Spherical Radial Basis Functions, Theory and Applications 3319179381, 978-3-319-17938-4, 978-3-319-17939-1, 331917939X

Table of contents :
Front Matter....Pages i-x
Motivation and Background Functional Analysis....Pages 1-28
The Spherical Basis Function Method....Pages 29-58
Error Bounds via Duchon’s Technique....Pages 59-83
Radial Basis Functions for the Sphere....Pages 85-95
Fast Iterative Solvers for PDEs on Spheres....Pages 97-119
Parabolic PDEs on Spheres....Pages 121-138
Back Matter....Pages 139-143

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SPRINGER BRIEFS IN MATHEMATICS

Simon Hubbert Quôc Thông Lê Gia Tanya M. Morton

Spherical Radial Basis Functions, Theory and Applications 123

SpringerBriefs in Mathematics Series editors Nicola Bellomo, Torino, Italy Michele Benzi, Atlanta, USA Palle E.T. Jorgensen, Iowa City, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Vienna, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA G. George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

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Simon Hubbert Quôc Thông Lê Gia Tanya M. Morton •

Spherical Radial Basis Functions, Theory and Applications

123

Tanya M. Morton MathWorks Cambridge UK

Simon Hubbert School of Economics, Mathematics and Statistics Birkbeck, University of London London UK Quôc Thông Lê Gia School of Mathematics The University of New South Wales Sydney, NSW Australia

ISSN 2191-8198 SpringerBriefs in Mathematics ISBN 978-3-319-17938-4 DOI 10.1007/978-3-319-17939-1

ISSN 2191-8201

(electronic)

ISBN 978-3-319-17939-1

(eBook)

Library of Congress Control Number: 2015937749 Mathematical Subject Classification: 33C55, 35R01, 35Q86, 41A05, 41A24, 42A16, 42B37, 65D05, 65D15 Springer Cham Heidelberg New York Dordrecht London © The Author(s) 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)

Simon dedicates this book to his partner Michelle and his two daughters Nancy and Clara. Thong dedicates this book to his wife Tram Linh. Tanya dedicates this book to her husband Chris and her two sons Kai and Eldin.

Preface

In recent years mathematicians and researchers within the approximation theory community have become increasingly interested in using tools from approximation theory to develop numerical methods for problems set on spheres. Given that the two-dimensional sphere serves as a model for the surface of the earth, these problems are often generated by real-world applications. For instance, as more and more satellites are launched into space, the acquisition of global data is becoming more widespread and hence there is now increased demand for spherical data processing solutions. Another important application is in the quest to improve weather forecasting models through the development of efficient algorithms for solving partial differential equations (PDEs) posed on the surface of the sphere. There is also need for spherical approximation in areas other than geoscience and meteorology. For instance, one may want to construct a smooth surface that encloses a cloud of scattered points in Euclidean space, this can be achieved using spherical approximation solutions. This has numerous applications, for example, to model 3D objects for computer graphics, or to create a boundary model representing the safe operating envelope of internal combustion engines. This book serves to provide an introduction to the theory and applications of spherical (radial) basis functions (SBFs), which represent one of the most promising emerging technologies for solving spherical problems. SBFs are closely related to the more famous family of radial basis functions (RBFs) which are already wellestablished tools for solving data fitting problems and PDEs over regions in Euclidean space. RBFs have a much longer history than SBFs and so, consequently, much more is known about them, indeed [Buh03; Fass07; Wen05] are three excellent textbooks devoted to their theoretical properties and their practical implementations. Our primary aim in this book is to present enough practical and theoretical details to enable the reader to implement SBF techniques to solve real problems and also, if desired, to pursue further theoretical studies in this exciting area. In Chap. 1 we set out our motivation for studying SBFs and provide the background tools from functional analysis which will be used throughout the book. In Chap. 2 we demonstrate how key ideas and concepts from the interpolation theory of RBFs in Euclidean space can be recast into the spherical setting and, vii

