. takes values in the unit ball of L (0, T ; L ). Hence, it is bounded M k,h }k,h is in Ls Ω , Lκ (0, T ; Lκ ) (s > 2), and consequently, the sequence {M  κ  2 κ uniformly integrable on the space L Ω , L (0, T ; L ) . Now assertion (ii) immediately follows from (2.88) and the uniform integrability of the sequence M k,h }k,h . Assertion (iii) follows from (ii), and a standard subsequence argu{M ment. M k,h }k,h has In the next lemma we will show that the limit of the sequence {M unit length P -a.s., and a proper limit when the time converges to zero.   Lemma 2.31. Assume M k,h (0) → m 0 in L2 Ω , L2 for (k, h) → 0. We have m| = 1 Lebesgue almost everywhere on [0, T ] × O, and P -a.s. P -a.s. |m m(t) − m 0 L2 = 0 . lim m t↓0

Mk,h Proof. Step 1: The sphere constraint. Recall that |M  (t, x )| =2 1 for all Mk,h (t)| = 1 for x ∈ Eh and all t ∈ [0, T ], P-almost surely. Therefore, Ih |M all t ∈ [0, T ]. Hence, by the interpolation inequality (2.6), there exists C > 0 such that for every h > 0 and every K ∈ Th , + +  + 2 + + 2 +2 +|M Mk,h | − 1 +L2 (K) Mk,h | − 1+L2 (K) ≤ Ch +∇ |M + +2 +2

+ + M+ M+ M+ ≤ 4Ch +(M k,h ) ∇M k,h L2 (K) ≤ 4Ch ∇M k,h L2 (K) , where the last inequality follows from assertion (ii) in Corollary 2.12. Therefore, by Lemma 2.22, (ii),    M k,h |2 → 1 in L2 Ω , L∞ 0, T ; L2 ) |M (k, h → 0) . (2.89) Step 2: Behavior when t ↓ 0. Fix ε > 0. Let us fix ω ∈ Ω such that (2.88) holds for this particular ω . Note that the measure P of the set of such ω is equal to 1. By convergence (2.88), there exists δ > 0 such that for all (k, h) with |k| + |h| < δ we have M k,h (t, ω ) − m (t, ω )L2 ≤ sup M

t∈[0,T ]

ε . 3

(2.90)

Choose (k, h), |k| + |h| < δ such that M k,h (0) − m 0 L2 ≤ M

ε . 3

(2.91)

159

Convergent discretization of SLLG

Since M k,h (·, ω ) ∈ C([0, T ], L2 ), there exists t0 ≡ t0 (ω ) > 0 such that M k,h (t, ω ) − M k,h (0)L2 ≤ M

ε 3

∀ t ∈ [0, t0 ] .

(2.92)

Thus, by the triangle inequality we conclude from inequalities (2.90)–(2.92) that m(t, ω ) − m0 L2 ≤ ε ∀ t ∈ [0, t0 ] , m and the result follows. Next, we identify the limits of the deterministic integrals in Scheme 2.9. For this aim, by employing Corollary 2.30, (ii), and estimate (2.10) we easily conclude that ∞    T    lim E q M k,h × e × e , ϕ ds k,h→0

0

= E

=1 ∞ T 

 0

h

q

    m × e × e , ϕ ds

∀ ϕ ∈ W1,2 (O, R3 ) .

=1

Because of part (ii) in Lemma 2.22, we can also assume     h (M M k,h )+  Y I h M k,h × Δ in L2 Ω ; L2 (0, T ; L2 ) ,(2.93)     h (M M k,h × Δ M k,h )+ )  Z in L2 Ω ; L2 (0, T ; L2 ) .(2.94) I h M k,h × (M The following two results identify the processes Y, Z from (2.93) and (2.94). Property (iii)3 of Lemma 2.22 turns out to be useful to prove the following result. Lemma 2.32. Let us denote ·, ·1 = (·, ·), and ·, ·2 = (·, ·)h . For T > 0, M k,h }k,h be the sequence from Lemma 2.29. Then, for r = 1, 2, for every let {M ϕ ∈ C2 (O, R3 ), and every t ∈ [0, T ], (i)

lim E



k,h→0

t

   ! h (M M k,h )+ , ϕ ds I h M k,h × Δ r

0

= (ii)

lim E

k,h→0

n  p=1



t 0

   t  ∂m m ∂ϕ ϕ , × m ds , E 0 ∂xp ∂xp

   !  h (M M k,h )+ , ϕ ds I h M k,h × M k,h × Δ r

n t    m ∂(m m × ϕ) ∂m = E , × m ds . ∂xp 0 ∂xp p=1

160

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Proof. Step 1: Assertion (i). Consider first the case r = 2, and fix ϕ ∈ Mk,h }k,h is a piecewise affine interpolation C2 (O, R3 ). Since the sequence {M M j }, we may infer from the definition (2.86) of in time of Vh -valued iterates {M M k,h }k,h that the sequence {M M k,h }k,h is C [0, T ]; Vh -valued. the sequence {M Using the definition (2.8), equality a × b, c = −b, a × c which holds for h , we obtain all a, b, c ∈ R3 , and the definition (2.11) of the Laplacian Δ     h (M h (M M k,h )+ , ϕ M k,h )+ M k,h × Δ = − M k,h × ϕ , Δ h

h

n    ∂ ∂ ϕ × M k,h ], M k,h )+ . = I h [ϕ (M ∂xp ∂xp p=1

In order to control the effects due to interpolation in the last term, we benefit from the use of piecewise finite elements, satisfying ∇2M k,h |K = 0 for all K ∈ Th . Consequently, by using the standard interpolation estimate (2.6) for every K ∈ Th , and putting things together again, we get  ∂   ∂  M k,h )+ , I h − Id] ϕ × M k,h  (M [I  ∂xp ∂xp  ∂M  M k,h ϕ ∂ ∂ϕ ϕW2,2 , ≤ Ch M k,h L2  L2  L2 + ϕ ∂xp ∂xp ∂xp and after integration over the interval (0, t) and then taking the expectation, this term tends to zero for k, h → 0. Hence we need to show that the difference on the left-hand side below converges to zero. For this aim let us first notice that  ∂ϕ M k,h )+  ∂(M m  ∂ ∂m  ϕ m ϕ M E  × , [ϕ × k,h ], −  ∂xp ∂xp ∂xp ∂xp  ∂   ∂m m   m − M k,h ) , ≤ E  ϕ × (m  ∂xp ∂xp  ∂   ∂    M k,h )+  := I + II . +E  ϕ × M k,h , m − (M ∂xp ∂xp

We proceed separately with the terms I, II.   m | ∈ L2 Ω ; L∞ (0, T ; L2 ) . Hence we By Corollary 2.30, (i) we have | ∂m∂x(ω,·) p infer that  T  ∂ϕ m   ∂ ∂m  ϕ m−M M k,h )+ϕ ϕ× m−M M k,h ), I≤E ×(m (m  ds → 0 (k, h → 0) ,  ∂xp ∂xp ∂xp 0 while Corollary 2.30, (ii) is employed for the second part and Corollary 2.30, (i) for the first one.

161

Convergent discretization of SLLG

For the term II we easily obtain from Corollary 2.30, (i) and Lemma 2.22, (i) that E



T 

  ϕ ∂   ∂ϕ M k,h )+  ds → 0 m − (M × M k,h ,  ∂xp ∂xp

0

(k, h → 0) .

M k,h ∂  ∂M   M k,h )+ . The crucial term in II then comes from ϕ × m − (M , ∂xp ∂xp By using the identity M k,h )+ − m ] + m + M k,h = [(M

 1 M k,h )− − (M M k,h )+ , (M 2

(2.95)

the term corresponding to the first difference on the right-hand side vanishes, thanks to a × b, a = 0, and Corollary 2.30, (i). For the term corresponding to the second contribution in (2.95), we may conclude that E





T 

  m ∂ ∂m   M k,h )+ − m ]  ds → 0 , [(M  ϕ× ∂xp ∂xp

0

(k, h → 0) ,

because of Corollary 2.30, (i), and Lemma 2.22, (iii)3 . It remains to show that 1  E lim (k,h)→0 2



T  0

   ∂ ∂    M k,h )− − (M M k,h )+ , m − (M M k,h )+ ]  ds = 0 . (M [m  ϕ× ∂xp ∂xp

By Lemma 2.22, (ii)1 , (iii)3 , and Corollary 2.30, (i), we obtain the following upper bound for the expression on the left-hand side above: 1/2   T   1 ϕL∞ E M k,h )+ 2L2 ds ∇ m − (M . . . ≤ ϕ 2 0   T 1/2   M k,h )− − (M M k,h )+ 2L2 ds × E ∇ (M , 0

which converges to 0 when (k, h) → 0 by Lemma 2.22, (iii)3 . Therefore it remains to consider the case r = 1, i.e. to show that (formally)     h (M m in L2 Ω ; L2 (0, T ; L2 ) . M )+  m × Δm (2.96) I h M × Δ k,h

k,h

For this purpose, we compute     h (M M k,h )+ , ϕ I h M k,h × Δ      h (M h (M M k,h × Δ M k,h )+ ], ϕ − I h [M M k,h × Δ M k,h )+ ], ϕ = I h [M h   +  Mk,h ) , ϕ =: A1 + A2 . + M k,h × Δh (M h

162

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Since in view of (2.10),

+  + + + h (M M k,h )+ +L2 +∇ϕ ϕ +L2 |A1 | ≤ Ch +I h M k,h × Δ + + + + h (M M k,h )+ +h +∇ϕ ϕ +L 2 , ≤ Ch +M k,h × Δ

and hence by the result for r = 2 we infer that (formally)     t   t h (M M k,h × Δ M k,h )+ ], ϕ ds = E m × Δm m, ϕ ) ds . lim E (m I h [M k,h→0

0

0

This concludes the proof of assertion (i) in Lemma 2.32. Step 2: Assertion (ii). Consider r = 2, and fix ϕ ∈ C2 (O, R3 ). Using the definition (2.8) of the scalar product (·, ·)h we get the following identity     h (M M k,h )+ , ϕ = M k,h × M k,h × Δ h   +  Mk,h × Δh (M Mk,h ) ], I h [M M k,h × ϕ ] = − I h [M   h (M M k,h × Δ M k,h )+ ], I h [M M k,h × ϕ ] + I h [M    h (M M k,h × Δ M k,h )+ ], I h [M M k,h × ϕ ] − I h [M h

=: B1 + B2 .

Because of (2.9) and (2.10), and a stability property of the Lagrange interpola tion operator on each K ∈ Th (use (2.6), Theorem 2.12, (ii), and ∇2M k,h K = 0 for all K ∈ Th ) we may continue as follows, + + + + h (M M k,h × Δ M k,h )+ ]+L2 +∇I M k,h × ϕ ]+L2 I h [M |B2 | ≤ Ch +I h [M (2.97)   + + h (M M k,h )+ +h ∇2ϕ L2 + ϕ ϕW1,∞ ∇M M k,h L2 . ≤ Ch +M k,h × Δ Hence, in view of Lemma 2.22, (ii)1 and (ii)2 , we infer that B2 → 0 as (k, h) → 0. Moreover,   h (M M k,h × Δ M k,h )+ ], [I I h − Id][M M k,h × ϕ ] B1 = I h [M   h (M M k,h × Δ M k,h )+ ], M k,h × ϕ ] + I h [M   h (M M k,h × Δ M k,h )+ ], ϕ . =: B1,1 − M k,h × I h [M Therefore, as in (2.97), we infer that h (M I h [M M × Δ M )+ ]L2 ∇[M M |B1,1 | ≤ Ch I k,h

k,h

k,h

× ϕ ]L2 → 0

(k, h → 0) .

Let now r = 1. Because of (2.96), and Corollary 2.30, (ii), we then have for (k, h) → 0 that (formally)   h (M M k,h × Δ M k,h )+ ]  m × (m m × Δm m) M k,h × I h [M in L2 Ω ; L2 (0, T ; L2 ) . This concludes the proof of assertion (ii) in Lemma 2.32.

163

Convergent discretization of SLLG

2.2.3

Existence of a solution to the SLLG equation

The aim of this section is to prove that the process m constructed in Section 2.2.2 is a weak martingale solution of problem (2.47). For every (k, h) > 0, we define the following F -martingale on the new probability space (Ω , F , P ), X k,h (t)

:=

t 0

  M k,h )− × dWk,h I h (M (s)

∀ t ∈ [0, T ] .

(2.98)

M k,h )− Let us recall Definition 2.4 to define the piecewise constant processes (M M k,h )+ for M k,h . Because of Lemma 2.29, we have for all t ∈ [0, T ], and (M M k,h )± = M ± (M k,h ◦ φk,h ,

M k,h = M k,h ◦ φk,h .

and

In a similar fashion we define the following F-martingale on the former probability space (Ω, F, P), Xk,h (t) :=

t 0

  I h M− k,h (s) × dW(s)

∀ t ∈ [0, T ] .

(2.99)

 Let us denote by Qk,h = Qk,h (t); t ∈ [0, T ] the quadratic variation process of the last martingale. Thus, for ψ 1 , ψ 2 ∈ L2 (O, R3 ) we have, 



ψ 1 , ψ 2 := Qk,h (t)ψ

t ∞ 0 =1

   − M− M q I h [M × e ], ψ [M × e ], ψ I ds 1 2 h

k,h k,h

∀ t ∈ [0, T ] . In Definition 2.4, (i) we defined the process M k,h by piecewise affine extension M j }Jj=0 . Another natural definition would be to use of the random variables {M integrals; cf. Figure 2.3. To be precise, in analogy with formula (2.69), one can #k,h by the following formula for all t ∈ [tj , tj+1 ), define a process M #k,h (t, ·) := M j + M =



t tj

j fk,h ds

+

j M j + fk,h (t − tj ) +

t

gjk,h dW(s)

tj ∞ 

gj k,h

(2.100)

 √  q β (t) − β (tj ) ,

=1

where we use the decomposition (2.70) to define the L2 -valued random vector

164

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

j fk,h , and the T2 (K, H)-valued random operator gjk,h by

  j hM j+1 ) − M j+1/2 × Δ hM j+1 I h αM M j+1/2 × Δ M j+1/2 × (M fk,h = −I +

∞    11  1 A j1 − k A j2 + A j4 , q I h M j+1/2 × e × e + 2 k 2

∞ 

gjk,h =

=1

  j gk,h = I h M j × e .

j gk,h ⊗ e ,

=1

We will also use the following notation, fk,h (s) :=

J−1 

1[tj ,tj+1 ) (s)fjk,h

∀ s ∈ [0, T ] ,

1[tj ,tj+1 ) (s)gjk,h

∀ s ∈ [0, T ] .

j=0

gk,h (s) :=

J−1  j=0

Note that in view of (2.100), and Definition 2.4, (i), #k,h (t) = M j+1 = M k,h (tj+1 ) lim M

t↑tj+1

(0 ≤ j ≤ J − 1) .

(2.101)

#k,h is a continuous process on [0, T ], P-a.s.. From (2.100) we find out Hence, M that for t ∈ [0, T ], #k,h (t) M

=

M0 +

t ∞ √ 0 =1

t   #− (s) × e dβ (s) + q I h M fk,h (s) ds k,h 0

:= M 0 + Xk,h (t) (2.102) t   − + #− (s), M #+ (s) ds , + F k,h Wk,h (s), Wk,h (s); M k,h k,h 0

− + where we define Wk,h (s) := Wk,h (tj ) if s ∈ [tj , tj+1 ), and similarly Wk,h (s). 1 1/2 0 1 Moreover, with M = 2 (M + M ), by (2.69) and (2.70),    h M1 ) − M1/2 × Δ  h M1 I h αM1/2 × (M1/2 × Δ F k,h W0 , W1 ; M0 , M1 ) := −I

   1 q I h M1/2 × e × e 2 ∞

+

=1 ∞

 1  |β1 − β0 |2 − k  1/2 I h (M × e ) × e q + 2 k

=1

(2.103)

165

Convergent discretization of SLLG

−

∞      √  h M1 ) + M1/2 × Δ  h M 1 × e q β1 − β0 I h αM1/2 × (M1/2 × Δ

=1

    Δj β m 1 Δ j β m 2 1  √ + I h (M1/2 × em1 ) × em2 . qm1 qm 2 2 k m1 =m2

Note that

∞ √ t  0

=1

  #− (s) × e dβ (s) is equal to the following Itô inteq I h M k,h

gral with respect to the process dW: t  −  # (s) × dW(s) . Ih M k,h 0

Using Definition 2.4, (i), equalities (2.80) and (2.69), and the definition of Aj }j is in (2.70), we obtain the following identity for M k,h at times t ∈ {A [tj , tj+1 ),      +  M (t) × Δ M (t), Φ (t), Φ + α(t − t ) − M k,h (t) − M − j k,h h k,h k,h h h    +  Mk,h (t) × ΔhM k,h (t)], Φ + M k,h (t) × [M h   t − tj = M k,h (t) × Δj W, Φ (2.104) k h   t − tj M− = k,h (t) × Δj W, Φ k h ∞       k Mk,h (t) × e ] × e , Φ + A j , Φ h . + q [M 2 h

=1

#k,h . The following result bounds the difference between processes M k,h and M Lemma 2.33. For each β ∈ (0, 12 ) there exists a random variable Kβ > 0 of finite exponential moment (i.e. E[exp(δKβ )] < ∞ for some δ > 0), such that for all pairs (k, h), #k,h (t) − M k,h (t)L∞ ≤ Kβ k β . sup M

t∈[0,T ]

Proof. We use (2.102) and (2.104) to deduce the following identity, #k,h (t) − M k,h (t) = M

∞ 

j gk,h Z k (t)

∀ t ∈ [tj , tj+1 ) ,

(2.105)

=1

where, for t ∈ [tj , tj+1 ), √  q β (t) − β (tj ) − Z k (t) = =

 t − tj  β (tj+1 ) − β (tj ) tj+1 − tj  t − t √  tj+1 − t √ √ √ j q β (t) − q β (tj ) − q β (tj+1 ) − q β (t) . k k

166

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

In other words, with Z k defined by, for t ∈ [tj , tj+1 ), Z k (t) = = one has Z k (t) =



 t − tj  W(tj+1 ) − W(tj ) tj+1 − tj  t−t   tj+1 − t  j W(t) − W(tj ) − W(tj+1 ) − W(t) , k k W(t) − W(tj ) −

∞ 

=1

Z k (t)e and identity (2.105) can be written as

#k,h (t) − M k,h (t) = gj Z k (t) M k,h

∀ t ∈ [tj , tj+1 ) .

(2.106)

The last equality can also be written in a more compact form #k,h (t) − M k,h (t) = M

J−1 

j 1[tj ,tj+1 ) (t)gk,h Z k (t)

∀ t ∈ [0, T ] .

(2.107)

j=0

Because of part (i) of Lemma 2.22 and estimate (2.7) there exists a constant C > 0 such that for almost all ω ∈ Ω, sup

j sup gk,h L ∞ ≤ C .

k,h>0 0≤j≤J

Since for any β ∈ (0, 12 ) the Wiener process W has K-valued Hölder continuous trajectories with constant β, and the C β ([0, T ]; K)-norm of W is exponentially integrable [44, Thm. 2.6], by Lemma 2.5 and equality (2.106) we infer that there exists an exponentially integrable random variable Kβ such that #k,h (t) − Mk,h (t)L∞ = sup M

t∈[0,T ]

≤C

sup

sup

j∈{0,...,J−1} t∈[tj ,tj+1 )

sup

sup

j∈{0,...,J−1} t∈[tj ,tj+1 )

Z k (t)K ≤ C Z

#k,h (t) − Mk,h (t)L∞ M

Kβ β k . 2β

Hence the result follows. #k,h is known for every finite (k, h) > It is from (2.100) that the continuity of M 0, P-a.s.; the following result asserts Hölder continuity of it for every α ∈ (0, 12 ), which follows from (2.105), and the corresponding property of the Wiener process. Lemma 2.34. If α ∈ (0, 12 ) and r ∈ (1, 4), then for each pair (k, h) > 0, P-a.s.   #k,h ∈ W α,r 0, T, L2 . M

167

Convergent discretization of SLLG

j, Proof. Let us fix α ∈ (0, 12 ) and r ∈ (1, 4). In particular, we have gk,h (ω) ∈ 2 3 L (O, R ) for all 0 ≤ j ≤ J, 1 ≤  < ∞, P-a.s.. From identity (2.107) and #k,h − M k,h is a Hölder continuous process for Lemma 2.5 we infer that M whole follows since every α ∈ (0, 12 ) on the   interval [0, T ]. The  result 2then  α 2 α,r 2 α,r C ([0, T ]; L ) ⊂ W 0, T ; L , and M k,h ∈ W 0, T ], L P-a.s., by Theorem 2.12.