viii

Preface

in doing so, we introduce the notion of SBFs and we show how they can be used to provide unique solutions (SBF interpolants) to data fitting problems on the sphere. Furthermore, we also reveal a simple variational framework for SBF interpolation and show how this can be used to analyse the accuracy of a particular SBF interpolant to a given target function. In Chap. 3 we pursue the error analysis in much greater detail. Specifically, we present the technical ingredients of an error bounding strategy which we then use to provide much improved error estimates for SBF interpolation. In Chap. 4 we test the theory and present the results of numerical experiments for the SBF method for solving data fitting problems on the sphere. In the final two chapters of the book we move away from data fitting applications and concentrate more on investigating how SBF approximations can be used to solve PDEs on spheres. In Chap. 5 we focus more on computational issues and propose a preconditioning strategy to speed up the iterative solution of an elliptic PDE. Finally, in Chap. 6, we examine the inhomogeneous heat equation as an example of a parabolic PDE. Here we develop a collocation solution method and provide a full error analysis when the time variable is discretized using either the backward Euler or the Crank-Nicolson method. In summary, the material covered in this book is aimed at graduate students and researchers in mathematics and related fields such as the geophysical sciences and statistics. We have tried to make the exposition as clear and as self-contained as possible and have made efforts to ensure that technical details are explained in a friendly and readable style. We hope this will encourage the reader to delve deeper and discover more. January 2015

Simon Hubbert Quôc Thông Lê Gia Tanya M. Morton

Contents

1

Motivation and Background Functional Analysis . 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hilbert Space Theory . . . . . . . . . . . . . . . . . . 1.5 Spherical Harmonics and Fourier Analysis. . . . 1.6 Sobolev Spaces in Euclidean Space . . . . . . . . 1.7 Sobolev Spaces on the Unit Sphere . . . . . . . .

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1 1 1 3 8 10 14 21

2

The 2.1 2.2 2.3 2.4 2.5 2.6

Spherical Basis Function Method . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . A Brief History of the RBF Method . . . . The Spherical Basis Function Method . . . Framework for Pointwise Error Estimates Pointwise Error Estimate I . . . . . . . . . . . Pointwise Error Estimate II . . . . . . . . . .

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29 29 30 39 47 50 52

3

Error Bounds via Duchon’s Technique . . . . . . . . . . . . . . . . . . . . . 3.1 Duchon’s Recipe for the Sphere . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Error Bounds for SBF Interpolation . . . . . . . . . . . . . . . .

59 59 76

4

Radial Basis Functions for the Sphere . . . . . . . . . . . . . . . . . . . . . . 4.1 Duchon Splines for the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical Investigation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 91

5

Fast 5.1 5.2 5.3

Iterative Solvers for PDEs on Spheres Introduction . . . . . . . . . . . . . . . . . . . . The Weak Formulation of the PDE. . . . The Additive Schwarz Method . . . . . . .

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103 104 111 113

Parabolic PDEs on Spheres. . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Homogeneous Semi-discrete Problem . . . . . . . . . . 6.3 The Inhomogeneous Semi-discrete Problem . . . . . . . . . 6.4 Time Discretization Using the Backward Euler Method. 6.5 Time Discretization Using the Crank-Nicolson Method . 6.6 Numerical Experiments on S2 . . . . . . . . . . . . . . . . . .

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121 121 122 125 128 132 135

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

6

A Subspace Decomposition Algorithm . . . . . . . . An Upper Bound for the Condition Number κðPÞ An Overlapping Additive Schwarz Algorithm . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Motivation and Background Functional Analysis

1.1 Introduction We open this chapter by establishing the basic notation that will be used regularly throughout the book. We then provide some motivation behind what we aim to achieve in this book, namely to develop the theory and explore the applications of spherical basis functions (SBFs). In order to achieve this goal we shall require a collection of tools and techniques from functional analysis and the remainder of the chapter is devoted to this. We begin this task with a brief review of Hilbert space theory, important because Hilbert spaces provide a natural setting within which we can analyse the SBF method. We then move on to discuss spherical harmonics (the spherical analogue of classical polynomials) and show how these provide a Fourier analysis for the sphere. We then describe the structure and properties of Sobolev spaces in both Euclidean space and on the sphere. Sobolev space are the Hilbert spaces to which the functions we wish to approximate belong. We will provide a careful examination of specific Sobolev spaces that are defined locally on the sphere.

1.2 Notations Throughout the book, unless stated otherwise, the letters x, y will denote points in Euclidean space Rd and the letters ξ, η will denote points on the (d −1)−dimensional unit sphere S d−1 := {x ∈ Rd : x = 1},

(1.1)

where  ·  denotes the Euclidean norm. We let ωd−1 denote the surface area of S d−1 which is calculated (see [AH12, Sect. 1.3]) via:

© The Author(s) 2015 S. Hubbert et al., Spherical Radial Basis Functions, Theory and Applications, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-17939-1_1

1

2

1 Motivation and Background Functional Analysis

 ωd−1 =

d

S d−1

dωd−1 (ξ) =

2π 2   for d ≥ 2, Γ d2

(1.2)

where dωd−1 is the surface element of S d−1 and where Γ (·) denotes the Gamma function defined by 

∞

Γ (y) :=

t y−1 e−t dt, y ∈ [0, ∞).

(1.3)

0

We remark that (1.2) will also be used for d = 1 where we define ω0 := 2. A set Ω in Rd will be called a domain if it is a non-empty, open and connected subset of Rd . For instance, the open Euclidean ball centred at x ∈ Rd with radius r > 0, given by B(x, r ) := {y ∈ Rd : d(x, y) = x − y < r }, is an example of a bounded domain in Rd and will feature heavily in this work. The spherical equivalent of B(x, r ) is the open geodesic ball G(z, θ), centred at z ∈ S d−1 with radius θ ∈ (0, π), and is defined by G(z, θ) := {ξ ∈ S d−1 : g(z, ξ) < θ}, where g : S d−1 × S d−1 → [0, π] is the geodesic metric, given by g(ξ, η) = cos−1 (ξ T η), ξ, η ∈ S d−1 .

(1.4)

Given a domain Ω ⊂ Rd , we define C(Ω) to be the collection of all real-valued bounded and uniformly continuous functions on Ω, such that  f ∞ = sup | f (x)| < ∞. x∈Ω

Furthermore, we define L p (Ω), with p ∈ [1, ∞], to be the set of all real-valued Lebesgue-measurable functions on Ω, such that   f L p (Ω) :=

1/ p Ω

| f (x)| p dx

< ∞,

if p ∈ [1, ∞),

or,  f L∞ (Ω) :=

inf

sup | f (x)| < ∞, if p = ∞.

ω: meas ω=0 x∈Ω\ω

To describe derivatives of multivariate functions we use the multi-index notation, i.e., we fix α = (α1 , . . . , αd )T , with α j ∈ N0 = {0, 1, . . .}, to be a multi-index

1.2 Notations

3

vector and then, if f (x) is a smooth function of x = (x1 , . . . , xd )T ∈ Rd , we denote its α−order derivative as Dα f (x) =

 ∂ |α| f αj. αd (x) where |α| = α1 ∂x1 · · · ∂xd j=1 d

Furthermore, for any domain Ω in Rd , we let C k (Ω) := { f ∈ C(Ω) : Dα f ∈ C(Ω) for every α ∈ Nd0 with |α| = k} denote the space of k−times differentiable bounded functions on Ω. The standard space of test functions on Ω is given by D(Ω) = {γ ∈ C ∞ (Ω) : γ is compactly supported in Ω}

(1.5)

and the space of rapidly decaying test functions is defined by S(Rd ) = {γ ∈ C ∞ (Rd ) : xβ Dα γ ∈ L2 (Rd ) for all α and β ∈ Nd0 }.

(1.6)

Finally we let Πk (Rd ) := span{x → xα = x1α1 · · · xdαd : |α| ≤ k, x ∈ Rd }

(1.7)

denote the space of d−variate polynomials of degree at most k.

1.3 Motivation The subject of this book belongs to the branch of mathematics known as approximation theory, an area which tackles real-life problems. Suppose, for instance, that we are able to measure the depth of the ocean over a number of locations throughout an area of interest. A typical approximation problem is then to use this information to construct an approximate computer image of the ocean floor. In this setting the terrain of the ocean floor is our target function (or surface in this case) and our job is to build an approximation from a certain approximation space; polynomials, rational functions, splines, wavelets and radial basis functions are just a selection of possible function spaces we could use. The wide variation in the choice of approximation spaces, the target functions we try to approximate, the method we employ and the ways of measuring a goodness-of-fit are all ingredients which provide a diverse and rich theory. The problem of reconstructing a picture of the ocean floor from a sample of measurements is a classic example of an interpolation problem, since here we require our computer image to fit the true measurements at the given locations. This is a