 # Define now a family of processes M k,h k,h by an analog of the formula (2.86), i.e.,

# (t) := M #k,h (t) ◦ φk,h M k,h

∀ t ∈ [0, T ] .

(2.108)

It follows then from (2.98), (2.102), and Lemma 2.29 that for every t ∈ [0, T ], t t   + # (t) = M 0 + # M f (s) ds + I ) (s) × dW (s) ( M h k,h k,h k,h k,h 0 0 t   # )− (s), (M # )+ (s) ds = F k,h (Wk,h )− (s), (Wk,h )+ (s); (M k,h k,h 0

M 0 + X k,h (t) . +M

(2.109)

According to Lemma 2.33, there exists an exponentially integrable random variable Kα defined on the probability space P such that # (t) − M (t)L∞ ≤ K k α . sup M α k,h k,h

(2.110)

t∈[0,T ]

 # Let us now list some additional properties of the family M k,h k,h . 1 1 Corollary 2.35. If α ∈ (0, ), r ∈ (1, 4) satisfy α > , and κ ∈ [1, ∞), then 2 r (i) for all indices k, h,       # : Ω → Lκ 0, T ; Lκ ∩ W α,r 0, T ; L2 ∩ C [0, T ]; L2 , M k,h is a measurable map and #k,h ) = L(M # ) L(M k,h

on

      Lκ 0, T ; Lκ ∩ W α,r 0, T ; L2 ∩ C [0, T ]; L2 .

(ii) P -almost surely for k, h → 0     (2.111) Lκ 0, T ; Lκ ∩ C [0, T ]; L2 .   #k,h ∈ W α,r 0, T ; L2 ) P-a.s.. Hence, by the definition Proof. By Lemma 2.34, M # the property (i) is satisfied. of M k,h Property (ii) is a direct consequence of inequality (2.110) and of Lemma 2.29. # → m M k,h

in

168

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

In the sequel, let Fk,h (corr. F k,h ) denote the natural filtration generated by # ). # k,h (corr. M the process M k,h Lemma 2.36. For each pair (k, h) > 0, the process X k,h is an L2 -valued square integrable F k,h -martingale. Moreover, the quadratic variation process of the martingale X k,h is equal to a process Q k,h , which is defined for all times t ∈ [0, T ] and all ψ 1 , ψ 2 ∈ L2 (O, R3 ) via   ψ1, ψ 2 Qk,h (t)ψ (2.112) ∞    t  # )− × e ], ψ 1 I h [(M # )− × e ], ψ 2 ds . q I h [(M := k,h k,h 0

=1

Proof. We begin with a trivial observation that by definition (2.99) the process Xk,h is a continuous square integrable L2 -valued Fk,h -martingale. Let us fix s, t ∈ (0, T ] such that s < t, and ψ ∈ L2 . Let us choose arbitrary {si ; i = 1, · · · , K} such that 0 ≤ s1 ≤ s2 ≤ · · · ≤ sK ≤ s. Let {ϕi : L2 → R; i = 1, · · · , K}, be bounded continuous functions. Since Xk,h is a square integrable L2 -valued Fk,h -martingale, we infer that E



Xk,h (t) − Xk,h (s), ψ

K '

  # k,h (si ) = 0 . ϕi M

(2.113)

i=1

Let us recall that from equalities (2.102) and (2.109) we have that the following representations of the processes Xk,h and X k,h are valid for all t ∈ [0, T ], #k,h (t) − M 0 Xk,h (t) = M t   #− (s), M #+ (s) ds , − F k,h W− (s), W+ (s); M k,h k,h 0

# (t) − M 0 X k,h (t) = M k,h t   # )− (s), (M # )+ (s) ds . − F k,h (Wk,h )− (s), (Wk,h )+ (s); (M k,h k,h 0

Let us also recall, see Lemma 2.29 and Corollary 2.35, that the laws on C0 ([0, T ]; # k,h ) and (W , M # ) are equal. Since the map K) ×C([0, T ]; L2 ) of (W, M k,h k,h Λ = Λsi ,ϕi ,s,t,FF k,h defined by formula Λ : C0 ([0, T ]; K) × C([0, T ]; L2 )  (w, x) → (2.114) t     K F k,h w− (s), w+ (s); x− (s), x+ (s) ds ϕi (x(si )) i=1 , x(t) − x(s) − ∈ RK × L 2

s

169

Convergent discretization of SLLG

is we infer that the laws on RK × L2 of the random variables  continuous, K  # k,h (si ) , ϕi M i=1      # (si ) K , X (t) − X (s) are also equal. Xk,h (t) − Xk,h (s) and ϕi M k,h k,h k,h i=1 Thus, in view of equality (2.113) we infer that E





(X k,h (t)

−

X k,h (s), ψ )

K '

  # (si ) = 0 . ϕi M k,h

i=1

  This implies that E X k,h (t)−X k,h (s)F k,h (s) = 0, i.e. X k,h is a F k,h -martingale. It remains to show formula (2.112) for the quadratic variation of the martingale X k,h . Because of (2.99), the quadratic variation process Qk,h := Xk,h  of the martingale Xk,h satisfies, see e.g. [87] 



ψ1, ψ 2 Qk,h (t)ψ



:=

∞ 

q

=1

t  #k,h )− × e ], ψ 1 · I h [(M 0

  #k,h )− × e ], ψ 2 ds · I h [(M

∀ t ∈ [0, T ] .

In view of [87, Theorem 22.8], the process Xk,h ⊗ Xk,h − Xk,h  is a T1 (L2 )valued martingale. As in the first part of the proof, we fix numbers s, t ∈ (0, T ] such that s < t, functions ψ 1 , ψ 2 ∈ L2 , choose arbitrary {si ; i = 1, . . . , K} such that 0 ≤ s1 ≤ s2 ≤ · · · ≤ sK ≤ s and finally choose a collection of bounded continuous functions {ϕi : L2 → R; i = 1, . . . , K}. Thus we infer that E

K '

        # k,h (si ) ϕi M Xk,h (t), ψ 1 ψ 2 , Xk,h (t) − Xk,h (s), ψ 1 ψ 2 , Xk,h (s)

i=1

−

∞ 

=1

q

t    #k,h )− × e ], ψ 1 I h [(M #k,h )− × e ], ψ 2 dr = 0 . I h [(M s

(2.115) # k,h ) and (W , M # ) Since the laws on C0 ([0, T ]; K) × C([0, T ]; L2 ) of (W, M k,h k,h are the same, the maps Λsi ,ϕi ,s,t,FF k,h and Γ := Γs,t,FF k,h defined respectively by formula (2.114) and the formula below Γ : C([0, T ]; L2 )  x →

∞ 

=1

q

t   x− × e ], ψ 1 I h [x x− × e ], ψ 2 dr ∈ R, I h [x s

170

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

are continuous, the laws on RK × R of the following two random variables    # k,h (si ) K , (Xk,h (t), ψ 1 )(ψ ψ 2 , Xk,h (t)) ϕi M i=1  #k,h ) , ψ 2 , Xk,h (s)) − Γ(M − (Xk,h (s), ψ 1 )(ψ    # (si ) K , X (t), ψ 1 ψ ψ 2 , X k,h (t) ϕi M k,h k,h i=1  # ) ψ 2 , X k,h (s) − Γ(M − X k,h (s), ψ 1 ψ k,h are also the same. Thus, in view of equality (2.115) we infer that E

  # (si ) (X (t), ψ 1 )(ψ ψ 2 , X k,h (t)) − (X k,h (s), ψ 1 )(ψ ψ 2 , X k,h (s)) ϕi M k,h k,h

K ' i=1

−

∞ 

q

t    # )− × e ], ψ 1 I h [(M # )− × e ], ψ 2 dr = 0 . I h [(M k,h k,h s

=1

This proves that the process X k,h ⊗ X k,h − Q k,h , where the process Q k,h is defined by the formula Q k,h (t)

=

∞ 

=1

q

t 0

# )− (s) × e ] ⊗ I h [(M # )− (s) × e ] ds I h [(M k,h k,h

∀t ≥ 0,

is a T1 (L2 )-valued F k,h -martingale. Since the T1 (L2 )-valued process Q k,h is continuous, increasing and F k,h -adapted, we infer, see for instance [87, Theorem 22.8], that it is equal to the quadratic variation process of the martingale X k,h . The proof is now complete. Next, we show convergence of the quadratic variation process for (k, h) → 0. Lemma 2.37. Let t ∈ [0, T ], then for all ψ 1 , ψ 2 ∈ L2 (O, R3 ), P -a.s.     ψ 1 , ψ 2 → Q (t)ψ ψ1, ψ 2 Qk,h (t)ψ (k, h → 0) , where, for t ∈ [0, T ] and ψ 1 , ψ 2 ∈ L2 (O, R3 ), 





ψ1, ψ 2 = Q (t)ψ

t ∞ 0 =1

   q m (s) × e , ψ 1 m (s) × e , ψ 2 ds .

(2.116)

Proof. It is enough to apply the convergence result (2.111) from Corollary 2.35, in conjunction with the Lebesgue dominated convergence theorem.

171

Convergent discretization of SLLG

Let m be the process in Subsection

constructed   2.2.2 and Corollary 2.35. Note # that both sequences M k,h k,h and M k,h k,h converge to the same process m in an appropriate sense (compare Corollary 2.35 and Lemma 2.29). Recall that the convergence results in Lemma 2.32 identify limits of the deterministic M k,h }k,h for (k, h) → 0; it is because of (2.101), integrals for the sequence {M and Definition 2.4 that #k,h , M k,h = M j+1/2 = M

#k,h )+ Mk,h )+ = M j+1 = (M and (M

∀ t ∈ [tj , tj+1 ) ,

 # so that Lemma 2.32 applies to the sequence M k,h k,h as well. In other words, if the processes Y, Z are defined in (2.93), (2.94), then for (k, h) → 0, we have   # ×Δ # )+  Y h (M Ih M k,h k,h   # × Δ # )+ )  Z h (M I h Mk,h × (M k,h k,h

  in L2 Ω ; L2 (0, T ; L2 ) ,   in L2 Ω ; L2 (0, T ; L2 ) .

by the random variables For t ∈ [0, T ], we denote by Ft the σ-field generated  m(s); s ≤ t}. Denote by F := {F t ; t ∈ [0, T ] the natural augmentation of {m the filtration F := {Ft ; t ∈ [0, T ]}. Define the following L2 -valued process X which is a natural candidate for the martingale part of the process m , t   X (t) := m (t) − m 0 − F m (s) ds ∀ t ∈ [0, T ] , (2.117) 0

where ∞    1  m) := −α m × m × Δm m + m × Δm m + F (m q m × e × e . 2

(2.118)

=1

Note that by Lemma 2.32 we have with the above processes Y and Z that ∞

m) = −α Z + Y + F (m

 1  q m × e × e . 2

(2.119)

=1

The two auxiliary Lemmata 2.38 and 2.39 are needed below: together with the martingale representation theorem, they ensure the existence of a K-valued Q-Wiener process on some extended probability space, such that the process X is an Itô integral with respect to that new Wiener process of the process m (s)×· — appearing in formula (2.116). It is important in both these lemmata that F is the natural augmentation of the filtration generated by the process m. Lemma 2.38. The process X defined by the formula (2.117) is a L2 -valued square integrable F-martingale.

172

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Proof. As in [111], in view of [51, p. 75] it is enough to show that the process X is an F-martingale. Let us fix t, s ∈ [0, T ] such that s ≤ t, and n ∈ N. We have to show that for any partition 0 ≤ s1 < s2 < . . . sn ≤ s, and any bounded and continuous functions {hi : L2 → R; i = 1, . . . , n}, as well as any ϕ ∈ L2 (O, R3 ) the following equality holds n   '  E X (t) − X (s), ϕ hi m (si ) = 0 .

(2.120)

i=1

First of all, by (2.117) and (2.119) we see that E



X (t) − X (s), ϕ

n '

  hi m (si )

i=1

t n '      F m (r) dr, ϕ hi m (si ) = E m (t) − m (s) − s

i=1

t t  = E m (t) − m (s) + α Z(r) dr − Y(r) dr s

1 − 2

(2.121)

s

t n ∞ '     q m (r) × e × e dr, ϕ hi m (si ) . s =1

i=1

Next we infer from Lemma 2.32 and (2.88) (since α > 1r ) that E

n  '   hi m (si ) m (t) − m (s), ϕ i=1

n  '    # (s), ϕ # (t) − M m (s h ) , = lim E M i i k,h k,h h,k→0

E



t

Z(r) dr, ϕ s

= lim E h,k→0

E



n '

i=1

  hi m (si )

i=1



t s



h

n  '      + # # #  dr, ϕ I h M k,h × M k,h × Δh (M k,h ) hi m (si ) , h

t

Y(r) dr, ϕ s

= lim E h,k→0



n '

  hi m (si )

i=1  t

 s

i=1

n   '   # × Δ # )+ dr, ϕ h (M Ih M hi m(si ) . k,h k,h h

i=1

173

Convergent discretization of SLLG

Finally, by (2.93) we have E



∞  t  s =1

n '     q m (r) × e × e dr, ϕ hi m (si )

= lim E k,h→0

= lim E k,h→0

∞  t  s =1 ∞  t  s =1

i=1

n   '    # q M k,h × e × e , ϕ dr hi m (si ) h

i=1

n   '      # × e × e , ϕ dr m (s q I h M h , ·) . i i k,h h

i=1

Now, we use (2.104), in combination with Lemma 2.23 to conclude (2.120). Hence the proof of Lemma 2.38 is complete. We apply Theorem 2.6 to show the following second auxiliary lemma. Lemma 2.39. The quadratic variation of the continuous L2 -valued square integrable F-martingale X defined by the formula (2.117) is equal to the process Q defined by (2.116). Proof. Lemma 2.39 is a consequence of Theorem 2.6, with E = H = L2 and pro# }k,h and {X }k,h on the probability space P defined respectively cesses {M k,h k,h in the formulae (2.108) and (2.98). Below we will show that all assumptions of Theorem 2.6 are satisfied. Let us notice that assumption (2.30) is satisfied in view of Lemma 2.37. Moreover, assumption (2.24) is a consequence of Corollary 2.35. To show that assumption (2.25) is satisfied let us recall that by formula (2.109) the process {X k,h } has the following representation # (t) − M 0 − Y (t) X k,h (t) = M k,h k,h where Yk,h (t) :=



t 0

(t ≥ 0) ,

  # )− , (M # )+ ds F k,h (Wk,h )− , (Wk,h )+ ; (M k,h k,h

∀t ≥ 0.

  # } converges in C [0, T ]; L2 P -a.s. by part (ii) of Since the sequence {M k,h }. Define Corollary 2.35, we only have to deal with the sequence {Yk,h Yk,h (t) :=

t 0

  #k,h )− , (M #k,h )+ ds F k,h (Wk,h )− , (Wk,h )+ ; (M

∀t ≥ 0.

By the estimates in part (ii) of Lemma 2.22, it is sufficient to show that   sup E Yk,h 2W 1,2 (0,T ;L2 ) < ∞ . (2.122) k,h>0

174

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

First, by elementary calculations we have   E Yk,h 2W 1,2 (0,T ;L2 ) T     #k,h )− , (M #k,h )+ 2 2 ds . F k,h (Wk,h )− , (Wk,h )+ ; (M ≤ C(T ) E F L 0

Now by definition (2.103) the term F k,h is equal to the sum of five terms, which we will denote (A), (B), . . ., (E) and estimate separately. By (2.5), (2.9), and part (i) of Theorem 2.11 we have T

J    hM j+1 2 ≤ C(α, T ) , (2.123) M j+1/2 × Δ E (A)2L2 ds ≤ C(α2 + 1)k M h j=0

0

where the last estimate follows from part (ii)2 of Theorem 2.11. For the second term, the part (i) of Theorem 2.11 and (2.5), (2.9) imply that T

  E (B)2L2 ds ≤ CT sup e 2L∞ (Tr Q)2 .

(2.124)

0

The third and fifth term can be analyzed similarly. It remains to consider the fourth term. By the Cauchy-Schwarz inequality, (2.5), (2.9), and part (i) of Theorem 2.11 we have T

J ∞ +      √  + E (D)2L2 ds = k E + q β (tj+1 ) − β (tj )

0

Ih ×I



j=0

=1

 +2  hM j+1 ) + M j+1/2 × Δ hM j+1 × e + M j+1/2 × Δ M j+1/2 × (M αM + 2

J ∞   2  √  1 E q 1 β (tj+1 ) − β 1 (tj ) ≤ Ck

1 =1

j=0

×E

∞  √

+ + hM j+1 ) M j+1/2 × Δ M j+1/2 × (M q 2 + αM

2 =1

 +2  hM j+1 × e + M j+1/2 × Δ +M 2 + h

≤ Ck 2

J 

1

hM j+1 2 M j+1/2 × Δ (Tr Q 2 )2 (α2 + 1) sup e 2L∞ M h

j=0 1 2

≤ C(α, Tr Q , T )k ,

L

175

Convergent discretization of SLLG

where the last estimate follows from part (ii)2 of Theorem 2.11. That concludes the proof of the estimate (2.122). Let us recall that [96, Proposition B.2] the embedding W 1,2 (0, T ; L2 ) ⊂ C([0, T ]; Hw ) is compact, where Hw denotes the L2 space endowed with the weak topology. Consequently, the bound (2.122) implies that the sequence of probability laws {L(Yk,h )} is tight on C([0, T ]; Hw ). Since the space C([0, T ]; Hw ) is not metrizable, we cannot apply directly the Prohorov theorem (i.e. Theorem 2.2). To overcome this difficulty, we embed C([0, T ]; Hw ) into RN . Let Q be a countable set consisting of all elements of a certain fixed orthonormal basis of H, and let us denote by {(sj , ej )}∞ j=1 a sequence consisting of all elements of the set (Q ∩ [0, T ]) × Q. Define next a continuous map π : C([0, T ]; Hw ) → RN

∞ by the formula π(U) := (U(sj ), ej )H j=1 . Since the space RN is metrizable   and the sequence of laws {L π(Yk,h ) } is tight on RN , by the Prohorov theorem (Theorem 2.2), there exists a subsequence {π(Ykn ,hn )}∞ n=1 such that kn + hn converge to zero and the sequence {L(π(Ykn ,hn ))}∞ n=1 converges weakly to a N certain Borel probability measure μ on R . It follows from Lemma 2.29 that the sequence π ◦ Yk,h ◦ φk,h converges in RN P -a.s.. Let us recall that Yk,h = Yk,h ◦φk,h . We have shown that P -a.s. for all s ∈ Q∩[0, T ] the sequence Yk,h (s) 

(s) k,h for s ∈ Q∩[0, T ] converges weakly in H. The limit of the sequence Yk,h is  identified  by Lemma 2.32. Thus we can conclude that P -a.s. the sequence X k,h (s) converges to X (s) in Hw for all s ∈ Q ∩ [0, T ]. In order to (k,h)   , extend the convergence to all t ∈ [0, T ], it is enough to notice that X k,h (k,h)   X are martingales which are uniformly bounded in L2 Ω , C([0, T ], L2 ) , and apply the dominated convergence theorem for conditional expectations. Hence the proof of assumption (2.25) is finished. Because of part (i) of Lemma 2.22, there exists a constant C > 0 such j that for almost all ω ∈ Ω, supk,h, supj∈{0,...,J} gk,h L2 ≤ C. Hence, in view of definition (2.112), the assumption (2.32) is satisfied. Finally, assumption (2.31) is satisfied in view of (2.108) and Theorem 2.12. The remaining arguments which are needed to show the validity of all assumptions of Theorem 2.11 are now standard. According to the martingale representation Theorem 2.4, we may now enlarge the probability space which is given by theSkorokhod theorem 2.3 (as well as the filtration F ) so that on  a new P := Ω , F , F , P there exists a K-valued F -adapted Q-Wiener process W such that

X (t) =



t 0

 m (r) × dW (r)

∀ t ∈ [0, T ] .

176

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

This, together with (2.117) and (2.118) implies that for all t ∈ [0, T ], P -almost surely, identity (2.51) holds. The proof of Theorem 2.11 is now complete.

2.2.4

A convergent discretization of the SLLG equation which uses random walks

The goal of this section is to show the convergence of Scheme 2.10, where the increments of the Wiener process are replaced by general, not necessarily Gaussian, random variables {ξξ j }Jj=0 ; cf. Definition 2.14. In what follows, we will show that the solutions to Scheme 2.10 have the same stability properties as those from Scheme 2.9. Then we will be able to identify the limits of the deterministic integrals exactly as we have done in Subsection 2.2.2. Finally, we use a martingale embedding theorem [52, Theorem 1.6] which will allow us to embed the approximation of the diffusion term by Scheme 2.10 (which is a time discrete martingale) into a time continuous martingale, and identify the limit of the stochastic integral. The following definition is taken from [52, p. 80]. Definition 2.40. A probability space (Ω, F, P) is called an extension of a probability space (Ω, F, P), if there exists a map π : Ω → Ω such that π −1 (F) ⊂ −1 F, and

P = P ◦ π  . If F = {Ft ; t ∈ [0, T ]} is a filtration−1on Ω, a filtration F = F t ; t ∈ [0, T ] on Ω is called an extension  of F, if π (Ft ) ⊂ F t for all −1 t ∈ [0, T ]. For the extension π (Ft ); t ∈ [0, T ] we will use the same notation F. Similarly, if X is a random variable on Ω, the extended random variable X ◦ π is usually again denoted by X. Definition 2.41. Assume that X is a random variable on (Ω, F, P), and that G is a sub-σ-algebra of F. Define the conditional variance of X given G by    2  Var X|G := E X − E [X|G] |G . (2.125) The next result is a generalization of [52, Theorem 1.6]. Theorem 2.13. Let T > 0, J ∈ N, and Ik = {tj }Jj=0 be an equi-distant partition of size k := TJ > 0. Assume that H is a separable Hilbert space, Fk := {Ftj }Jj=0 , that Pk := (Ω, F, Fk , P) is a filtered probability space, and M kj }Jj=0 is an H-valued square integrable Fk -martingale. Then there M k = {M exist an extended probability space (Ω, F, P) defined by the map π : Ω → Ω, a filtration F = {F t ; t ∈ [0, T ]} such that {F tj ; j ≥ 0} is an extension of Mk (t, ·); t ∈ {Ftj ; j ≥ 0}, and a continuous, square integrable F-martingale {M [0, T ]} on Ω such that (i) M k (tj ) = M kj ◦ π

(j = 0, . . . , J).

177

Convergent discretization of SLLG

(ii) For every 1 ≤ j ≤ J and any x, y ∈ H j−1       1  k  M kl+1 − M kl , x M kl+1 − M kl , y E M (tj )x, y |F = 3 H H H l=0

+

2 3

j−1 

      E M kl+1 − M kl , x M kl+1 − M kl , y Ftl . H

l=0

H

(2.126)

Moreover, there exists a constant c ∈ R such that, if in addition M k is an L4 -integrable Fk -martingale, then there exists a continuous, L4 -integrable Fmartingale M k satisfying (i) and (ii), and also for every x ∈ H   k   Var M (tj )x, x |F H

=

2 45 +

j−1  l=0

4 9

j−1  k 4 4   8   k M l+1 − M kl , x H + E M l+1 − M kl , x H Ftl 45 l=0

−c

j−1  

 2  2 E M kl+1 − M kl , x H Ftl

l=0

j−1   k 2  2   M l+1 − M kl , x H E M kl+1 − M kl , x H Ftl +c

−c

l=0 j−1 

 k   3   M l+1 − M kl , x H E M kl+1 − M kl , x H Ftl

(2.127) (j = 1, . . . , J) .

l=0

Furthermore, if for some p > 2   p M kj+1 − M kj 2p E M H ≤ C(p)k then

  p Mkt+k − M kt 2p E M H ≤ C (p)k

(j = 0, . . . , J − 1) , (t ∈ [0, T − k]) .

(2.128)

Proof. If H is one-dimensional then the parts (i), (ii) and formula (2.127) have been shown in [52, Theorem 1.6]. To show inequality (2.128) we consider two cases: (i) t = tj , for some j = 0, . . . , J − 1. The inequality (2.128) immediately follows from part (i) of the theorem. (ii)t ∈ (tj , tj+1 ), for some j = 0, . . . , J. We apply the triangle inequality to deduce that 1 1      2p 2p  2p k k Mkt+k − M kt 2p M E M ≤ E M − M  tj+1 H t+k H 1    2p Mktj+1 − M kt 2p + E M . (2.129) H

178

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

+ Since Mk is a F-martingale and the function  · 2p H : H → R is convex, we deduce that the first term in the sum (2.129) can be estimated as follows: +    +   +E M kt − M kt F t+k +2p Mkt+k − M ktj+1 2p E M = E H j+2 j+1 H    2p  k k Mtj+2 − M tj+1 H F t+k ≤ E E M   p Mktj+2 − M ktj+1 2p = E M H ≤ C(p)k .

Similarly, we can estimate the second term on the right-hand side of (2.129) and the result follows. The general case of a separable Hilbert space H follows from the one-dimensional case by constructing a sequence of extensions of probability spaces to embed projections of M k to an increasing sequence of finite dimensional subspaces of H (spanned by the elements of some fixed orthonormal basis of H), and taking an inductive limit afterwards. Corollary 2.42. In the framework of Theorem 2.13 we have for all x ∈ H E

   2     M k (tj )x, x H − E M k (tj )x, x |F   4  2 ≤ ( + c) E M kl+1 − M kl , x H . 3 j−1 l=0

Proof. The result follows from formulae (2.125), (2.127) and elementary properties of the conditional expectation. Now we are able to show the following generalization of Theorem 2.11. We use again the notation from Section 1.1. Theorem 2.14. Let O ⊂ Rn , n ≤ 3 be a polyhedral bounded domain, J ∈ N and T = tJ > 0. For every finite (k, h) > 0, let Th be a regular triangulation of O, and Ik be an equi-distant partition of [0, T ]. Let {ξξ j }Jj=0 be a K-valued Q-random walk on a filtered probability space Pk , such that Q ∈ T1 (K) satisfies conditions (S1 ) and (S2 ) from page 135. Assume that m 0 ∈ W1,2 (O, R3 ) with M 0 (x )| = 1 for m0 | = 1 Lebesgue almost everywhere, and M 0 ∈ Vh such that |M |m 0 1,2 3 all  ∈ L, as well as M → m 0 in W (O, R ) for h → 0. Then there exists a

179

Convergent discretization of SLLG

  M j }Jj=0 ⊂ L2 Ω; W1,2 (O, R3 ) of Scheme 2.10 that satisfies solution {M (i) (ii)

M j (x )| = 1 |M for all  ∈ L , and all 1 ≤ j ≤ J , P-almost surely , J    j 2 hM j+1 2 ≤ CT , M L 2 + k M j+1/2 × Δ E sup ∇M M h

(iii) E

1≤j≤J

j=1

J−1  j=0

1 M j+1 − M j 4h + M j+1 − M j 2h + M M k  M j+1 − M j ]2L2 ≤ CT , +∇[M

(iv) for each 0 ≤ j ≤ J, the map M j : Ω → W1,2 (O, R3 ) is Ftj -measurable. Moreover, there exist a filtered probability space P = (Ω , F , F , P ), a K-valued Q-Wiener process W on it, an F -adapted process m : Ω → L2 (OT , R3 ) such that P , W , m is a weak martingale solution of problem (2.47), and a # : [0, T ] × O × Ω → R3 such that continuous process M k,h # (tj ) = M j M k,h

(j = 0, . . . , J) ,

and for any κ ∈ [1, ∞), α ∈ (0, 12 ), r ∈ (1, 4), P -almost surely # → m M k,h

in

      Lκ 0, T ; Lκ ∩W α,r 0, T ; (W1,2 )∗ ∩C [0, T ]; L2

(k, h → 0) .

We will need the following reformulation of Scheme 2.10. Let {e }∞

=1 be an orthonormal basis of K such that Qe = q e ( ∈ N); see p. 117. Now we apply Theorem 2.13 to embed the K-valued Q-random walk {ξξ j }Jj=0 into a continuous square integrable martingale denoted here by V k on a filtered probability space Pk . Let us note that Scheme 2.10 can be rewritten in terms of V k as follows: Scheme 2.15. Let V k be a K-valued continuous square integrable martingale M 0 (x )| = 1 for on a filtered probability space Pk , and M 0 ∈ Vh be such that |M k k V k (tj ) = ξ j ◦π, all x ∈ Eh . For every j ≥ 0, and increments Δj V := V (tj+1 )−V j+1 such that Pdetermine a Vh -valued Ftj+1 -measurable random variable M almost surely holds for all Φ ∈ Vh   hM j+1 ], Φ M j+1 − M j , Φ )h + αk M j+1/2 × [M M j+1/2 × Δ (M (2.130) h    j+1/2  hM j+1 , Φ = M j+1/2 × Δj V k , Φ . ×Δ −k M h h Proof of Theorem 2.14. The estimates shown in Lemmata 2.22 and 2.24 can be deduced in the same manner as before — by simply replacing everywhere Δj W by Δj V k . Hence we can define piecewise affine processes M k,h and show that

180

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

the estimates from Theorem 2.12 hold in the present case as well. Consequently, Mk,h } on the space wecan show tightness of thelaws of the processes {M  the α,r  2 κ 1,2 ∗ 2 0, T ; (W ) ∩ C([0, T ]; L ) as in Lemma 2.26. Hence, L 0, T ; L ∩ W we can apply the Skorokhod representation Theorem 2.3, and may identify the limits of the deterministic integrals as in Lemma 2.32. It only remains to

k,h of identify the stochastic integral. For this purpose, we define the analog X the process Xk,h defined in Subsection 2.2.3 as follows: (i) Set for all j = 0, . . . , J

k,h (tj ) = X



  l I h M− k,h (tl ) × ξ .

(2.131)

l 1 we have 8 (t) − M (t)p ∞ sup M k,h k,h L

t∈[0,T ]

≤

sup

sup

j=0,...,J−1 t∈[tj ,tj+1 )

(t) − X

(tj )p ∞ = . . . X k,h k,h L

is an F -martingale, by invoking the Jensen inequality Furthermore, since X k,h k,h we have  +  +p  + +

. . .= sup sup +E X k,h (tj+1 ) − Xk,h (tj )Fk,h (t) + ∞ j=0,...,J−1 t∈[tj ,tj+1 )

≤

sup

sup

L

 

(tj+1 ) − X

(tj )p ∞ F (t) E X k,h k,h k,h L 

j=0,...,J−1 t∈[tj ,tj+1 )

≤ sup

sup

t∈[0,T ] j=0,...,J−1

  

(tj+1 ) − X

(tj )p ∞ F (t) E X k,h k,h k,h L

 $ %     

(tj+1 ) − X

(tj )p ∞ F (t)  . ≤ sup E sup X k,h k,h k,h L  j=0,...,J−1 t∈[0,T ]  Thus, we have shown that 8 (t) − M (t)p ∞ sup M k,h k,h L

t∈[0,T ]

 $ %      p 



≤ sup E sup Xk,h (tj+1 ) − Xk,h (tj )L∞ Fk,h (t)  .  j=0,...,J−1 t∈[0,T ] 

Taking the expectation and applying the Doob martingale inequality then leads to % $   p p

(tj+1 ) − X

(tj ) ∞ . 8 (t) − M (t) ∞ ≤ E E sup M sup X k,h

t∈[0,T ]

L

k,h

j=0,...,J−1

k,h

k,h

L

Now we can use identity (2.131), and the estimates from Lemmata 2.22 and 2.24 to conclude the proof of the analog of Lemma 2.33. The rest of the proof of Theorem 2.14 is similar to the proof of Theorem 2.11. The only difference is in the proof of the convergence of the quadratic variation

as (k, h) → 0, i.e. in the proof of Lemma 2.37. of the martingale X k,h 2 Let ψ 1 , ψ 2 ∈ L . To obtain a corresponding result, we need to show the ψ 1 , ψ 2 for pointwise in time convergence of the quadratic variation Q k,h (t)ψ 2

to the quadratic variation ψ 1 , ψ 2 ∈ L of the martingale X k,h







ψ1, ψ 2 = Q (t)ψ

t  ∞ 0

=1

   q m (s) × e , ψ 1 m (s) × e , ψ 2 ds ,

182

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

of the limiting process X as (k, h) → 0. The proof of this result will consist of the following three steps. Step 1: Reduction of the convergence on the whole interval [0, T ] to the convergence at nodal points of the partitions {Ik ; k > 0}. The formulae (2.126) and (2.127) allow us to calculate the conditional expectation and the conditional variance of the quadratic variation only at the nodal points of the partition Ik . Thus we need to find a way to reduce the convergence of the quadratic variation on the whole interval to the convergence at the nodal points. Notice that the quadratic variation Q k,h is a non-decreasing process by definition. Therefore, we have the following lemma: Lemma 2.43. Let {Ik }k>0 be an equi-distant partition of [0, T ] with mesh size k, t1k , t2k ∈ Ik , and t1k ≤ t ≤ t2k . Then for k, h > 0 we have       ψ 1 , ψ 2 ≤ Q k,h (t)ψ ψ 1 , ψ 2 ≤ Q k,h (t2k )ψ ψ1, ψ2 Qk,h (t1k )ψ ∀ ψ 1 , ψ 2 ∈ L2 . (2.134) Assume that t1k ↑ t, and t2k ↓ t as k → 0 in the Lemma 2.43. Consequently, it is enough to show the convergence of estimates from above and below in the formula (2.134) to the formula (2.116). Step 2: Approximation of the quadratic variation of the weak martingale solution of the SLLG equation by the conditional expectation of the quadratic variation of the solution of Scheme 2.15. For this aim let us define Gk,h :=    T  

σ {X (tl ); l = 0, . . . , } and fix j = t . By the identity (2.126) from k,h

k

k

Theorem 2.13 we deduce that    ψ 1 , ψ 2 Gk,h E Q k,h (tj )ψ =

j−1 ∞        1  √ 8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × em , ψ 2 qn qm ξ l,n ξ l,m I h M 3 l=0 n,m=0 j−1 ∞

       2  8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × en , ψ 2 . kqn I h M + 3 l=0 n=0

Hence, by elementary calculations we infer that     ψ 1 , ψ 2 Gk,h E E Q k,h (tj )ψ −

j−1  ∞  l=0 n=0

      2  8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × en , ψ 2 kqn I h M

183

Convergent discretization of SLLG j−1     1    l,n l,m √ 8k,h (tl ) × en , ψ 1 · = E ξ ξ qn qm I h M 9 l=0 n=m     8k,h (tl ) × em , ψ 2 · Ih M

+

j−1  ∞ 

      2  8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × en , ψ 2 (|ξ l,n |2 − k)qn I h M

l=0 n=0

≤ ... Furthermore, by the triangle inequality we may continue ≤

j−1     4    l,n l,m √ 8k,h (tl ) × en , ψ 1 · E ξ ξ qn qm I h M 9 l=0 n=m  2    8k,h (tl ) × em , ψ 2 · Ih M

    4    l,n 2 8k,h (tl ) × en , ψ 1 · + E (|ξ | − k)qn I h M 9 l=0 n=0   2    8k,h (tl ) × en , ψ 2 · Ih M j−1 ∞

= (A) + (B) .

(2.135)

Now we will show that the term (B) in (2.135) is of order O(k). We have j−1 ∞     4   8k,h (tl ) × en , ψ 1 (|ξ l,n |2 − k)(|ξ l,m |2 − k)qn qm I h M (B) = E 9 l=0 n,m=0           8k,h (tl ) × en , ψ 2 I h M 8k,h (tl ) × em , ψ 1 I h M 8k,h (tl ) × em , ψ 2 . · Ih M

Consequently, the tower property and the in   expectation,

of the conditional dependence of random variables ξ l,n ; l = 1 . . . , Tk , n ∈ N imply that in the last sum the only non-zero terms are for n = m. Therefore, j−1 ∞     4   l,n 2 8k,h (tl , ·) × en , ψ 1 2 · (B) = E (|ξ | − k)2 qn2 I h M 9 l=0 n=0      8k,h (tl ) × en , ψ 2 2 . · Ih M

Since 0 ≤ j ≤ conclude that

T k,

and since iterates of Scheme 2.15 are S2 -valued, we can (B) ≤ C

T 2 ψ 1 L2 ψ ψ 2 L2 . k Tr Q2 ψ k

184

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

We infer that the second term (B) in the sum (2.135) converges to zero as k → 0. The term (A) can be treated similarly. The corresponding calculations are left to the reader. Summing up, we have shown that for k → 0,     ψ 1 , ψ 2 Gk,h E E Q k,h (tj )ψ −

j−1  ∞ 

       2  8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × en , ψ 2 kqn I h M → 0.

l=0 n=0

On the other hand, the term j−1  ∞ 

       8k,h (tl ) × en , ψ 1 I h M 8k,h (tl ) × en , ψ 2 kqn I h M

l=0 n=0

is P-a.s. convergent to the quadratic variation t  ∞ 0

   q m (s) × e , ψ 1 m (s) × e , ψ 2 ds

=1

8k,h converges to m in C([0, T ], L2 ) of the limiting martingale X . Indeed, P-a.s. M as (k, h) → 0. Finally, we can conclude   that  there exists  a subsequence (k, h) t  ψ 1 , ψ 2 Gk,h P-a.s. converges to convergent to zero such that E Qk,h ( k k)ψ   ψ 1 , ψ 2 for all t ∈ [0, T]. the quadratic variation Q (t)ψ    ψ 1 , ψ 2 is apIt remains to show that the quadratic variation Q k,h ( kt k)ψ      ψ 1 , ψ 2 Gk,h when k proximated by its conditional expectation E Q k,h ( kt k)ψ converges to zero. This is the purpose of the following step.

with Step 3: Approximation of the quadratic variation of the martingale X k,h its conditional expectation. We apply Corollary 2.42 to deduce that     2  t t ψ 1 , ψ 2 − E Q k,h ( ψ 1 , ψ 2 Gk,h k)ψ k)ψ E Q k,h ( k k t ] [ 4 ∞ k '      √ 8k,h (tl ) × en , ψ 1 . (2.136) E qni ξ l,ni I h M ≤C i l=0 n1 ,n2 ,n3 ,n4 =1

i=1

Since the iterates of Scheme 2.15 are S2 -valued, we can estimate each term in the sum on the right-hand side of formula (2.136) as follows, E

4 ' √ i=1

4 '     √ 8k,h (tl ) × en , ψ 1 ≤ Ck 2 ψ 1 L 2 . qni ξ l,ni I h M qni ψ i i=1

(2.137)

185

Convergent discretization of SLLG

By combining (2.136) with (2.137) we infer that E



    2  t t ψ 1 , ψ 2 − E Q k,h ( ψ 1 , ψ 2 Gk,h Q k,h ( k)ψ k)ψ k k   1 4 ψ 1 L2 Tr Q 2 k , ≤ CT ψ

and the proof of Step 3 is finished. Hence, the proof of convergence of the

is complete. quadratic variation of the martingale X k,h Remark 2.44. A natural question which comes out of the proof of Theorem 2.14 is why it is necessary to use Theorem 2.13 twice: would it not be more natural to use the continuous extension V k of the K-valued Q-random walk

j J

k,h ξ j=0 to define the process X

k,h := X



t 0

  V k (s) I h M− k,h (s) × dV

∀ t ∈ [0, T ] ,

(2.138)

and then follow the proof as above? A problem in this approach appears when

k,h (defined by we try to show that the quadratic variation of the process X formula (2.138)) 

ψ1, ψ2 Q k,h (t)ψ



:=

∞ 

=1

q

t 0

 # )− × e ], ψ 1 · I h [(M k,h

    # )− × e ], ψ 2 d V · I h [(M k,h k,h, (s) ∀ t ∈ [0, T ] ∀ ψ 1 , ψ 2 ∈ L2 ,

(2.139)

V k,h , e )H ,  ∈ N) converges to the quadratic variation of the (where V k,h, := (V weak martingale solution of the SLLG equation given by the formula (2.116). The integral in the formula (2.139) should be understood as a Young integral (see Subsection 2.1.3). Consequently, we would need to use Corollary 2.13 of the Love-Young inequality (Theorem 2.1) to show such a convergence result. Unfortunately, we have  unable to show that the rate of convergence of the  been quadratic variation V k,h is fast enough to apply Corollary 2.13. The best available estimate for the numbers p and q from the Love-Young inequality was that they both are very close to 2 from above, and consequently the condition 1 1 p + q > 1 is not satisfied. It remains an open problem to realize this approach. Remark 2.45. Another approach to the proof of Theorem 2.14 is to work with discrete martingales altogether. It has been realized for the stochastic Navier-Stokes equation in the proof of convergence of Scheme 2 in [26].

186

2.3

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Computational studies

In Chapter 1 we focused on dynamic and asymptotic properties of ferromagnetic systems with a finite number of spins. In the following we will perform computational studies of the infinite-dimensional problem (2.1) using the discrete approximation via Scheme 2.9: for every fixed mesh size h > 0, this scheme approximates the N -spin system (1.1) with N = O(h−n ) for O ⊂ Rn , and the theoretical results from Chapter 1 are applicable. However, the analytical results from Chapter 1 only hold for a finite number of spins, and a corresponding generalization for h → 0 is not clear. Therefore, it is not known whether these results remain valid for the weak martingale solutions from Definition 2.20, that are constructed as a limit of the discrete iterates of Scheme 2.9 or Scheme 2.10 for k, h → 0. In addition, the convergence of Scheme 2.9 is only shown for sufficiently smooth noise, while it is the space-time white noise that is often relevant in applications. The subsequent computational experiments aim to provide insight into the features of the infinite dimensional model that are not known theoretically so far.