4

1 Motivation and Background Functional Analysis

typical real-world problem since there is no known mathematical representation of the ocean floor. We hope, however, that if our approximation method is tried and tested, i.e., it is known to perform well over a wide range of known test cases, then we can be confident that it will deliver faithful representations in other settings too. Thus, the sense in which we want the approximation to be “good” is important. It could, for instance, be purely aesthetic. For example in the case of rendering the ocean floor it may be enough to decide whether the surface is visually pleasing. However, most commonly we use a quantitative measure such as an L p norm to address the quality of approximation. The set up of the most general form of the interpolation problem is described as follows. Firstly, we fix the region where we want to examine our approximation, i.e., a suitable domain Ω lying in d−dimensional Euclidean space Rd . Our ultimate aim is to reconstruct a target function (defined on Ω) from only a finite set of sample N N that are prescribed at a given a set of locations X = {xi }i=1 values { f i := f (xi )}i=1 inside Ω. In order to gain the best results we require that the locations should fill the domain Ω in a uniform way, as opposed to being too concentrated in particular areas whilst leaving wide holes in others. To account for this we assign a density measure to the set X in Ω by defining the number h to be the radius of the largest Euclidean ball that can be positioned in Ω but which does not overlap any of the locations (this definition will made more precise in the next chapter). Armed with a sufficiently dense set of locations one can set about finding a suitable interpolant i.e., a function sh : Ω → R for which sh (xi ) = f i for i = 1, . . . , N. The usual approach is to search a certain data-dependent approximation space, Sh say, for the interpolant sh which is uniquely identified by computing a number of free parameters so that sh agrees with the target function f at the data locations and possibly meets further constraints or has particular properties required by the application. Once we have established that a unique interpolant can be found from our approximation space we then turn to deeper and more interesting questions regarding its accuracy and this is where theoretical error bounds are required. The general recipe for developing error bounds typically requires some additional information. For example, if we know that the target function belongs to a certain function space then it is usual to derive error bounds of the form sh − f  ≤ B(h) f ,

(1.8)

where B(h) is a function converging to zero as h → 0, the error sh − f is measured in a suitable norm (L p for instance) and the size of f is measured in the norm of the space to which it belongs. Radial basis function (RBF) approximation spaces are well-known to deliver high quality solutions to the interpolation problem. Specifically, the RBF approach specifies a suitable basis function φ : [0, ∞) → R (the RBF) and proposes a solution of the form

1.3 Motivation

5

sh (x) =

N 

λi φ(x − xi ) + p(x),

(1.9)

i=1

with λi real coefficients,  ·  the Euclidean norm and p a suitable polynomial (which is not needed in all cases). This approach is extremely flexible: it places no geometrical demands on the data locations, it does not become more elaborate as the space dimension increases and it is easy to implement on a computer. The candidate RBFs are chosen to be (conditionally) positive definite, a property which guarantees the uniqueness of (1.9). Furthermore, using tools from Fourier theory and functional analysis, a sound theoretical framework has been established within which RBF error bounds can be derived. When the data locations are known to lie on the surface of the sphere it seems natural to take this information into account. This leads us to consider a specialization of the RBFs. We no longer use the Euclidean metric with a univariate RBF φ, instead we use the geodesic metric g(x, y) = cos−1 (xT y) in conjunction with a univariate SBF ψ : [0, π] → ∞. The spherical analogue of (1.9) takes the form sh (x) =

N 

λi ψ(g(x, xi )) + Y(x),

(1.10)

i=1

where Y is a suitable spherical harmonic (the spherical analogue of a polynomial). We say that (1.10) is the SBF interpolant to the given data. A key aim of this book is to provide the reader with the technical machinery required to develop theoretical error bounds for SBF approximation. Mathematically speaking, we aim to determine the behaviour of the function B(h) appearing in (1.8). For many of the most commonly used SBFs B(h) will be a polynomial, B(h) = Ch p , where C and p are positive constants independent of h. In this case we say that the method converges to f at a rate of order p. We will set out to carefully explain how to derive the convergence order, or estimates of it. This is a technical process which, as we shall see, will depend upon the smoothness of the SBF, the dimension of the sphere and the norm which we choose to measure the error. The exploration of theoretical convergence orders is driven by the desire to predict the numerical errors that are seen when example problems are coded in mathematical software. The convergence orders that we derive are the best that are currently known in the literature but, apart from a few special cases, the numerically observed orders are usually higher than the theory suggests. This indicates that the theory still has some catching up to do and so there is still work to be done. In addition to the classical interpolation problem on the sphere, we will also demonstrate how to solve partial differential equations (PDEs) using the so-called collocation technique. Collocation is a comparatively simple technique which requires that the SBF approximation satisfies the PDE exactly at a finite set of locations on the sphere. Collocation in Euclidean space and on spheres is an emerging