2.3.1

Numerical implementation

Below we discuss the numerical approximation of the space-time white noise which we employ in the subsequent numerical experiments. Next, we propose a space-time adaptive algorithm that allows us to reliably resolve small spatio-temporal scales which arise during the formation of singularities. The space-time adaptivity also improves the accuracy of the the space-time white noise approximation.

Discretization of the space-time white noise Discretization strategies for space-time white noise have been proposed in [2, 19, 11]. In the computations below we adopt the discretization approach from [11]. The space-time white noise is approximated as (0 ≤ j ≤ J − 1) # Δj W(x) =



∈L

ϕ (x) = Δj W (n + 1)−1 |supp ϕ |

∀x ∈ O,

(2.140)

 h , and Δj W = where ϕ (xm ) = δ m determines the nodal basis function ϕ ∈ V

3 (Δj β1 , Δj β2 , Δj β3 ) for  ∈ L are independent R -valued Brownian increments. To generate discrete realizations of Brownian increments we employ random number generators from the GNU Scientific Library [58]. The expected values of computed quantities are computed using the well-known Monte Carlo method, see for instance [35].

Section 2.3

Computational studies

187

Note that the spatial correlation for the noise approximation (2.140) of nodes x , xm ∈ Eh of the finite-element mesh Th is given by     # m ), Δj W(x # ) = kδm (n + 1)|supp ϕ |−1 = O kδm h−n , E Δj W(x where δm denotes the Kronecker delta, while the space-time correlation of the white noise is given by the Dirac delta distribution. In the numerical experiments with the space-time white noise we replace the multiplicative noise term in (2.48) by   # Φ , ν M j+1/2 × Δj W, h where ν ≥ 0 is the noise intensity.

Adaptive mesh refinement and time-step control The simulations in Section 2.3.3 below are performed using the following spacetime adaptive algorithm. Algorithm 2.46. Start with a uniform mesh Th0 with mesh size h = h0 , choose the initial time-step k 0 > 0, constants c1 > c2 > 0, and set j = 0, M 0 = m 0 . (i) Compute M j+1 using Scheme 2.10 with noise (2.140) for Th ≡ Thj , k ≡ k j and record the number of iterations m∗j of the Newton nonlinear solver which are required to compute M j+1 up to a prescribed tolerance. (ii) (a) If m∗j > 15 set k j+1 = k j /2; else set k j+1 = k j .   M j+1 |K  > c1 /hK mark K for refinement; if (b)Loop over all K ∈ Thj : if ∇M  ∇M M j+1 |K  < c2 /hK mark K for coarsening. (c) Obtain Thj+1 from Thj by local refinement/coarsening of the elements K ∈ Thj marked in step (b). (iii) Stop if j + 1 = J; otherwise set j ← j + 1 and go to (i). The local mesh refinement and coarsening in the above algorithm is performed using a bisection algorithm, and are based on the finite element code ALBERT [100]. The constants c1 /c2 determine how much refinement/coarsening is performed on each time level; for the deterministic computations we choose c1 = 0.5, for the stochastic calculations we set c1 = 1; we fix c2 = 0.05 in both cases.

188

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

  M j (x )| = 1 we get that ∇M M j |K  ≤ Remark 2.47. Due to the property |M ∗ C ∗ h−1 K , where the constant C only depends on the geometry of the mesh, such 3 √ M j L∞ ≤ C ∗ h−1 that C ∗ ≈ 2 2 in our experiments. Consequently, ∇M min , for M j (x )|K | × hK ≤ C ∗ will attain values hmin = minK hK . The quantity 0 < |∇M close to C ∗ in the neighborhood of singularities. This motivates the criterion for mesh refinement and coarsening in step (ii)-(b) of Algorithm 2.46. The nonlinear systems at each time-step of Scheme 2.10 are solved using the Newton Algorithm 1.48. In order to eliminate round-off errors the linear systems in each iteration of the Newton algorithm are solved using a direct linear solver (Gaussian elimination).

2.3.2

Effects of the space-time white noise in 1D and 2D

We consider the SLLG equation (2.1) on a 1D domain O = (0, 1) and for T = 50, with Dirichlet boundary conditions m(t, 0) = m(t, 1) = (0, 0, 1); we take A = 1, K = 0, ν = 2. The problem is computed for a range of mesh sizes h = hp = 2−p (p = 4, 5, 6, 7, 8, 9), with space-time white noise and ν = 2. The probability density for the third, and the first two components of the computed solution at the point x = 0.5 are depicted in Figure 2.5: we observe that the distribution for the spin is invariant with respect to the mesh size h > 0. In Table 2.2 (left) we display the time step k used for each mesh size hp , as well as T /k 1  k∇Mj 2 along a single trajectory. the average exchange energy ET = 2T j=1

As expected, the energy values of the discrete iterates of the midpoint scheme with space-time white noise depend on the mesh size h > 0; cf. [11]. We observe that asymptotically the energy grows by a factor ≈ 1.5 as the mesh size is halved. For comparison we include similar calculations for a 2D problem with T = 10, O = (0, 1)2 , for periodic boundary conditions and external field Hext = (0, 0, 1); the remaining parameters for the computations were A = 1, K = 0, ν = 0.5. The effects of the space-time noise in the 2D setting appear stronger than in the 1D case; Table 2.2 (right) shows that the energy increases approximately by a factor 3 after each level of mesh refinement. The graphs in Figure 2.6 shows that the asymptotic probability density in 2D is becoming more diffuse for decreasing mesh size which suggests that the effects of the noise in 2D become more relevant with mesh refinement. However, we stress that these computational studies in 2D are limited by their computational complexity and may hence not be conclusive.

Section 2.3

189

Computational studies

p

k

ET

4

0.01

8.3

p

k

ET

5

0.002

14.1

3

0.001

12.1

6

0.001

26.5

4

0.0005

35.7

7

0.0005

42.8

5

0.00025

99

8

0.00025

61.5

6

0.000125

259

9

0.0001

98

Table 2.2. Computed time-asymptotic energies on refined meshes for 1D (left) and 2D (right).

3

0.7

h_4 h_5 h_6 h_7 h_8

2.5

mx h_5 my h_5 mx h_6 my h_6 mx h_7 my h_7 mx h_8 my h_8

0.65 0.6 0.55

2 0.5 0.45 1.5 0.4 0.35 1 0.3 0.25

0.5

0.2 0

0.15 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 2.5. Probability distribution for the third and first two components of the 1D solution for hp = 2−p (p = 4, 5, 6, 7).

6

h_3 h_4 h_5 h_6

5

4

3

2

1

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 2.6. Probability distribution for the third component of the 2D solution for hp = 2−p (p = 4, 5, 6).

190

2.3.3

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

Discrete blow-up of the SLLG equation with space-time white noise

As is detailed in Section 2.1.6, it is known that a solution of the harmonic map heat flow problem may develop a singularity in finite time. Moreover, there is strong numerical evidence for this phenomenon to occur for the LLG equation; see Figure 2.2. Extensive numerical studies of a discrete blow-up for the SLLG equation have been performed in [11, 9]. Below, we complement the existing studies to provide further insight into the blow-up phenomena.

Discrete blow-up for the Landau-Lifshitz-Gilbert equation for fixed spatial resolution We investigate the effects of noise-induced perturbations on the blow-up of the numerical solution for (fixed in time) finite element meshes refined towards the singularity. For x ∈ O = (−0.5, 0.5)2 we consider the initial condition

0

m (x) =

⎧ 2 2 2 2 ⎪ ⎪ ⎨ (4x, B − 4|x| )/(B + 4|x| ) ⎪ ⎪ ⎩

for |x| ≤ 0.25,

(0, 0, −1)

(3.6x/|x|,

B2

−

0.92 )/(B 2

+

for 0.25 < |x| ≤ 0.45,

0.92 )

else ,

(2.141) where B = (1 − 2|x|)4 . The initial data (2.141) will evolve analogically as in Figure 2.2; a discrete blow-up will occur at a finite time t∗ = tj ∗ , where the gradient of the solution will reach its maximum value: for each fixed mesh size h we get that (see also Remark 2.47) ∗

M j L∞ ≈ ∇M

√

2

3/2 −1

h

.

(2.142)

We set A = 1, K = 0, α = 5 and k = 5 × 10−5 , and consider the space-time white noise (2.140) for different values of ν. We compute on a finite element mesh which is adapted towards the center of the domain, with minimum mesh size hmin = 1/64 and maximum mesh size hmax = 1/32, see Figure 2.7 (left); The evolution for ν = 0 leads to a discrete blow-up at the center at time ∗ M j L∞ ≈ 181 (j ∗ = t∗ /k), see Figure 2.8 (left). Figure 2.9 t∗ ≈ 0.03 with ∇M displays the evolution of the computed solution for one realization of the spacetime white noise for ν = 0.707 and ν = 2, respectively. We observe that the noise breaks the symmetry of the solution. The evolution of the middle spin for ν = 0, ν = 0.707 and ν = 2 is displayed Figure 2.10. The evolution exhibits a discrete blow-up for the subcritical noise intensity ν = 0.707, see Figure 2.8. For ν = 2 there is no formation of a blow-up since the strong noise does not

Section 2.3

191

Computational studies

Figure 2.7. Mesh with hmin = 1/64 (left) and hmin = 1/128 (right).

200

220

300

nu=0

nu=0.707

180

200

160

180

140

160

120

140

100

120

80

100

60

80

40

60

20

40

0

20

nu=2 250

200

150

100

50

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

M j L∞ for ν = 0 (left), ν = 0.707 (middle), ν = 2 Figure 2.8. Evolution of tj → ∇M (right) for one realization of the space-time white noise (hmin = 1/64).

Figure 2.9. Evolution of the computed solution with space-time white noise with ν = 0.707 at times t = 0, 0.015, 0.0268, 0.0225 (top), ν = 2 at times t = 1 × 10−4 , 2 × 10−4 , 4 × 10−4 , 8 × 10−3 (bottom) for hmin = 1/64.

192

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

allow for pointwise singularities to persist. The experiments show that beyond a certain critical value of the noise intensity ν the evolution is dominated by noise-induced gradients which are proportional to (2.142). To complement the

Figure 2.10. Evolution of tj → M j (x∗ ), x∗ = (0, 0) for ν = 0 (brown), ν = 0.707 (black) (left) and for ν = 2 (right) for one realization of the space-time white noise (hmin = 1/64).

above studies we show in Figure 2.11 the evolution of expected values as well as of individual realizations for ν = 0.5 with hmin = 1/32, 1/54, 1/128. The results 350

(*) h=1/32 (*) h=1/64 (*) h=1/128 (**) h=1/32 (**) h=1/64 (**) h=1/128

300

250

(*) h=1/32 (*) h=1/64 (*) h=1/128 (**) h=1/32 (**) h=1/64 (**) h=1/128

600

500

400 200

300 150

200

100

100

50

0

0 0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

M j L∞ (left) and tj → ∇M M j 2L2 (right): exFigure 2.11. Evolution tj → ∇M pectation (**), and solution for one realization of the space-time white noise (*), hmin = 1/32, 1/64, 1/128.

show blow-up of the solutions for individual realizations of the space-time white noise while the expected values remain smooth. The smooth evolution of the expectation can be seen in Figure 2.12 where we display a typical evolution of the expected value of the computed solution around the center of the domain for h = 1/32; the expectation for the middle spin remains parallel with the z-axis and the spin switches by changing its length.

Section 2.3

193

Computational studies

M j ]. Figure 2.12. Smooth evolution of tj → E[M

Space-time white noise with adaptive mesh refinement As has been pointed out in [19], it is due to the finite numerical correlation of the space-time noise approximations that it is essential to employ mesh refinement in order to correctly capture the effects of space-time white noise in numerical simulations. Moreover mesh adaptivity allows correct simulation of the unbounded gradients of the solution, see Remark 2.47. In the subsequent simulation, we employ the space-time adaptive Algorithm 2.46 for each realization of the space-time white noise. We choose the following parameters for 3000

250

nu=0. nu=0.707 (1) nu=0.707 (2) nu=0.707 (3) nu=0.707 (4)

2500

nu=0. nu=0.707 (1) nu=0.707 (2) nu=0.707 (3) nu=0.707 (4)

200

2000 150 1500 100 1000

50 500

0

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

M j L∞ (left) and exchange tj → 21 ∇M M j 2L2 Figure 2.13. Evolution of tj → ∇M (right), blow-up dynamics for the adaptive Algorithm 2.46 for ν = 0 and ν = 0.707.

220

75

nu=0.707 (1) nu=0.707 (2)

200

nu=0.707 (1) nu=0.707 (2)

70

180

65

160 60 140 55 120 50 100 45 80 40

60

35

40 20

30 0

0.05

0.1

0.15

0.2

0.25

0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

M j L∞ (left) and exchange tj → 21 ∇M M j 2L2 Figure 2.14. Evolution of tj → ∇M (right), dynamics without singularities for the adaptive Algorithm 2.46 for ν = 0 and ν = 0.707.

194

Chapter 2

The stochastic Landau-Lifshitz-Gilbert equation

the computation: A = 1, α = 5, k 0 = 2 × 10−4 , h0 = 2−5 , ν = 0.707; for the initial condition we take (2.141). As can be seen in Figure 2.13 (left), even with space-time adaptivity the numerical solution of the deterministic and stochastic LLG may exhibit a discrete blow-up behavior. Figure 2.13 (left) shows the M j L∞ ≈ 3000, however we were able to compute the gradient value of maxj ∇M up to the magnitude of 1012 when both, minK hK and k are in the region of round-off errors. Figure 2.13 (right) shows that due to the adaptive mesh refinement, the space-time white noise (2.140) injects additional energy into the system. Despite the fact that the amount of injected energy is growing as hmin

Figure 2.15. Zoom on the computed solution (top) and the adapted mesh (bottom) at times t = 0, 0.0130305, 0.0134703, 0.0138609 at x = (0.5, 0.5) for one realization of the space-time white noise, blow-up dynamics.

decreases, the solution still exhibits discrete blow-up dynamics for most realizations of the Wiener process. This indicates that for in the majority of case the perturbation effects of the space-time white noise do not prevent (path-wise) the formation of singular discrete dynamics. We display in Figure 2.15 a detail of a blow-up formation which is computed with Algorithm 2.46, together with the corresponding finite-element mesh near the point x = (0, 0). For approximately 10% of the realization of the space-time white noise we do not observe a formation of singularities, see Figure 2.14. Instead the magnetization evolves towards a spatially uniform state (a minimum for the exchange energy), by a different switching mechanism Figure 2.16. The alternative dynamics is enabled by the space-time noise, which allows the system to overcome the energy barrier that separates the different switching mechanisms in the deterministic case. In addition, the singularities form at different positions in time and space

Section 2.3

Computational studies

195

Figure 2.16. Magnetization for one realization of the space-time white noise with smooth dynamics at time t = 0, 0.1472, 0.1772, 0.1972, 0.2372.

for different realizations of the space-time white noise (see Figure 2.13). ThereM L∞ ] does not exhibit a singular behavior and fore the expected value E[∇M remains smooth, cf. Figures 2.11 and 2.12.

Chapter 3

Effective equations for macrospin magnetization dynamics at elevated temperatures In the article [59], D.A. Garanin derived a macroscopic equation called the Landau-Lifshitz-Bloch equation to approximately describe within the mean m] of a single spin field approximation the macroscopic magnetization m = E[m m at elevated temperatures, where denotes the solution of the SLLG equation (2.1). A practical motivation to derive macroscopic equations from the SLLG equation is to e.g. simulate the multi-scale (macroscopic) magnetization dynamics, where the SLLG equation (2.1) would require too long simulation times. Typical applications include thermally assisted recording, where local heating of ferromagnetic media alleviates magnetization reversal in hard disc drives; see Figure 3.1, (a). The approach in [59] to take into account thermal fluctuations is based on the Kolmogorov equation for the SLLG equation, and a moment closure approximation for higher moments to resolve nonlinear effects; see also Remark 3.16 below. The following remark is to motivate the macroscopic model (3.5). Remark 3.1. We are going to calculate the constant ΛN in formula (6) on page 3, and show that a standard moment closure approximation for the SLLG equation (2.1) leads to exponential convergence to zero of the magnetization as the temperature rises. We follow here [29]. We will consider the simplest m|2 dx m) = 12 O |∇m case when the energy consists of the exchange energy E(m 3 M n k nk only. Let O be a three-dimensional torus, and Hthm = n=1 k=1 e f β˙ , ∞,3 where {β nk }k,n=1 are independent one-dimensional Brownian motions, where {en }3n=1 denotes a standard orthonormal basis in R3 , and {f k }∞ k=1 stands for ∞,3 2 n k an orthonormal basis in L (O). Note that the family {e f }k,n=1 forms an orthonormal basis in L2 (O, R3 ). Hence, Hthm constructs white noise as M tends to ∞. We fix M ∈ N. Thus the SLLG equation (2.47) in this case can be written as follows, M 3     m = m × m m − αm m × (m m × m m) dt + ν dm m × ◦en f k dβ nk ,

(3.1)

n=1 k=1

where the stochastic integral is again understood in the Stratonovich sense.

Chapter 3

Effective equations for macrospin magnetization dynamics

197

Now the Itô correction term can be calculated as follows 3 M M  ν2   n n k 2 2 m × e ) × e |f | dt = −ν m (m |f k |2 dt . 2 n=1 k=1

(3.2)

k=1

Thus equation (3.1) can be rewritten in Itô form as follows % $ M  2 k 2 m = m × m m − αm m × (m m × m m) − ν m |f | dt dm

(3.3)

k=1

+ν

M 3  

m × en f k dβ nk .

n=1 k=1

m] satisfies the system By taking the expectation we deduce that m = E[m     ∂m m × Heff ) = ΛM m + E m × Heff − α E m × (m ∂t where ΛM = −ν 2

M  k=1

|f k |2 = −2ακB τ

M 

on R+ × O ,

(3.4)

|f k |2 ∼ −ακB τ M ,

k=1

where the second equality follows from the fluctuation–dissipation relation (1.3) on page 7. A moment closure approximation (i.e., the assumption that the expectation of the product is equal to the product of expectations) leads to the conclusion that m(t)]| = |m(t)| = e−ΛM t ∼ e−ακB τ M t . |E[m Thus we see that a direct moment closure approximation does not allow us to understand the Curie temperature phenomenon; see Figure 3.1, (b) for an observed profile τ → m(τ  ). In this chapter, we will use a different approach for a similar macroscopic model which accounts for thermal effects. It is a deterministic PDE which has a simple geometric motivation to describe the magnetization dynamics at elevated temperatures (O ⊂ Rn , n = 1, 2, 3, and T > 0), ∂m α = κ m + m × Heff − m × (m × Heff ) ∂t m ∂m =0 ∂n m(0, ·) = m0

on OT := (0, T ) × O , on (0, T ) × ∂O ,

(3.5)

on O ,

where Heff ≡ Heff (m) = −DE(m), and the Landau-Lifshitz energy E is given in (5). This model is again considered within the mean-field approximation. It

198

Chapter 3

Effective equations for macrospin magnetization dynamics

is an augmented LLG equation which has been proposed in [14] to allow for space-time   variations of the modulus of the magnetization |m(t, x)| = m(t, x) ≡ m  τ (t, x) according to a given temperature field τ : OT → R. Here, the saturation magnetization m  ≡ m(τ  ) follows Landau’s phenomenological powerlaw at temperatures τ below the Curie temperature τC (see Figure 3.1, (b))  τ β . (3.6) m(τ  )=m 0 1 − τC Heff beam spot

m × Heff m × (m × Heff )

m ˜ m ˜0 1

Position

write width

disk motion

κm

thermally stable

written bits write core

m writable

Temperature

τc

0

τ (c)

(b)

(a)

Figure 3.1. (a) Thermally assisted magnetic recording, (b) temperature dependent saturation magnetization m,  (c) longitudinal and transverse relaxation of magnetization.

The exponent β > 0 is found by experimental or theoretical evidence. Then, we use the mutual orthogonality of the m, m × Heff , and m × (m × Heff ) to describe the magnetization dynamics at elevated temperatures τ via (3.5). The (longitudinal) relaxation function κ(t, x) = κ  τ (t, x), τt (t, x) in (3.5) is set up to take into account both, the temperature function as well as its time derivative, and is chosen such that the corresponding modulusof the  solution |m| satisfies thepower-law (3.6). In addition to |m(t, x)| = m t, x in OT we  2 ask that κ  ∈ C [0, τC ) × R satisfies the following natural requirements:

κ (τ, τt ) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0

for τt = 0 ,

≤0

for τt ≥ 0 ,

>0

for τt < 0 .

(3.7)

Scalar multiplication of (3.5) with m then leads to the following ordinary differential equation for every fixed x ∈ O, 1d 2 m = κ m2 2 dt

∀t > 0,

m2 (0, ·) = |m0 |2 .

(3.8)

Hence, by the chain rule, κ (τ, τt ) = −

d τt d ln m  2 (τ ) = −τt ln m(τ  ). 2 dτ dτ

(3.9)

Chapter 3

Effective equations for macrospin magnetization dynamics

199

Therefore, κ  satisfies the requirements in (3.7). For successive times 0 ≤ t1 ≤ t2 ≤ T we then have  t2        2 2  τ (t1 , ·) exp 2 κ  τ (s, ·), τt (s, ·) ds in O . (3.10) m  τ (t2 , ·) = m t1

Moreover, inserting the power-law (3.6) in (3.9) then leads to κ (τ, τt ) = −

βτt . τC − τ

(3.11)

Shrinking, extension, and conservation of the magnetization saturation m by means of heating, cooling, and constant temperature may then be described via (3.5), (3.11) for a given field τ : OT → [0, τC ); see Figure 3.1, (c). In Section 3.1, we will construct and approximate the local strong solution (n = 2) of system (3.5) where κ is given by formula (3.8). The problem is different from the standard LLG system since solutions of (3.5) are mappings m : OT → R3 where the target manifold is smoothly varying in space and time, i.e. |m(t, x)| = m(t, x) for a given m : OT → R+ \ {0}. In particular, equation (3.5) reduces to LLG for the isothermal case. The additional Bloch term in (3.5) is expected to cause avoidance, delay, acceleration, or the new formation of singular behavior if compared to solutions of LLG with the stationary target S2 ; see [13], and the computational studies in Section 3.4. Below, we will study two discretizations of (3.5). The first one is the computationally efficient linear Scheme 3.3 introduced in Section 3.2. It is constructed such that the iterates converge with rates to a strong solution of (3.5) for n ≤ 2. The inherent projection steps ensure that |Mj | = m(tj , ·) at every nodal point x ∈ Eh . As it will be clear from equation (3.21), the scheme may be interpreted as a semi-implicit penalization method. The second discretization is the nonlinear Scheme 3.5 from Section 3.3, where iterates satisfy a discrete energy principle and for n ≤ 3 construct a weak solution of (3.5). However, the iterates only approximate the target manifold of the solution even for the nodal points x ∈ Eh . A linearization of the second scheme via a simple fixed point strategy with a proper stopping criterion is proposed in Subsection 3.3.1. In this scheme, the iterates inherit the properties of those of Scheme 3.5 both, with respect to stability and (unconditional) convergence. Computational studies are reported in Section 3.4 which include a comparison of approximate solutions of the macro  m scopic model (3.5) with macroscopic quantities E f (m ) from Algorithm 2.10 for the SLLG equation (2.1).

200

3.1

Chapter 3

Effective equations for macrospin magnetization dynamics

Construction of local strong solutions for the augmented LLG equation in two space dimensions

Let O ⊂ Rn , n = 1, 2 be a bounded polyhedral domain, and Heff = Δm. Throughout this section, we assume:   (A1) The given function m ∈ C2 OT ; R+ \{0} satisfies ∂n m = 0 on [0, T ]×∂O. φ) = The κ ∈ C2 (OT ) is then determined by (3.8). We denote E(φ function 1 2 φ |∇φ | dx. We will show the existence of a local strong solution for (3.5), by 2 O following arguments from [105], and [69], which have been successfully employed for m ≡ 1. Definition 3.2. Fix T > 0 and m0 ∈ W1,2 (O, R3 ) such that |m0 | = m(0, ·) Lebesgue almost everywhere in O. Suppose that (A1) is valid.   Let Heff = Δm.   A function m ∈ L2 0, T ; W2,2 (O, R3 ) such that mt ∈ L2 0, T ; L2 (O, R3 ) is called a strong solution to problem (3.5) if and only if (i) for all t ∈ [0, T ], |m(t, ·)| = m(t, ·) Lebesgue almost everywhere in O.   (ii) for all φ ∈ W1,2 O, R3 and all t ∈ [0, T ] 

m(t, ·), φ



 L2

−

t

n t   φ ∂m ∂φ ds = − , × m ds 2 L ∂xp ∂xp L2 p=1 0 n t   × φ] ∂m ∂[ m +α , m × m 2 ds . ∂xp L 0 ∂xp

κ m, φ

0

  + m0 , φ L2



p=1

Consider the reformulation of equation (3.5) in the following form α mt − αm Δm − κ m − |∇m|2 m+ m α  2 Δm m − m × Δm = 0 on OT , + 2m ∂m =0 on (0, T ) × ∂O , (3.12) ∂n m(0, ·) = m0 on O .     A function m ∈ L2 0, T ; W2,2 (O, R3 ) such that mt ∈ L2 0, T ; L2 (O, R3 ) is similarly called a strong solution of (3.12) if the equations are valid in a weak sense. Lemma 3.3. Suppose (A1). Then a function m : OT → R3 is a strong solution of (3.5) if and only if it is a strong solution of (3.12).

Section 3.1

201

Construction of local strong solutions for the augmented LLG

We sketch the main arguments to verify this assertion in the following proof. Proof. Step 1. Let us assume that a function m : OT → R3 is a strong solution of (3.5). By using the vector cross product formula, and |m| = m in OT we infer that Leb-a.e. in OT m × (m × Δm) = m, Δmm − |m|2 Δm 1 = −|∇m|2 m + (Δm2 )m − m2 Δm . 2

(3.13)

Inserting this identity into (3.5)1 proves that m : OT → R3 solves (3.12) as well. m|2 , and Step 2. Let m : OT → R3 be a strong solution of (3.12). Set R = |m observe m, m t  , Rt = 2m

 ∂m m ∂R , = 2 m, ∂xp ∂xp

m, Δm m + 2|∇m m |2 . ΔR = 2m

By (3.12), and a vector cross product formula, this implies that Lebesgue almost everywhere in OT , Rt − αm ΔR − 2κ R =

  α 2α m|2 R − m2 − (Δm2 )R . |∇m m m

Setting Z = R − m2 leads to

  α   m|2 − (Δm2 ) Z . Zt − αm ΔZ − 2κ Z = 2κm2 − ∂t m2 + 2|∇m m

The first bracket on the right-hand side vanishes, owing to (3.8). Next we multiply by Z, integrate, and observe that ∂n Z = 0 on (0, T ) × ∂O to infer that √ 1d Z2L2 + α  m∇Z2L2 2 dt     2α α  2 2 2 m| − |∇m Δm , Z = 2κ + + α ∇m, ∇Z, Z 2 . m m L L2 Because of 0 < m, and interpolation of L4 (O) between L2 (O) and W1,2 (O), we arrive at the following estimate for the L1 (0, T )-integrable function C (t) = m(t, ·)4L4 , and some 0 < c ≤ minOT m, C∇m d Z2L2 + αc ∇Z2L2 ≤ C Z2L2 . dt Since Z(0, ·) = 0 Lebesgue almost everywhere in O, by applying the Gronwall m| = m lemma we infer that Z = 0 Lebesgue almost everywhere in OT . Hence, |m Lebesgue almost everywhere in OT . Finally, using again (3.13) we conclude that m satisfies (3.5) as well.

202

Chapter 3

Effective equations for macrospin magnetization dynamics

For m ≡ 1 and n = 2, the global strong solution for small initial energies E(m0 ) ≡ 12 ∇m0 2L2 , or the local strong solution for finite initial energies has been constructed in [69], and [105, Sect. 1.4]. Below, we sketch corresponding arguments for m satisfying (A1), and n = 2. Further details can easily be added from these references. First, we find a priori bounds, under the assumption that a solution m : OT → R3 of (3.12) is smooth. Therefore, let us assume for the time being that the data O, m0 , κ, and m are smooth.   Lemma 3.4. Assume (A1). There exists 0 < C ≡ C mC2 , m−1 C , α < ∞ such that for any smooth solution m of (3.12) we have t     mt (s, ·)2L2 ds + E m(t, ·) ≤ C 1 + E(m0 ) (0 ≤ t ≤ T ) . 0

Proof. By multiplying equation (3.12) with mt , and integrating over O we get   1   1d  m, |∇m|2 − ∂t m, |∇m|2 − ∇m ∇m, mt , − (mΔm, mt ) = 2 dt 2    d  2 κ,m − ∂t κ, m2 , 2 (κ m, mt ) = (3.14) dt     |∇m|2 , ∂t m2 α = α |∇m|2 , ∂t m , 2m    Δm2 α , ∂t m2 −α = − Δm2 , ∂t m . 4m 2 Next, we bound | (m × Δm, mt ) |: taking the cross product of (3.12) with m, and using (3.13) and (3.12) lead to m × mt = αm m × Δm + m × (m × Δm) 1 = αm m × Δm − m2 Δm + (Δm2 ) m − |∇m|2 m 2   m 1 m = m α+ m × Δm + κ m − mt . α α α

(3.15)

α mt leads us to Multiplying the last equation by m   2 0 = α + 1 (m × Δm, mt ) + (mκ, ∂t m) − mt 2L2 ,

and hence



T 0

1 |(m × Δm, mt )| ds ≤ CT + 1 + α2



T 0

mt 2L2 ds .

Putting things together, we arrive at  T  1 α 1 2 mt (s, ·)L2 ds + m(T, ·)|∇m(T, ·)|2 dx 1− 2 1 + α2 2 O 0 T α ≤ m(0, ·)|∇m(0, ·)|2 dx + C 1 ∇m(s, ·)2L2 ds + C 2 , 2 O 0

Section 3.1

Construction of local strong solutions for the augmented LLG

for some positive constants   C 1 ≡ C 1 α, mC1

and

203

  C 2 ≡ C 2 mC2 .

The assertion of the lemma then follows from applying the Gronwall lemma. Let us denote E(m; ω) = 12 ω |∇m(x)|2 dx for ω  O. Let also Bρ (x0 ) ⊂ O be a ball centered at x0 ∈ O of radius ρ > 0. By a slight abuse of (earlier) notation, we have that E(m) coincides with E(m; O). The following technical lemma is taken from [105, p. 274]; see also [69, Lemma 3.1]. Lemma 3.5.  There exist constants C, ρ0 > 0 such that   for every T > 0 and any ϕ ∈ L∞ 0, T ; W1,2 (O, R3 ) ∩ L2 0, T ; W2,2 (O, R3 ) , and any ρ ∈ (0, ρ0 ]

T 0

  ϕ4L4 ds ≤ C ess sup(s,x)∈OT E ϕ (s, ·); Bρ (x) · ∇ϕ   T T ϕ2L2 ds . ∇2ϕ 2L2 ds + ρ−2 ∇ϕ · 0

0

The following lemma asserts that the local energies at a fixed time can be controlled in terms of those at earlier times. This result may be obtained by a localized energy argument in the proof of Lemma 3.4. A proof for m ≡ 1 is given in [69, Lemma 3.4] and can easily be generalized to our current setting. Lemma 3.6. Assume (A1), and that m : OT → R3 is a smooth solution of (3.12). Let us assume that ρ ∈ (0, ρ0 ]. There exists C ≡ C(mC2 , m−1 C , α) > 0 such that for every x0 ∈ O      T E m(T, ·); Bρ (x0 ) ≤ E m0 ; B2ρ (x0 ) + C 2 1 + E(m0 ) . ρ We can now follow the general argumentation from [105, p. 274], which we include for the convenience of the reader: For m0 ∈ C∞ (O, R3 ) and any given ε1 > 0, let ρ1 > 0 be the maximal number such that   sup E m0 ; B2ρ1 (x) < ε1 . x∈O

Let 0 < T1 < T be the time such that any smooth solution m : OT → R3 of (3.12) with initial value m0 : O → R3 satisfies   sup E m(s, ·); Bρ1 (x) ≤ 2ε1 . (3.16) (s,x)∈OT1

Note that in view of Lemma 3.6 we may let T1 =

ε1 ρ21 CE(m0 ) .

204

Chapter 3

Effective equations for macrospin magnetization dynamics

Let {φi } be smooth cut-off functions subordinate to a cover of O by balls B2ρ1 (xi ) with finite overlap, and such that 0 ≤ φi ≤ 1, and |∇φi | ≤ ρ21 , such  that i φ2i = 1. We may then compute for all t ∈ [0, T1 ],  4 |∇m(t, ·)|4 φ2i dx ∇m(t, ·)L4 = i

O





 ≤ C sup E m(t, ·); B2ρ1 (xi ) |∇2 m(t, ·)|2 dx + ρ−2 1 E(m0 ) i O   ≤ Cε1 |∇2 m(t, ·)|2 dx + ρ−2 (3.17) 1 E(m0 ) O

by Lemma 3.5, and (3.16). We are now in a position to verify the following result. Lemma 3.7. For ε1 > 0 sufficiently small, there exist T1 > 0 and ρ1 > 0 such that the existing smooth solution m of (3.12) satisfies

  t  m2W2,2 ds ≤ C E(m0 ) + 1 1 + 2 ρ1 0   with a constant C ≡ C mC2 , m−1 C , α > 0. t

(0 ≤ t ≤ T1 ) ,

Proof. Multiplying (3.12) with −Δm and integrating over O we get     α √ 1d ∇m2L2 +  mΔm2L2 ≤ C 1+∇m2L2 +  ∇κ⊗m, ∇m  + ∇m4L4 , 2 dt 2   for some C ≡ C mC2 (OT ) , α > 0. For sufficiently small ε1 > 0, by (3.17), since 0 < c ≤ m, we can control the last term on the right-hand side by the second one on the left-hand side. Integration in time, and Lemma 3.4 then show the assertion. To sum up, for every ε1 > 0 sufficiently small we have found bounds for smooth solutions m : OT1 → R3 of (3.12) in  ϕ)+ V (OT1 ; R3 ) = ϕ : OT1 → R3 ; ess sup[0,T1 ] E(ϕ

T1 0

 ϕ2W2,2 +ϕ ϕt 2L2 ds < ∞ . ϕ

It is not difficult to verify uniqueness of solutions m : OT1 → R3 of (3.12) which are in this class V (OT1 ; R3 ); see [69, Theorem 3.9] for the case m ≡ 1. In order to prove the existence of a local smooth solution m : OT1 → R3 to (3.12) for smooth data O, m0 , κ, and m, we may then exploit the strong parabolicity property of our problem, and follow the argument used in [69, Theorem 3.12 and Lemma 3.10] for m ≡ 1.

Section 3.1

Construction of local strong solutions for the augmented LLG

205

Theorem 3.1. Assume (A1), and that a smooth map m0 : O → R3 satisfies |m0 | = m(0, ·). Then there exist T1 > 0, and a unique, smooth solution m : OT1 → R3 of (3.12). Let us go back to the following data. Assume O ⊂ Rn to be a bounded Lips  2 + 2 chitz domain, that m ∈ C OT ; R \ {0} , κ ∈ C (OT ), and m0 ∈ W1,2 (O, R3 ). Then, strong solutions may be constructed by approximating those data by smooth ones, i.e. for every r ∈ N big enough, smooth data Or , mr , κr , and m0,r satisfy m − mr C2 (OT ) + κ − κr C2 (OT ) + m0 − m0,r W1,2 ≤

1 . r

(3.18)

By Theorem 3.1, a smooth, unique, local solution mr ∈ C∞ (OT1 , R3 ) of (3.12) exists for each r ∈ N big enough, with the following uniform bound with respect to r > 0,   sup E mr (s, ·); Bρ (x) ≤ ε1 ∀ ρ ∈ (0, ρ1 ] , (s,x)∈OT1

for any ε1 > 0. In particular, note that T1 ≡ T1 (m0 ) does not depend on r > 0 by Lemma 3.6. Similar to [69, Lemma 3.7], a consequence from Lemmata 3.4 and 3.7, and the Aubin-Lions lemma is that there exist m : OT1 → R3 , and a subsequence (not relabeled) {mr } such that for r → ∞,   mr  m in W1,2 OT1 , R3 ,   (3.19) mr  m in L2 0, T1 ; W2,2 (O, R3 ) ,   mr → m in L2 0, T1 ; W1,2 (O, R3 ) .   In particular, m ∈ C [0, T1 ]; W1,2 (O, R3 ) because of (3.19)1,2 . Now take any smooth vector field φ : O → R3 ; for any time t ∈ [0, T1 ], the smooth solution mr to (3.12) satisfies t t      mr (t) − m0,r , φ 2 − α κr mr , φ L2 ds mr Δmr , φ 2 ds − L L 0 0 t    2 1 1 Δmr 2 =α |∇mr | mr , φ 2 − mr , φ 2 ds 2 mr L L 0 mr t   + mr × Δmr , φ L2 ds . 0

We may now use (3.19) and (3.18) to pass to the limit for r → ∞ and find that t       m(t) − m(0), φ 2 − α mΔm, φ L2 − κm, φ L2 ds L 0 t t     |∇m|2 1  Δm2 m, φ 2 − m, φ 2 ds + =α m × Δm, φ L2 ds . m 2 2m L L 0 0

206

Chapter 3

Effective equations for macrospin magnetization dynamics

We proved the following result. Theorem 3.2. Let O ⊂ R2 be a convex polygonal domain, m0 ∈ W1,2 (O, R3 ),  and m ∈ C2 OT ; R+ \ {0} be given, such that |m0 | = m(0, ·) Lebesgue almost everywhere in O. There exist T1 > 0, and a unique, local strong solution m ∈ V (OT1 ; R3 ) of (3.12). Moreover, this strong solution is also a local strong solution of (3.5) by Lemma 3.3. Remark 3.8. 1. Throughout the analysis we assume that 0 < m. For m ≥ 0, the above estimates are not valid any more; from a physical viewpoint, those points (t, x) ∈ OT where m = 0 would take the Curie temperature τC . 2. Equation (3.5) describes m : OT → R3 such that for all t ≥ 0 holds 0 < |m(t, ·)| = m(t, ·) Leb-a.e. in O. We show that the unit vector field m := m m : OT → S2 then solves m ∂m m × Heff ) on OT . = m × Heff − α m × (m ∂t

(3.20)

Note that Heff ≡ Heff (m); hence, (3.20) is a modification of the deterministic LLG equation, describing damped = precession of m around the applied field Heff . To verify (3.20), recall m(t, x) = |m(t, x)|2 , and use elementary calculus to find ! 1 1 ∂m(t, x) ∂m(t, x) ∂|m(t, x)|2 = = , m(t, x) . ∂t 2m(t, x) ∂t m(t, x) ∂t We use this identity, and (3.5) to then compute  1  ∂m ∂m  ∂ m (t, x) = − m (t, x) m ∂t m2 ∂t ∂t 1  = κ m + m × Heff − α m × (m × Heff ) (t, x) m !   1 − 3 κ m + m × Heff − α m × (m × Heff ), m m (t, x) . m Because of a × b, a = 0, and m = |m| we then arrive at (3.20). Let m satisfy (A1): to construct a stable, convergent discretization which satisfies |Mj | = m(tj , ·) > 0 even only at mesh points (tj , x ) ∈ Ik × Eh is a challenging task. In the following, we study two discretizations of (3.5), which are based on the two equivalent formulations of problem (3.5): • Scheme 3.3: This linear scheme is based on the reformulation (3.12). At the end of each iteration step, vectors are projected onto the sphere of radius m(tj , x ), for every 1 ≤ j ≤ J, and every  ∈ L. The iterates converge to a (local) strong solution of (3.5) at optimal rates (see Section 3.2).

Section 3.2

Convergence with optimal rates for Scheme A

207

• Scheme 3.5: This nonlinear scheme is a discretization in space and time of the reformulation (3.23) of (3.5), where the midpoint rule, reduced integration in space, and the discrete Laplacian are used. Its motivation comes from the numerical analysis of LLG for m ≡ 1, where the iterates take values in the target at every mesh point x ∈ Eh , and the convergence to a weak solution is known (see also Chapter 2). As it will be shown in Section 3.3, the modulus of the iterates of Scheme 3.5 at each node (tj , x ) ∈ Ik ×Eh only approximates m(tj , x ), and only in the limit for vanishing discretization parameters takes values of m accordingly. The iterates are shown to unconditionally converge to a weak solution of (3.23). The proof of this fact is based on a discrete energy estimate; see Section 3.3.

3.2

Convergence with optimal rates for Scheme A

Assume that n ≤ 3, and O ⊂ Rn is a bounded polyhedral domain. Moreover, assume that assumption (A1) on p. 200 is valid. A general space-time discretization of problem (3.12) produces iterates {Mj } ⊂ Vh , where |Mj (x )| = m(tj , x ) for 1 ≤ j ≤ J and  ∈ L. This is the motivation for the following projection scheme, which generalizes the corresponding one for m ≡ 1 in [98, p. 139] to the present case. 8 0 ∈ Vh . 1. Let j = 1, 2, . . . , J. Find Scheme 3.3. Assume that M0 ≡ M j 8 ∈ Vh such that for all Φ ∈ Vh M      1 8j 8 j , ∇Φ 8 j , ∇m(tj , ·) ⊗ Φ Φ = α ∇M M − Mj−1 , Φ + α m(tj , ·)∇M k h     α  8 j−1 |2 − 1 Δm2 (tj , ·) + κ(tj , ·) M 8j , Φ |∇M + m(tj , ·) 2 n   j  8 Φ 8 j−1 × ∂ M , ∂Φ M . − ∂xp ∂xp p=1

2. Compute Mj ∈ Vh by the formula Mj (x ) = m(tj , x )

8 j (x ) M 8 j (x )| |M

( ∈ L) .

The well-posedness of the Scheme 3.3 easily follows from the Lax-Milgram 8 j (x )|} for theorem for k ≤ k0 (κC ) sufficiently small, and positivity of {|M h ≤ h0 sufficiently small. If we combine both steps, we obtain the following

208

Chapter 3

Effective equations for macrospin magnetization dynamics

8 j : O → R3 , equation for the iterates M 

  m(tj−1 , ·)  8 j−1  1  8 j , ∇Φ Φ 1− M , Φ + α m(tj , ·)∇M 8 j−1 | k h |M n    8j Φ  8 j−1 × ∂ M , ∂Φ 8 j , ∇m(tj , ·) ⊗ Φ − (3.21) M = α ∇M ∂xp ∂xp p=1    α    1 8 j−1 |2 − Δm2 (tj , ·) + κ(tj , ·) M 8j , Φ . |∇M + m(tj , ·) 2 

8j , Φ dt M

+ h

It is due to the second term that the projection Scheme 3.3 may be considered as a semi-explicit penalization method. The following theorem asserts optimal convergence rates for Scheme 3.3 on quasi-uniform meshes Th when a strong solution of (3.12) exists, which is true at least locally by Theorem 3.2. Its proof uses an inductive argument, which has been developed in [98, Sections 4.2 & 4.3] (n = 2), and used in [41] (n = 3) for the case m ≡ 1. We skip details since arguments from [98, Chapter 4] may easily be adopted to the present case m > 0. Theorem 3.4. Let O ⊂ R2 be a bounded convex polygonal domain, let m0 ∈ W2,2 (O, R3 ) be such that |m0 | = m(0, ·) on O, and assume that (A1) is valid. Let T = tJ > 0 be such that the strong solution of (3.12) exists. Let Ik = {tj }Jj=0 be an equi-distant mesh which covers [0, T ], and Th be a quasi-uniform triangulation of O. Assume that numerical parameters are sufficiently small, i.e., +     k ≤ k0 T, α, mC2 , +m−1 C , h ≤ h0 T, α, mC2 , m−1 C , (3.22)   and √1k = o h1 . Suppose m0 − M0 W1,2 ≤ C(k + h). There exists a constant   C ≡ C T, α, mC2 , m−1 C > 0 independent of k, h > 0, such that iterates 8 j } ⊂ Vh of Scheme 3.3 satisfy {M J   1/2 j 8 8 j 2 1,2 m(tj , ·) − M ≤ C(k + h) . max m(tj , ·) − M L2 + k W

1≤j≤J

Moreover,

j=1

+ + 8 j |+ 2 ≤ C(k + h) . max +m(tj , ·) − |M L

1≤j≤J

This result asserts the convergence with rates for iterates of Scheme 3.3 towards the strong solution of (3.12) in the case of sufficiently small discretization parameters which satisfy the (non-critical) mesh constraint below (3.22). We remark that no discrete energy identity is known to hold for these iterates, which

Section 3.3

209

Construction of a weak solutions via Scheme 3.5

is why an inductive perturbation argument is employed instead to transfer stability properties of the strong solution of (3.12) to the solution {Mj }Jj=0 ⊂ Vh ; see [98, Chapter 4] for details. Consequently, it is not clear whether the scheme remains stable in situations when strong solutions of (3.12) loose their regularity properties, and the above inductive arguments cease to hold. A different approach is pursued with Scheme 3.5 in the following section where weak solutions of (3.5) for n ≤ 3 are constructed in a practical manner. Its relevancy is apparent for non-regular solutions of (3.5): here, iterates of Scheme 3.5 satisfy a discrete energy bound, and convergence (without rates) towards a weak solution of (3.5) holds for vanishing discretization parameters.

3.3

Construction of a weak solutions via Scheme 3.5

Let O ⊂ Rn , for n ≤ 3 be a bounded polyhedral domain, and suppose that assumption (A1) on p. 200 holds. First we consider a reformulation of problem (3.5) which goes back to Gilbert for the case m ≡ 1; see e.g. [81]. As in (3.15), we take the cross product of (3.12) with m, and hence restate (3.5)1 (with Heff = Δm) as mt − κ m − (1 + α2 ) m × Δm +

α m × mt = 0 m ∂m =0 ∂n m(0, ·) = m0

on OT , on (0, T ) × ∂O , on O .

(3.23)

We introduce the notion of a weak solution of problem (3.23), which generalizes the concept of a strong solution introduced in Section 3.1. Definition 3.9. Suppose (A1). Let us assume that a map m0 ∈ W1,2 (O, R3 ) is such that |m0 | = m(0, ·) Leb-a.e. in O. A mapping m : OT → R3 is called a weak solution of (3.23), if and only if     (i) m ∈ L∞ 0, T ; W1,2 (O, R3 ) , mt ∈ L2 0, T ; L2 (O, R3 ) , and lim m(t, ·) − m0 L2 = 0 .

t→0

(ii) |m| = m is valid Leb-a.e. in OT ,   (iii) for all φ ∈ C∞ OT , R3 holds

 OT

n      φ! α ∂m ∂φ m×mt , φ dxds = 0 . , mt , φ −κ m, φ +(1+α2 ) m× + ∂xi ∂xi m i=1

210

Chapter 3

Effective equations for macrospin magnetization dynamics

(iv) the following energy inequality is valid for almost all 0 ≤ t ≤ T , t     α mt (s, ·) 2 E m(t, ·) + L2 ds ≤ C 1 + E(m0 ) . = 2 1+α 0 m(s, ·)   Note that (i) implies m ∈ C [0, T ]; L2 (O, R3 ) . Below we often drop the explicit dependence of functions on x ∈ O. In the remainder of this section, we will construct a weak solution of (3.23) by using a practical finite element based discretization. We refer to Subsection 2.1.1 in Chapter 2 for the used notation. Moreover, let mj := m(tj ), and κj := κ(tj ). Scheme 3.5. Let M0 ∈ Vh . For j = 0, . . . , J − 1 find Mj+1 ∈ Vh such that       Mj j+1 j+1 j+1/2 j+1 Φ Φ Φ dt M , h − κ M , +α × dt M , h mj+1/2 h   2 j+1/2 j+1  = (1 + α ) M × Δh M , Φ ∀ Φ ∈ Vh . h

Remark 3.10. 1. The linear third term is motivated by the identity   k j j+1 j+1/2 j+1 = M − dt M M × dt M × dt Mj+1 = Mj+1/2 × dt Mj+1 . 2 2. If compared to Scheme 3.3, the midpoint rule is used for the temporal discretization, and the reduced integration for the spatial discretization. Moreh is used. These modifications allow for a discrete over, the discrete Laplacian Δ energy estimate as stated in Lemma 3.12. 3. The formulation of Scheme 3.5 is not convenient for an error analysis to verify some rates in the presence of strong solutions for (3.23), which motivates to restate the scheme in a way which lead to (3.12). This strategy leads to a perturbed version of a discretization of (3.12), which is due to time-averaged quantities in Scheme 3.5. It is not clear whether rates of convergence may be obtained with methods similar to those proposed in Section 3.2. First, we verify the existence of a solution to Scheme 3.5. Lemma 3.11. Let T = tJ > 0, and suppose (A1). Then there exists k0 ≡ J−1

k0 (T, κC0 ) such that for all k ≤ k0 there exists Mj+1 j=0 ⊂ Vh which solves Scheme 3.5. Proof. Assume that j = 0, . . . J − 1, and Mj ∈ Vh is given. We define a continuous functional F : Vh → Vh by setting for R ∈ Vh     2 2α j R − Mj − I h κj+1 R − F(R) = R × R − M j+1/2 k km  h (2R − Mj ) . +(1 + α2 ) R × Δ

Section 3.3

211

Construction of a weak solutions via Scheme 3.5

For k −1 > maxOT κ, and all R ∈ Vh such that (1 − k maxOT κ)Rh ≥ Mj L2 , we have      2 R2h − (Mj , R)h − κj+1 R, R = F(R), R k h h   2 j Rh (1 − k max κ)Rh − M h ≥ k OT ≥ 0. Hence, the Brouwer fixed point theorem implies the existence of R∗ ∈ Vh such that F(R∗ ) = 0. Then, Mj+1 := 2R∗ − Mj solves Scheme 3.5. Next, we study the stability properties of Scheme 3.5. Lemma 3.12. Let the assumption of Lemma 3.11 be fulfilled. Then there exists a constant C ≡ C T, α, mC2 , m−1 C , κC2 ) > 0 such that   (i) max max |Mj (x )|2 + max max dt |Mj (x )|2  ≤ C , 1≤j≤J ∈L

(ii) (iii)

max

1≤j≤J

k

J 

1≤j≤J ∈L

∇Mj 2L2

+k

2

J 

∇dt Mj 2L2

+k

j=1

J−1 

dt Mj 2h ≤ C ,

j=1

h Mj+1/2 2 ≤ C . Mj+1/2 × Δ h

j=1

Proof. Step. 1. To prove assertion (i), let us fix  ∈ L, choose Φ = Mj+1/2 (x )ϕ in Scheme 3.5, and observe statement 1. of Remark 3.10, to find that dt |Mj+1 (x )|2 = 2κj+1 (x )|Mj+1/2 (x )|2   ≤ C |Mj+1 (x )|2 + |Mj (x )|2 .

(3.24)

Sum over j, restrict to k ≤ k0 sufficiently small, and apply the discrete version of the Gronwall lemma to conclude that max max |Mj (x )|2 ≤ C .

0≤j≤J ∈L

(3.25)

The remaining parts of assertion (i) now follow from the last inequality and (3.24). h Mj+1 in Step 2. In order to verify claim (ii), let us first choose Φ = −Δ Scheme 3.5 and estimate       h Mj+1 h Mj+1 = − I h κj+1 Mj+1/2 , −Δ − κj+1 Mj+1/2 , −Δ h h + + + +  j+1 j+1 +2 j +2 + + ≥ −Cκ C ∇M + ∇M L2 L2     − Mj+1/2 ⊗ ∇κj+1 , ∇Mj+1  + I ,

212

Chapter 3

Effective equations for macrospin magnetization dynamics

with the error term I, which can be controlled by the standard interpolation estimate on each K ∈ Th (see (2.6) in Chapter 2), and ∇2 Mj+1 K = 0 for all K ∈ Th ,      I := ∇ Id − I h κj+1 Mj+1/2 , ∇Mj+1 ≤

Ch κj+1 C2 Mj+1/2 W1,2 Mj+1 W1,2 .

Hence, for some constant C ≡ C(α, κC2 ) > 0, we have    1 k dt ∇Mj+1 2L2 + ∇dt Mj+1 2L2 ≤ C 1 + h) Mj+1 2W1,2 + Mj 2W1,2 2 2   Mj j+1  j+1 × dt M , Δh M . (3.26) +α mj+1/2 h   j+1 Next, we choose Φ = I h dmt M in Scheme 3.5, and get j+1/2 + α + + dt Mj+1 +2 √ + + = 1 + α2 mj+1/2 h

  j+1/2 α j+1 M j+1 κ , dt M (3.27) 1 + α2 mj+1/2 h   Mj+1/2  × Δh Mj+1 , dt Mj+1 . +α mj+1/2 h

Adding (3.26) and (3.27), and using the first statement in Remark 3.10, (3.25), and the Young inequality, we arrive at + +2   1 1 α + dt Mj+1 + +√ + ≤ C(1+h) 1+Mj+1 2 1,2 +Mj 2 1,2 . dt ∇Mj+1 2L2 + W W + + 2 2 1 + α2 mj+1/2 h (3.28) Assertion (ii) then follows as before by applying the discrete version of the Gronwall lemma for sufficiently small time steps k ≤ k0 .   h Mj+1 Step 3. Assertion (iii) now follows from choosing Φ = I h Mj+1/2 × Δ in Scheme 3.5, and assertions (i), (ii). Equality (3.24) is the discrete version of property (3.8) for iterates at every node x ∈ Eh . Hence, we can only expect that |Mj (x )|2 ≈ m2 (tj , x ) for all 1 ≤ j ≤ J, and all  ∈ L. We use Definition 2.4 in Chapter 2 to introduce the piecewise affine interpolation M ≡ M k,h ∈ C [0, T ]; Vh of the iterates {Mj }Jj=0 . Assertion (ii) of Lemma 3.12 may then be rewritten as follows, M+ (t)2L2 + k sup ∇M

0≤t≤T



T 0

Mt (s)2L2 ds + ∇M

α 1 + α2



T 0

Mt (s)2h ds ≤ C . M (3.29)

Section 3.3

Construction of a weak solutions via Scheme 3.5

213

By a compactness argument we infer that there exist a function m ∈ W1,2 (OT ; Mk,h } ⊂ C [0, T ]; Vh such that for R3 ), and a subsequence (not relabeled) {M k, h → 0   ∗ M k,h , M ± in L∞ 0, T ; W1,2 (O; R3 ) , k,h , M k,h  m in W1,2 (OT ; R3 ) ,

M k,h  m ± M k,h , M k,h , M k,h → m

2

(3.30)

3

in L (OT ; R ) .

In the above, we use the Aubin-Lions compactness lemma to obtain (3.30)3 , and Lemma 3.12, (ii) to get the corresponding results for M ± k,h and M k,h . Dropping the index (k, h) in M k,h , Scheme 3.5 may be rewritten in the following form: For Φ (t) := I hφ (t) and φ ∈ C∞ (OT ; R3 ) one has

T  0

M t − κ+ M + α

    M−  hM + , Φ × M t , Φ − (1 + α2 ) M × Δ ds = 0 . m h h (3.31)

Theorem 3.6. Let the assumption of Lemma 3.12 be satisfied, Th be a regular triangulation of the bounded Lipschitz domain O ⊂ Rn for n = 1, 2, 3, and limh→0 M0 − m0 W1,2 = 0. For every k, h > 0, let M k,h : OT → R3 denote the piecewise affine interpolation of an {Mj ;  j ≥ 0} which solves Scheme 3.5. There exists a convergent subsequence M k,h to a function m : OT → R3 in W1,2 (OT ; R3 ) which is a weak solution of (3.23). Proof. Step 1. Verification of conditions (i), (iii) of Definition 3.9. By (2.10) in Chapter 2, we have for almost all t ∈ [0, T ] that    M t , Φ ) − (M Mt , Φ )h  ≤ Ch M Φ L 2 Mt L2 ∇Φ φ ∈ Vh . This estimate, (3.30)2 , and the approximation property (2.6) in Chapter 2 for the Lagrange interpolation yields

T 0

Mt , Φ )h ds → (M

T 0

(mt , φ ) ds

(k, h → 0) .

(3.32)

Similarly,

T 0

(κ+M , Φ )h ds →



T 0

(κm, φ ) ds

(k, h → 0) .

(3.33)

By (2.10), (3.29), and Lemma 3.12, (i) we infer that

T 0

 M × M t , Φ h ds → m



T 0

 m × mt , φ ds m

(k, h → 0) .

(3.34)

214

Chapter 3

Effective equations for macrospin magnetization dynamics

The last term in (3.31) requires a more detailed study. We write      hM + , Φ = M × Φ , Δ  hM + − M×Δ h   h  + M × Φ ), ∇M M − ∇[M M × Φ ], ∇M M+ =: I − II . = ∇[Id − I h ](M  Since ∇2M K = 0 for every K ∈ Th , an approximation property of I h yields the estimate  M×Φ Φ)L2 (K) ∇M M+ L2 (K) ≤ Ch ∇M ML2 Φ ΦW2,∞ ∇M M + L 2 . I ≤ Ch ∇2 (M K∈Th

For the second term, we use a × b, a = 0, and the identity M (s) = M + (s) − k / Ik to verify 2 M t (s) for times s ∈ n n   Mt M +   Φ ∂M M+  ∂M ∂Φ k  ∂M M×Φ Φ], ∇M M+ = − Φ, II = ∇[M ×Φ M× , + . 2 ∂xp ∂xp ∂xp ∂xp p=1

p=1

The integral which corresponds to the leading term on the right-hand side vanishes for k, h → 0 because of (3.29),  k T  ∂M 1/2  k2 T Mt M+   ∂M  Mt 2L2 ds × Φ , ∇M · ds ≤   2 0 ∂xp ∂xp 2 0 M+ L2 (OT ) → 0 ΦC0 (OT ) ∇M (k → 0) . ·Φ Putting things together then yields for k, h → 0 that T n T    φ ∂m  ∂φ +  M × ΔhM , Φ h ds → − , m× ds . ∂xp ∂xp 0 0

(3.35)

p=1

We may now insert (3.32)–(3.35) into (3.31) to verify property (iii) of Definition 3.9 for m : OT → R3 . +2 + M(s)|2 − m2 (s)+L2 → 0 for k, h → 0. Step 2. Verification of sups∈[0,T ] +|M By the estimate (2.6) for the Lagrange interpolation, assertions (i) and (ii) in Lemma 3.12, and m ∈ C2 (OT ; R+ \ {0}), +2 + M(s)|2 − m2 (s)+L2 sup +|M s∈[0,T ]

+   + +2 + M(s)|2 − m2 (s) + M(s)|2 − m2 (s) − Ih |M ≤ 2 sup + |M 2 s∈[0,T ]

+  + + +2 M(s)|2 − m2 (s) + +C sup +Ih |M s∈[0,T ]

  ≤ C h2 + Ik,h ,

h

L

(3.36)

Section 3.3

215

Construction of a weak solutions via Scheme 3.5

where for ζ = O ϕx dx, and by (2.8), +    2 + +2 + M(s)|2 − m2 (s) + = sup M(s, x )|2 − m2 (s, x ) . ζ |M Ik,h := sup +Ih |M h

s∈[0,T ]

s∈[0,T ] ∈L

Step 2A. Ik,h → 0 for k, h → 0. For this aim we use the identity Mj+1/2 (x ) = Mj+1 (x ) − k2 dt Mj+1 (x ) in (3.24) to obtain for all  ∈ L,     dt |Mj+1 (x )|2 − |mj+1 (x )|2 = 2κj+1 (x ) |Mj+1 (x )|2 − |mj+1 (x )|2 + IIj, , (3.37) where IIj, = −dt |mj+1 (x )|2 + 2κj+1 (x )|mj+1 (x )|2   ! k2 j+1 j+1 2 j+1 j+1 |dt M (x )| − k M (x ), dt M (x ) +κ (x ) . 2 Note that

! k k2 −k Mj+1 (x ), dt Mj+1 (x ) = dt |Mj+1 (x )|2 + |dt Mj+1 (x )|2 . 2 2

(3.38)

A simple consequence of m ∈ C2 (OT , R+ \ {0}), Taylor’s formula, and (3.8) is that for all  ∈ L,     sup dt |mj+1 (x )|2 − 2κj+1 (x )|mj+1 (x )|2  → 0 (k → 0) . 1≤j≤J

Hence, by assertions (i) and (ii) of Lemma 3.12, we infer that k

J  

ζ |IIj, | → 0

(k, h → 0) .

(3.39)

j=0 ∈L

  We now multiply (3.37) by ζ |Mj+1 (x )|2 − |mj+1 (x )|2 and sum over all  ∈ L. Because of Lemma 3.12, (i), we obtain for some C > 0 that +   +2 +2   1 + dt +Ih |Mj+1 |2 − |mj+1 |2 +h ≤ C +Ih |Mj+1 |2 − |mj+1 |2 +h + ζ |IIj, | . 2

∈L

Summing over all iteration steps, (2.9), and using the discrete version of the Gronwall lemma yields that for sufficiently small k ≤ k0 that +  +  +2 + +Ih |M0 |2 − |m0 |2 +2 2 sup +Ih |Mj |2 − |mj |2 +L2 ≤ L 0≤j≤J

+k

J  

   ζ |IIj, | exp CtJ

j=0 ∈L

→ 0

(k, h → 0) .

216

Chapter 3

Effective equations for macrospin magnetization dynamics

M(·, x )|;  ∈ L} to all s ∈ [0, T ]: by using We may extend the supremum for {|M j+1 the identity M (s, ·) = M (s) + (s − tj+1 )dt Mj+1 (s) for all s ∈ [tj , tj+1 ), (3.38), and assertions (i), (ii) of Lemma 3.12, we have +   M(s)|2 − |m(s)|2 2L2 → 0 (k, h → 0) . sup +Ih |M s∈[0,T ]

Step 3. Verification of (ii) of Definition 3.9. Since M k,h → m in L2 (OT ; R3 ) for k, h → 0 by (3.30)2 , there exists a subsequence M k ,h → m Leb-a.e. in OT for k , h → 0. By the Lebesgue dominated convergence theorem and Lemma 3.12, we infer that + + + +2 +|m|2 − m2 +2 2 +|M Mk ,h |2 − m2 +L2 (O ) = 0 . =  lim L (O )  T

k ,h →0

T

This concludes the verification of condition (ii) in Definition 3.9.

3.3.1

Solving the nonlinear system in Scheme 3.5

We use the Brouwer fixed point theorem to show solvability of Scheme 3.5 in Lemma 3.11. However, a practical version of Scheme 3.5 requires an iterative solution of the arising nonlinear algebraic systems. In this subsection, we discuss a simple fixed point iterative algorithm which amounts to solving a sequence of linear problems (labeled by r ≥ 0) in each iteration step indexed by j ≥ 0, and gives a stopping criterion to step over to the following one. A key property of iterates from Algorithm 3.13 is that for every iteration level j ≥ 0, these iterates are uniformly bounded with respect to r ≥ 0. Consequently, there exists a convergent subsequence by the Bolzano-Weierstraß theorem, and the stopping criterion as stated below in (iii) will be met to terminate level j ≥ 0. A combination with Theorem 3.6 then shows that the iterates from this linear scheme may be used to construct a weak solution of (3.23). In the following Algorithm 3.13, we will solve the nonlinear system for Ξ := Mj+1/2 , Ξ − Mj ). After and hence replace the discrete time derivative dt Mj+1 by k2 (Ξ   h (2Ξ Ξ − Mj ), Φ h , and using the a linearization of the nonlinear term Ξ × Δ 2 j j+1 j j+1/2 = kM × M we obtain the following algorithm for identity M × dt M fixed k, h, ε > 0. Algorithm 3.13. (i) For j ≥ 0, set Ξ j+1,0 := Mj , and r := 0. (ii) Compute Ξ j+1,r+1 ∈ Vh such that for all Φ ∈ Vh holds   2 j+1,r+1 Mj 2α j+1,r+1 Ξ (Ξ , Φ )h − (κj+1Ξ j+1,r+1 , Φ )h + × Ξ , Φ k k mj+1/2 h   2 2 j+1,r+1 j+1,r j j h (2Ξ Ξ −(1 + α ) Ξ ×Δ − M ), Φ = (M , Φ )h . (3.40) k h

Section 3.3

217

Construction of a weak solutions via Scheme 3.5

(iii) Set

  h Ξ j+1,r − Ξ j+1,r+1 . Rj+1 := 2Δ

Ξj+1,r+1 − Mj , and go to (iv). Otherwise Stop if Rj+1 h ≤ ε, set Mj+1 := 2Ξ set r ← r + 1 and continue with (ii). (iv) Stop if j + 1 = J; otherwise set j ← j + 1 and go to (i). (iv) Set r ← r + 1 and go to (i). This algorithm generalizes a corresponding strategy for m ≡ 1 in [17]. The following theorem shows that all steps in Algorithm 3.13 are well-defined, that the algorithm terminates, and that iterates converge to a weak solution of (3.23). The key tool for a verification of the last property is Theorem 3.6. Theorem 3.7. Fix T = tJ > 0, let the assumptions of Theorem 3.6 be satisfied, 2 and choose k < min0≤j≤J−1 κj+1 , and ε, h > 0. For all 0 ≤ j ≤ J −1, and all C r ≥ 0, there exists a unique solution Ξ j+1,r+1 ∈ Vh of problem (3.40). Moreover, Ξj,r ; r ≥ 0} ⊂ Vh for each j ≥ 0 there exists a subsequence (not relabeled) {Ξ j such that the stopping criterion is met, and iterates {M ; j ≥ 0} ⊂ Vh satisfy the stability properties in Lemma 3.12. Finally, there exists a subsequence of iterates from Algorithm 3.13 which converges to a weak solution of (3.23) for k, h, ε → 0, in a sense which is made precise in Theorem 3.6. Proof. Step 1. Well-posedness and convergence. Fix 0 ≤ j ∗ ≤ J − 1. For every r ≥ 0, the left-hand side of (3.40) defines a continuous bilinear form ∗ Ξj +1,r+1 , Φ ) = k2 (Mj , Φ ) for ar : Vh × Vh → R, such that (3.40) reads as ar (Ξ ∗ all Φ ∈ Vh . Taking Φ = Ξ j +1,r+1 in (3.40), and restricting to k < κj ∗2+1  we C infer that  ∗  2  j ∗ +1,r+1 2 ∗ ∗ Ξ − κj +1 C Ξ h > 0 ar Ξ j +1,r+1 , Ξ j +1,r+1 ≥ k

(r ≥ 0) .

Hence, by the Lax-Milgram theorem we infer the unique solvability of problem

≡ C(t

J , mC2 ) > 0 (3.40) for every r ≥ 0. Suppose that for some constant C one has

. sup sup |Mj (x )| ≤ C (3.41) 0≤j≤j ∗ ∈L

Fix  ∈ L, r ≥ 0, and choose Φ = Ξj Ξj |Ξ

∗ +1,r+1

(x )| ≤

∗ +1,r+1

(x )ϕx in (3.40) to conclude that

1 1−

k j ∗ +1  C 2 κ

∗

|Mj (x )|

∀r ≥ 0.

(3.42)

By the Bolzano-Weierstraß theorem, we infer that there exists a convergent ∗ Ξj +1,r ; r ≥ 0}, and stopping criterion (iii) will subsequence (not relabeled) {Ξ

218

Chapter 3

Effective equations for macrospin magnetization dynamics ∗

∗

∗

∗

Ξj +1,r +1 − Mj be reached for some r∗ ≡ r∗ (j ∗ , h) ≥ 1. Then, Mj +1 = 2Ξ solves   ∗  1  j ∗ +1 ∗ ∗ M − Mj , Φ h − κj +1 Mj +1/2 , Φ h k   Mj 2α j ∗ +1/2 − ×M ,Φ (3.43) k mj+1/2 h  ∗  ∗ ∗ ∗ h (2Ξ Ξj +1,r − Mj ), Φ = 0 −(1 + α2 ) Mj +1/2 × Δ ∀ Φ ∈ Vh . h

Fix  ∈ L, and choose Φ = M

j ∗ +1/2

(x )ϕx . We obtain

 1   1  j ∗ +1 ∗ ∗ ∗ ∗ |M (x )|2 − |Mj +1 (x )|2 ≤ κj +1 L∞ |Mj +1 (x )|2 + |Mj (x )|2 . k 2 This inequality holds for all 0 ≤ j ≤ J − 1; similar to (3.24), this establishes

≡ C(t

J , mC2 ) > 0. that (3.41) is valid with a constant C Step 2. Convergence towards a weak solution of (3.23). Fix j ≥ 0, and let r∗ ≡ r∗ (j, h) be the finite number when the stopping criterion is met. According to equation (3.43), the iterates {Mj }Jj=0 solve the following perturbed version of Scheme 3.5, 



 j+1 j+1/2   Mj  − κ M , Φ + α × dt Mj+1 , Φ h (3.44) h h j+1/2 m     h Mj+1 , Φ = (1 + α2 ) Mj+1/2 × Rj+1 , Φ −(1 + α2 ) Mj+1/2 × Δ h

dt Mj+1 , Φ

h

  h Ξ j+1,r∗ (j) − Mj+1/2 . A slightly modifor all Φ ∈ Vh , with Rj+1 = 2Δ fied argument shows that the properties in Lemma 3.12 remain valid also for {Mj }Jj=0 , as is detailed in Step 3 below. Properties (i) and (iii) in Definition 3.9 of a weak solution of (3.23) follow as in Step 1 of the proof of Theorem 3.6; property (ii) in Definition 3.9 follows as in Steps 2 and 3 of the proof of Theorem 3.6.

Step 3. Stability results (i) – (iii) of Lemma 3.12 remain valid for {Mj }Jj=0 . Assertion (i) is immediate, while assertion (ii) follows from a slightly modified argument as in Step 2 of the proof of Lemma 3.12. But now the additional term   h Mj+1 , Rj+1 (1 + α2 ) Mj+1/2 × Δ h appears on the right-hand side of (3.26), as well as   −(1 + α2 ) Mj+1/2 × dt Mj+1 , Rj+1 h

 on the right-hand side of (3.27). Furthermore, choosing Φ = I h Mj+1/2 ×  h Mj+1 in (3.44) leads to the estimate Δ   h Mj+1 2 ≤ C dt Mj+1 2 + Rj+1 2 , Mj+1/2 × Δ h h h

Section 3.3

219

Construction of a weak solutions via Scheme 3.5

  with some constant C ≡ C m−1 C , α > 0. Let δ1 , δ2 > 0. These considerations then lead to the following inequality which corresponds to inequality (3.28), 1 1 α dt Mj+1 2 √ dt ∇Mj+1 2L2 +   2 2 1 + α2 mj+1/2     ≤ C 1 + h 1 + Mj+1 2W1,2 + Mj 2W1,2 h Mj+1 2 + δ2 dt Mj+1 2 +δ1 Mj+1/2 × Δ h h (1 + α2 )2  1 1  j+1 2 R h + + 4 δ1 δ2     ≤ C 1 + h 1 + Mj+1 2W1,2 + Mj 2W1,2 + C(δ1 + δ2 )dt Mj+1 2h  (1 + α2 )2 1 1  ( + ) Rj+1 2h . + Cδ1 + 4 δ1 δ2  −1  We may now choose δi ≡ δi m C , α > 0 sufficiently small to absorb the last but one term on the right-hand side on the left-hand side. By the discrete version of the Gronwall lemma we obtain for k ≤ k0 sufficiently small the estimate max Mj 2W1,2 +

1≤j≤J

  1 α j 2 j+1/2 h Mj+1 2 k M  + M × Δ d t h h 4 1 + α2 J

j=1



≤ M0 2W1,2 + Ck

J 

   Rj+1 2h exp CtJ .

j=1

Remark 3.14. 1. If m ≡ 1, it is easy to verify that the iterates Mj+1,r+1 := Ξj+1,r+1 − Mj then satisfy |Mj+1,r+1 (x )| = 1 for all  ∈ L. 2Ξ 2. Algorithm 3.13 is a constructive method to solve Scheme 3.5, and replace an argument based on the Brouwer fixed point theorem in the proof of Lemma 3.11. 2 3. Fix j ≥  0. Let Th be quasi-uniform, and k ≤ Ch for some C ≡ C α, tJ , κC1 > 0 sufficiently large. Then there exists Θ < 1 such that Ξj+1,r+1 − Ξ j+1,r h ≤ Θ Ξ Ξj+1,r − Ξ j+1,r−1 h Ξ

(r ≥ 1) .

(3.45)

By the Banach fixed point theorem, (3.45) implies the existence of a unique sequence {Mj }Jj=0 ⊂ Vh which solves Scheme 3.5. In order to prove (3.45), let us subtract two subsequent iterations of (3.40). ∗ ∗ ∗ Set Φ = Ej +1,r+1 := Ξ j +1,r+1 − Ξ j +1,r , and use (3.42) together with the

220

Chapter 3

Effective equations for macrospin magnetization dynamics

h Ej ∗ +1,r L2 ≤ Ch−2 Ej ∗ +1,r h (see the proof of Lemma 2.3) inverse estimate Δ to conclude that    ∗   2−kκj +1 C j ∗ +1,r+1 2 ≤ 2(1 + α2 ) Ξ j ∗ +1,r × Δ ˜ h Ej ∗ +1,r , Ej ∗ +1,r+1  E  h k h ∗

∗

∗

Ξj +1,r L∞ Ej +1,r h Ej +1,r+1 h ≤ (1 + α2 )Ch−2 Ξ

−2 Ej ∗ +1,r h Ej ∗ +1,r+1 h . ≤ (1 + α2 )2CCh 2



2(1+α )CC k Hence, (3.45) holds for time steps k such that 2−kκ j+1  h2 < 1. C 4. Let m ≡ 1. In Lemma 2.22, the existence of a discrete Markov chain M j }Jj=0 which solves Scheme 2.9 is proved by the Brouwer fixed point theorem; {M see also Theorem 2.14 for Scheme 2.10. Similar to Algorithm 3.13, it is possible to set up an implementable simple fixed point method with a pathwise stopping criterion for Schemes 2.9 and 2.10.

3.4

Computational experiments

We perform numerical experiments for the augmented Landau-Lifshitz equation (3.5) and its Gilbert reformulation (3.23). In the first part we study a modification of the standard μMag benchmark problem [93, no. 4]. In the second part we examine how well simulations from the SLLG equation (2.1) coincide with those for the macroscopic model (3.5).

3.4.1

μMag standard problem no. 4 with thermal effects

Dimensionless model with the Maxwell’s equations We briefly describe how the numerical schemes described before can be generalized to models which include Maxwell’s equation. Equation (3.23) with a general effective field takes the following dimensionless form α (3.46) mt = κ m + (1 + α2 ) m × Heff − (m × mt ) on OT , m 2

where Heff = AΔm + Ke, me + Hdem + Hext , κ = κ  1+α γm ˜ 0 , and the time is measured in dimensionless units of

1+α2 γm 0

seconds. The dimensionless exchange and

anisotropy constants A and K are related to material parameters as A =

 2A , μ0 m ˜ 20

  K,  γ are respectively, the saturation magnetization K = μ2K , where m  0 , A,  20 0m at zero temperature, the exchange constant, the uniaxial anisotropy constant, and the gyromagnetic ratio. The demagnetizing field Hdem is usually expressed in terms of a scalar mag → R, i.e. Hdem = ∇u : O  → R3 , where netic potential u : O

(∇u, ∇φ) = (m, ∇φ)

 R3 ) . ∀φ ∈ W1,2 (O;

(3.47)

Section 3.4

Computational experiments

221

Hence, one has to solve the system of coupled equations (3.46),(3.47). The  see  ⊂ Rn (n = 2, 3) has to be taken sufficiently large and O  O; domain O e.g. [10]. A generalization of Scheme 3.5 from Section 3.3 for the system (3.46), (3.47)  h ⊂ W1,2 (O;  R), while Vh ⊂ W1,2 (O; R3 ). is given below. We denote V   Scheme 3.8. Let M0 ∈ Vh . For j = 0, 1, 2, ..., J − 1 find Mj+1 , U j+1 ∈  h such that for all Φ × Φ ∈ Vh × V  h, Vh × V 

(∇U j+1/2 , ∇Φ) = (Mj+1/2 , ∇Φ) ,  (3.48) dt Mj+1 , Φ h − κj+1 Mj+1/2 , Φ h     Mj j+1/2 = +α × dt Mj+1/2 , Φ , −(1 + α2 ) Mj+1/2 × Heff , Φ h mj+1/2 h 

j+1/2

where Heff



h Mj+1/2 + Ke, Mj+1/2 e + ∇U j+1/2 + Hext . = AΔ

Solving the nonlinear system We extend Algorithm 3.13 to solve the nonlinear system (3.48) in Scheme 3.8 for Ξ j+1 := Mj+1/2 , and W j+1 := U j+1/2 . For fixed k, h, ε > 0 we obtain the following generalized fixed-point algorithm. Algorithm 3.15. (i) For j ≥ 0, set Ξ j+1,0 := Mj , W j+1,0 := U j−1/2 , and r := 0.    h , such that for all Φ ×Φ ∈ Vh × V  h, (ii) Compute Ξ j+1,r+1 , W j+1,r+1 ∈ Vh × V Ξj+1,r , ∇Φ) , (∇W j+1,r+1 , ∇Φ) = (Ξ 2 j+1,r+1 Ξ (Ξ , Φ )h − (κj+1Ξ j+1,r+1 , Φ )h − k   −(1 + α2 ) Ξ j+1r+1 × Hj+1,r , Φ + eff h   Mj 2 2α × Ξ j+1,r+1 , Φ = (Mj , Φ )h , + k mj+1/2 k h

(3.49)

hΞ j+1,r + Ke, Ξ j+1,r e + ∇W j+1,r+1 + Hext . where Hj+1,r := AΔ eff (iii) Once h (Ξ Ξj+1,r+1 − Ξj+1,r )||h ≤ ε , ||Δ Ξj+1,r+1 − Mj , U j+1/2 := W j+1,r+1 , and go to (iv). Otherwise set Mj+1 := 2Ξ set r ← r + 1 and go to (ii).

222

Chapter 3

Effective equations for macrospin magnetization dynamics

(iv) Stop if j + 1 = J; otherwise set j ← j + 1 and go to (i). Another discretization is Scheme 2.8 on p. 130, which is easily generalized to the augmented LLG equation (3.5).   Scheme 3.9. Let M0 ∈ Vh . For j = 0, 1, 2, ..., J − 1 find Mj+1 , U j+1 ∈    h such that for all Φ, Φ ∈ Vh × V  h, Vh × V (∇U j+1/2 , ∇Φ) = (Mj+1/2 , ∇Φ) ,       j+1/2 dt Mj+1 , Φ h − κj+1 Mj+1/2 , Φ − Mj+1/2 × Heff , Φ = −α j+1/2

where Heff

 Mj+1/2 mj+1/2

h

 j+1/2 × [Mj+1/2 × Heff ], Φ ,

h

h

is defined as before.

μMag standard problem no. 4 In our first experiment we compute the μMag standard problem no. 4, see [93], with m  0 ≡ 8 × 105 , i.e. we neglect the temperature effects in the model.  = [(−1.75, 2.25) × (−1.9375, 206) × The computations are done in R3 for O −6 (−1.536, 1.536)] × 10 , and O = [(0, 0.5) × (0, 0125) × (0, 0.003)] × 10−6 . The initial condition is an S-state, see Figure 3.3. 2D cuts at x3 = 0 through the

 at x3 = 0 (left) and zoom at the mesh for the Figure 3.2. Mesh for the domain O domain O at x3 = 0 (right).

 and O are depicted finite element meshes for the computational domains O in Figure 3.2. The remaining parameters for the computation were: A∗ = 1.3 × 10−11 , K ∗ = 0, α = 0.02, γ = 2.211 × 105 , μ0 = 1.25667 × 10−9 , μ0 m  0 Hext = (24.6, 4.3, 0)×10−3 , k = 0.02, h = 0.00390625×10−6 . In Figure 3.4 we compare the first two components of the spatially averaged magnetization from Scheme 3.5 and Algorithm 3.15 (with κ ≡ 0), and the results from [46].

Section 3.4

223

Computational experiments

The results are in good agreement, in particular the graphs for Scheme 3.9 and Algorithm 3.15 are almost indistinguishable.

Figure 3.3. S-state initial condition for the magnetization.

1

mx cross my cross mx d’Aquino et al. my d’Aquino et al. mx Gilbert my Gilbert

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

Figure 3.4. Evolutions tj →

20

40

1 Leb[O]

60

O

80

100

120

140

160

180

(Mj )1 dx and tj →

1 Leb[O]

O

(Mj )2 dx.

Thermally assisted switching The next experiment demonstrates the temperature effects on the switching. We take K ∗ = 500, e = (1, 0, 0), μ0 m ˜ 0 Hext = (−14, 2.4, 0) × 10−3 , and the remaining parameters for the computations are the same as in the previous experiment. We consider the power-law (3.6) with β = 0.5, τC = 1.5, and express κ according to (3.11). First, the sample is quickly heated and the switching field Hext is applied which causes the magnetization to switch to the opposite direction. At time t = 160 the external field is turned off, i.e. for t > 160 we set Hext = 0 and the sample is allowed to reach a steady state. The initial temperature distribution is displayed in Figure 3.5, the corresponding spatial distribution of the saturation magnetization m is displayed in Figure 3.6. After the magnetization has switched to the opposite direction, the sample is left to cool down and the magnetization remains oriented in the direction of the applied field. The time evolution of the averaged temperature of the sample is displayed in Figure 3.7. The numerical error in the magnitude of the magnetization due to the temporal variations of m is displayed in Figure 3.7 (right). The plots of spatially

224

Chapter 3

Effective equations for macrospin magnetization dynamics

Figure 3.5. Initial temperature x → τ (0, x).

m_s 0.827 0.683 0.538 0.393 0.249 Figure 3.6. Initial condition m0 , with m(0) = |m(0)|.

Section 3.4

225

Computational experiments

averaged first and second components of the magnetization in Figure 3.8 reveal that, for the model with constant magnitude m = m  0 ≡ 1, the external field Hext is not sufficiently strong to overcome the anisotropy effects and switch the magnetization in the opposite direction. This is in clear contrast to the thermally assisted case. 0.007

1.2

average temperature

|m| - m_s 0.006

1

0.005 0.8

0.004 0.6 0.003 0.4 0.002 0.2 0.001 0

156

158

160

162

0 160

164

162

164

166

168

170

Figure 3.7. Evolution of the averaged temperature in time (left) and evolution tj → |Mj | − m(tj )L∞ (right).

1.5

mx, no temperature my, no temperature mx, with temperature my, with temperature

1

0.5

0

-0.5

-1 0

100

200

300

400

500



1 1 Figure 3.8. Evolutions tj → Leb[O] (Mj )1 dx and tj → Leb[O] (Mj )2 dx for O O the temperature dependent LLG equation (3.5), and the constant magnitude LLG equation.

3.4.2

Comparison of the macroscopic model with the SLLG equation

Saturation magnetization Problem (3.5) requires functions κ and m, where the first may be determined from the latter; cf. (3.8). The saturation magnetization m  ≡ m(τ  ) is usually obtained from physical measurements as a spatially averaged quantity; see (3.6). An alternative approach to determine m  is to use simulations with the SLLG equation (2.1) in a simplified setting; for this purpose, recall the fluctuationdissipation relation (1.3) which relates the noise intensity to the temperature

226

Chapter 3

Effective equations for macrospin magnetization dynamics

 , where C  = 2αkB . We then identify m ν 2 = Cτ  ≡ m(τ  ) in (3.5), see also (3.6), with the time-asymptotic limit  1   m ν (t) dx  , E m  ν (τ ) = lim t→∞ Leb[O] O where m ν solves the SLLG equation (2.1) with space-time white noise of intensity ν = ν(τ ) as given above; by construction, this approach yields a function τ → m(τ  ) for τ ∈ [0, τC ) which describes equilibrium dynamics; see also [79]. To reduce the complexity of this approach to obtain the map τ → m  ν (τ ), simulations of Scheme 2.9 (Chapter 2) to approximate the SLLG equation (2.1) are done in a simplified 1D setting. The saturation magnetization m  ν is determined for a range of noise intensities which correspond to spatially uniform temperature values. It is due to the ergodic properties of Scheme 2.9 as discussed in Chapter 1 that the time-asymptotic expected value (for each ν ≥ 0) can be computed from long-time simulations; the computations are done for the following parameters T = 1000, O = (0, 1), A = 1, K = 0, α = 0.2, k = 0.01, h = 18 , and Dirichlet boundary conditions m (t, 0) = m (t, 1) = (0, 0, 1). The resulting (approximate) map τ → m  ν (τ ) is displayed in Figure 3.9. It can be seen that the saturation magnetization exhibits a power-law dependence on the temperature for lower values of ν as stated in (3.6), and exhibits a different behavior for larger values of the noise intensity. 1

saturation magnetization 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 0

0.5

1

1.5

2

2.5

Figure 3.9. Graph of τ →  m  ν (τ ) computed from the 1D SLLG equation (2.1) with different noise intensities ν = ν(τ ) > 0.

Effective dynamics via the macroscopic model (3.5) In the following series of computational experiments we study how the solution m] of the m of the macroscopic model (3.5) approximates the expected value E[m solution of the stochastic model (2.1) with the unscaled space-time white noise which is defined on a regular triangulation Th of O, cf. (2.140),  ϕ (x) Δj W , (3.50) Δj W(x) =

∈L

Section 3.4

227

Computational experiments

for real-valued finite element basis functions {ϕ } ∈L . The experiments below are not meant to provide a complete understanding of the macroscopic model (3.5), neither are they based on rigorous physical arguments: we attempt to provide a basic understanding on the relation between the stochastic model (2.1) and the macroscopic ones (3.23), (3.5) which can serve as a basis for more in-depth studies; see also the following Remark 3.16. Remark 3.16. Let us remark on the similarities between the model (3.5) and the LLB equation from [79, 59]. The latter has the form α|| ∂m α⊥ = 2 m, Heff  m + m × Heff − 2 m × (m × Heff ) . ∂t m m

(3.51)

A comparison of (3.5) with (3.51) reveals the following analogy between the parameters in the respective models, κ ←→

α|| m, Heff  m2

and

αη ←→

α⊥ , m

(3.52)

  where (dropping a constant there) α⊥ = α 1 − ττC . As discussed in Section 3.4.2, once m is known, by (3.8) the coefficient κ in the macroscopic equation (3.5) is determined. Furthermore, if m is given by (3.6), the expression (3.52)2 leads to   τ 1−β ≈ αm1−β . (3.53) αη ←→ α 1 − τC Expression (3.53) is in agreement with the experiments below, where we choose αη = αm−η for η = β − 1 > 0. The initial condition in all experiments below is m0 = (0, 0, 1), and in all cases an external field Hext = (0, 0.1, −1) is applied which will cause switching of the magnetization towards the direction (0, 0, −1). An approximate function   j M (x )|2 (tj , x ) → |mk,h (tj , x )|2 = E |M (1 ≤ j ≤ J,  ∈ L) M j ; j ≥ 0} of Scheme 2.9 is determined for the macroscopic from iterates {M model (3.5); see Subsection 3.4.2. Let |mk,h |2 ∈ C(OtJ ) denote its space-time interpolate, which is locally affine. Then (3.8) motivates to set (x ∈ O) κj+1 (x) :=

|mk,h (tj+1 , x)|2 − |mk,h (tj , x)|2 2k|mk,h (tj+1 , x)|2

(0 ≤ j ≤ J − 1) .

Note that a discrete analogue of (3.8) for Scheme 3.5 takes the form (see 3.24) dt |Mj+1 (x )|2 = 2κj+1 (x )|Mj+1/2 (x )|2

( ∈ L) .

(3.54)

228

Chapter 3

Effective equations for macrospin magnetization dynamics

Below, we take O = (0, 1)2 , and if not mentioned otherwise, the time step in the computations with Scheme 3.5 is set to k = 0.01. As has been suggested in the physical literature, see e.g. [85] and the references therein, the physical parameters, such as the damping parameter α or the anisotropy parameter K in the augmented model (3.5) should also depend on the saturation magnetization m.  Our experiments support this conjecture: in order to obtain agreement between the macroscopic model (3.5) and the SLLG equation (2.1), we need to consider a modified damping parameter αη = αm−η in (3.23) (resp. in (3.12)), where η > 0 is a suitably chosen constant and α is the usual damping constant; see also Remark 3.16. We start with the simplest setting: we consider a single spin with Heff = Hext . The saturation magnetization mk,h is determined from the expected value of the M j ; j ≥ 0} of Scheme 2.9, with 1000 realizations of the modulus of the iterates {M Wiener process. In Figure 3.10 we display the evolution of mk,h (Scheme 2.9) and |M| (Scheme 3.5), and the evolution of the individual components of the stochastic solution for one realization of the Wiener process. The remaining parameters for the computation were α = 0.2, η = 0, 0.75, 2, and ν = 0.2. In Figure 3.11 we display the evolution of the components of the magnetization computed for the macroscopic model (3.5), and the expected value of the magnetization for the stochastic model (2.1). As can be seen from Figure 3.11, the choice of the damping exponent η has significant effects on the overall evolution, and η = 0.75 provides the best approximation. For η = 2, there is too much damping in the solution, while for η = 0, i.e. when the damping parameter does not depend on m, the macroscopic model exhibits completely different dynamics from the dynamics obtained by the stochastic model. 1

1

(A) (B)

0.9

mx my mz

0.8 0.6

0.8

0.4 0.7 0.2 0.6 0 0.5 -0.2 0.4 -0.4 0.3

-0.6

0.2

-0.8

0.1

-1 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20



1 Figure 3.10. (left) Evolution of tj → Leb[O] mk,h (tj , x) dx (A) and tj → O 1 j Leb[O] O |M (x)|dx (B) in time. (right) Evolution of computed magnetization components via Scheme 2.9 at x = (0.5, 0.5) for one realization of the Wiener process.

To study a more complex problem we include exchange and anisotropy effects. We take Heff = Hext + AΔm + Km, ee, with A = 0.25, K = 0.25. Further, we consider a higher noise intensity ν = 0.4, α = 0.2, and based on experiments we choose η = 0.75. The domain for this experiment is the

Section 3.4

229

Computational experiments

1

1

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8 0.6 0.4

1

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8 0.6 0.4

0.6 0.4

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

-0.6

-0.6

-0.6

-0.8

-0.8

-1

-1 0

2

4

6

8

10

12

14

16

18

20

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.2

-0.8 -1 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Figure 3.11. Evolution of each component of spatial averages of the computed solution {Mj } of Scheme 3.5 for η = 0, 0.75, 2 (from left to right) and the solution M j ]} of Scheme 2.9 in time. {E[M

unit square O = (0, 1)2 . We consider a coarse discretization of the problem: O is divided into four triangles; the resulting finite element mesh contains 5 nodes. In Figure 3.12 we display the evolution of the spatially averaged saturation magnetization and of the spatially averaged modulus of the solution of the effective model, as well as the evolution of the magnetization at the point x = (0.5, 0.5), which is computed using Scheme 3.5 for one realization of the Wiener process. Figure 3.13 shows the evolution of the spatially averaged comM j ]; j ≥ 0} for the ponents of the computed solutions {Mj ; j ≥ 0} resp. {E[M macroscopic and the stochastic model, where the latter is computed using 100 and 1000 realizations of the Wiener process, respectively. We can see how the overall agreement between the two models improves for more realizations of the Wiener process. 1

1

(A) (B)

0.9

mx my mz

0.8 0.6

0.8

0.4 0.7 0.2 0.6 0 0.5 -0.2 0.4 -0.4 0.3

-0.6

0.2

-0.8

0.1

-1 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Figure 3.12. magnetization (left) Evolution of the spatially averaged saturation 1 1 j m (t , x) dx (A is Scheme 2.9) and t →  |M (x)|dx (B is tj → Leb[O] k,h j j Leb[O] O O Scheme 3.5) in time. (right) Evolution of components of magnetization at x = (0.5, 0.5) for one realization of the Wiener process.

To highlight the limitation of the macroscopic model (3.5), we present a case where it is more difficult to obtain a good agreement between the effective and the stochastic dynamics. We consider the previous problem with increased

230

Chapter 3

Effective equations for macrospin magnetization dynamics

1

1

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.6

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 0

5

10

15

20

0

2

4

6

8

10

12

14

16

18

20

Figure 3.13. Evolution of spatially averaged components of {Mj } to approximate M j ]} to approximate (2.1) for 100 (left) and 1000 (3.23) via Scheme 3.5, and of {E[M (right) realizations of the Wiener process (Scheme 2.9).

anisotropy constant K = 1. The graphs in Figure 3.14 show that the disagreement between the models grows as mk,h approaches the critical value 0, a state that corresponds to the Curie temperature of the material. The results can be slightly improved by changing the η coefficient in the damping parameter αη . Alternatively, one could also consider anisotropy and exchange parameters in the effective model which depend on m as well. 1

1

(A) (B)

0.9

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.8

0.6

0.7 0.4 0.6 0.2 0.5 0 0.4 -0.2 0.3 -0.4

0.2

-0.6

0.1 0

-0.8 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Figure 3.14. magnetization (left) Evolution of the spatially averaged saturation 1 1 j m (t , x) dx (A) in time and t →  |M (x)|dx (B). (right) tj → Leb[O] k,h j j Leb[O] O O Different spatially averaged components of the magnetization in time for K = 1.

Next, we test how the dynamics of the two models changes with respect to mesh refinement. We consider the previous example with exchange and anisotropy, i.e. Heff = Hext + AΔm + Km, ee, with A = 0.25, K = 0.25, η = 0.75, ν = 0.4, α = 0.2 on a refined mesh with 13 resp. 41 nodes (i.e. mesh sizes h = 12 resp. h = 14 ), respectively. For the computation on the finest mesh we employ a smaller time step k = 0.005. In Figure 3.16 we display the normalized solution m of the macroscopic model (3.23), as well as the m] for the stochastic model (2.1), which is expected value of the solution E[m approximated via Scheme 2.9 and M = 1000 realizations of the Wiener process. We observe that the agreement between the two models remains good after mesh refinement, see Figure 3.16.

Section 3.4

231

Computational experiments

1

1

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.6

E(mx) SLLG E(my) SLLG E(mz) SLLG mx my mz

0.8

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Figure 3.15. Evolution of different spatially averaged components of the magnetization in time for h = 12 (left) and h = 14 (right).

M j ] for the macroscopic Figure 3.16. Normalized solutions tj → Mj resp. tj → E[M model (3.5) (green arrows) resp. the SLLG equation (2.1) (red arrow) at times t = 7, 8.6, 11.7, 1.48, 20 (h = 41 ).

Notation

The following symbols are ordered according to their first appearance in the text.

Physical data and Functions number of spins of a finite spin ensemble; i indexes a single spin (1 ≤ i ≤ N ) C generic positive constant; C(a, b) denotes explicit dependence on a, b K, A, α physical constants: anisotropy, exchange, damping; p. 1 E sum of exchange Eexch , anisotropy Eani and exterior energy Eext e ∈ R3 easy axis of ferromagnet in anisotropy energy Eani O ⊂ Rn (1 ≤ n ≤ 3) bounded Lipschitz domain OT = (0, T ) × O   Heff = −DE effective energy. Heff = Heff,1 , . . . , Heff,N for N -spin ensembles; p. 1 ˙ with intensity ν, and white noise W; ˙ p. 1 Hthm = ν W, l N N ×N l J = (Jm )m,l=1 ∈ R , with exchange energy coefficients Jm between spins m and l; p. 1    N ·, ·, ·, · standard scalar product in R3 resp. R3  N | · |,  ·  induced norms in R3 resp. R3 τ , τC temperature in [0, τC ), with Curie temperature τC ; pp. 2, 197 T time a∧b := min{a, b}; p. 14  J ∈ L (R3 )N symmetric positive definite operator, such that N  J X)i = Jil Xl for all 1 ≤ i ≤ N ; pp. 24, 34 (J N

⊗ m, m  κ, κ  E(m; ω)

l=1

tensor product; p. 163 saturation magnetization m  : [0, τC ) → R+ in (3.6), and m = |m|; p. 197 longitudinal relaxation function κ  : [0, τC ) × R+ → R; p. 198 1 = 2 ω |∇m(x)|2 dx; p. 203

Section 3.4

Computational experiments

233

Probability spaces and Stochastic processes 

Ω, F, P



P μ P (t, x, A ) Pt , L Leb μ[f ] J (P ) W X Xi X , X ,δ μ ,δ Mj #k M Pk

k Y L, Lk , Lk Hjeff M k,h W Fk ξj J {ξ  }j=0 N m

probability space, with sample set Ω, filtration F, and  probability  measure P = Ω, F, F, P , with filtration F = {Ft ; t ≥ 0} invariant measure (1.2) of X which solves SDE (1.37); p. 7 transition kernel of X, with A ⊂ B(X); p. 11 generator L (with domain D(L)) of semigroup {Pt ; t ≥ 0} generated by X; p. 11 : B(X) → R+ Riemannian volume measure in X; p. 12 = X f (x) μ[dx]; p. 22 set of reversible measures of the semigroup P = {Pt ; t ≥ 0}; p. 22    N W1 , . . . , WN R3 -valued Wiener process driving anN -spin system;  p. 39 = X1 , . . . , XN (R3 )N -valued solution process of (1.37); p. 40 = (Xip )3p=1 (1 ≤ p ≤ 3) (R3 )N -valued processes which solve the SDE (1.56); p. 51 invariant measure of X ,δ of SDE (1.56); p. 51  j  Markov chain which solves Scheme 1.11; = M1j , . . . , MN p. 68   #k,1 , . . . , M #k,N adapted continuous process which = M interpolates M; p. 74   = Pjk j≥0 family of discrete transition semigroups (dependent upon the size of the mesh k) generated by Y solving Scheme 1.16; p. 79 N  = Yk,i i=1 continuous interpolation of Y from Scheme 1.16; p. 80  k, Z  k − Fk in the proof of Theorem 1.17; generators of X, Z p. 83   j j ; p. 86 = Heff (Yj ) = Heff,1 , . . . , Heff,N piecewise affine interpolation of M ; piecewise constants M± k,h , M k,h ; p. 109  √ = ∞

=1 q β (t)e K-valued Q-Wiener process, with 0 ≤ Q ∈ T1 (K); p. 115 = {Ftj; tj ∈ Ik } ⊂ F, and filtered probability space Pk = Ω, F, Fk , P ; p. 117 K-valued Q-random walk on Pk ; p. 117 quadratic variation process for N ∈ M2T (K); p. 120 magnetization process R+ × O × Ω → R3 which solves (2.47);

234

Chapter 3

Effective equations for macrospin magnetization dynamics

# M k,h

p. 129 Vh -valued iterates from Scheme 2.9 (p. 131) or Scheme 2.10; p. 133 M j }; piecewise affine, globally continuous interpolation of {M p. 155 M j }; p. 155 piecewise constant interpolation of {M = W ◦ φk,h Wiener process on P ; p. 157 Mk,h } with improved convergence properties; transform of {M p. 157 law of M k,h ; p. 157 probability spaces, generated via Skorokhod embedding and martingale representation theorems corr.; p.s 157, 176 stochastic integral in (2.98) with quadratic variation process Q k,h in (2.112); p. 163 continuous interpolation of M in (2.100) which generates Fk,h ; p. 163 # k,h via (2.108) which generates F ; transformation of M

X F Vk 8 k,h M

X k,h Gk,h m m {Mj ; j ≥ 0}

pp. 167, 168 martingale part of the process m ; p. 171 natural augmentation of filtration F generated by m ; p. 171 martingale which interpolates {ξξ j }Jj=0 on Pk ; p. 179 # k,h for Scheme 2.10; p. 180 equivalent of the process M equivalent of the process X k,h for Scheme 2.10; p. 180    

(tj ); j = 0, . . . , T } ; p. 182 = σ {X k,h k magnetization R+ × O → R3 which solves (3.5); p. 197 =m m ; p. 206 Vh -valued iterates from Scheme B; p. 210

M j ; j ≥ 0} {M M k,h M± k,h , M k,h Wk,h M k,h Mk,h ) L(M P , P X k,h # k,h M

k,h

Sets, spaces, and operators  2 N 2 S  , S  X, B(X) Bb (X) C(X) Br (a)

1C

Cm (X)

(product of N ) unit spheres (S2 ⊂ R3 ) smooth manifold X ⊂ Rd and σ-field of Borel sets B(X); p. 11 Borel measurable bounded functions X → R; p. 11 continuous functions X → R; p. 12  N = {x ∈ R3 ; x − a ≤ r}; p. 12 characteristic function for the set C ; p. 14 m-times differentiable maps X → R; p. 41

Section 3.4

Computational experiments

L2 (O; R3 ) Wm,p (O; R3 ) C(O;R3 )  W s,p 0, T ; E N s,p 0, T ; E) E, E0 , E1 K, H T1 (H) T2 (K, H)

235

R3 -valued square integrable functions f : O → R3 ; p. 106 standard Sobolev space of R3 -valued functions f : O → R3 ; p. 107 continuous R3 -valued functions f : O → R3 ; p. 107 fractional Sobolev space; p. 111 Nikolskii space; p. 111 Banach space; pp. 111, 114 separable Hilbert spaces; p. 115 space of trace-class operators; p. 116 ⊂ L (K, H) Hilbert-Schmidt operators from K to H; p. 116

Time discretization and Finite elements Ik Kj+1 dt ϕj+1 ϕj+1/2 Δj W Th Eh Vh h V ϕ (·, ·)h ,  · h Ih h Δ Δj W

= {tj }Jj=0 . Mesh of size k > 0 covering the interval [0, T ]; p. 26

 ≡ Kj+1 (Yj ) = W ∈ (R3 )N : Wi , Yij  = 0 for all 1 ≤ i ≤ N ; p. 26 = k1 (ϕj+1 − ϕj ); p. 27 = 12 (ϕj+1 + ϕj ); p. 27  = Δj W1 , . . . , Δj WN , where Δj Wi = Wi (tj+1 ) − W (tj ); p. 68 regular triangulation of size h > 0 of the bounded domain O ⊂ Rn ; p. 106 = {x :  ∈ L} set of nodal points of Th ; p. 106 ⊂ W1,2 (O; R3 ); FE space; p. 107 analog of FE-space Vh for scalar-valued functions; p. 107  h nodal basis function, with ϕ (xm ) = δ m for , m ∈ L ∈V scalar product based on reduced integration, induced norm; p. 107 Lagrange interpolation operator; p. 107 discrete Laplacian; p. 108 = (Δj β1 , Δj β2 , Δj β3 ) for  ∈ L, and R-valued Wiener processes {βp ; 1 ≤ p ≤ 3}; pp. 186, 226

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