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1 Motivation and Background Functional Analysis

mesh-free technique for solving differential equations. The method is called meshfree since the locations do not need to be interconnected as they do, for example, in finite element techniques. In the remainder of this chapter, following this section, we will collect together all of the crucial mathematical tools and concepts from functional analysis that will be used frequently throughout the book. In Chap. 2 we present the SBF interpolation method as a solution to the spherical data fitting problem. To motivate this we provide a brief history of the more established RBF theory. Specifically, we review the concepts of (conditional) positive definite RBFs and show how these can be used to provide unique interpolants to scattered data in any dimension. We also outline the Fourier theoretic framework within which RBF error bounds can be derived. We emphasize the so-called Duchon strategy (named after its discoverer) which sets out a recipe for delivering very accurate RBF error bounds by essentially gluing together local pieces of the approximation. With this in place we then turn our attention to the spherical data fitting problem. Here we study new concepts of (conditional) positive definite SBFs and, just as in the RBF setting, we show how these SBFs guarantee the existence of a unique SBF interpolant (1.10). We then demonstrate how tools from Fourier and functional analysis on the sphere enable us to build a framework for deriving SBF interpolation error bounds. We demonstrate the power of this framework by presenting two different routes to a point-wise error bound. In both approaches it is assumed that the data locations are densely scattered across the unit sphere. The first error bound we encounter relies on information at every data point whereas the second is able to provide the same strength result by using only local information, i.e., using data points located within a small region of the point where we are examining the error. Both error bounds show that it is possible to annihilate spherical harmonics of a certain degree by using a linear combination of point evaluations and both approaches show that this can be done in a stable way. The second error bound is the stronger result from the perspective that it can be derived using only local information, however, as we shall see, the first approach also has its merits. Specifically, with the first approach the relation between the density of the points and the degree of spherical harmonics to be annihilated is made explicit, a fact which proves crucial to the error analysis. In the second approach the degree of spherical harmonics to be annihilated is held fixed at a number that is connected to the SBF employed. The error estimates derived in Chap. 2 are rough and ready and far from optimal and this motivates Chap. 3 whose purpose is to improve our understanding of the SBF method by delivering much improved error bounds. The process here, once again, borrows ideas from RBF theory. Specifically we demonstrate how the RBF Duchon-strategy (bringing local error estimates together to produce improved global error bounds) can be recast to the sphere; here the local-information error bound from Chap. 2 plays an important role. The subject matter of this chapter is rather technical dealing, as it does, with delicate arguments of how functions (with prescribed zeros) that are defined on a spherical cap can be continuously extended onto the whole sphere. We have tried to deliver this material in an easy going manner, providing motivation and explanations for the approach we have adopted. The payoff for setting

1.3 Motivation

7

up this technical framework is revealed at the conclusion of Chap. 3 where new error bounds are derived (measured in the L p −norm) which represent a dramatic improvement to those we reached in Chap. 2. In Chap. 4 we test the theory. Specifically we appeal again to RBF theory and take the family of Duchon-splines (a parameterized collection of conditionally positive definite RBFs), we restrict them to the unit sphere and compute their associated spherical Fourier expansions. Using this information we are then able to appeal to our theoretical results (from Chap. 3) to predict the rate at which a certain spherical Duchon-spline interpolant will converge to a given target function. A range of numerical experiments are then performed on the circle and the two-dimensional sphere to assess how well the theory explains the numerical findings. We also investigate the performance of the SBF method when the data lie on some open subset of the sphere, where we use the semicircle and the hemisphere as prototype domains. Here we find that the accuracy is badly affected by the presence of the boundary and, based on our numerical findings, we propose a conjecture on exactly how to quantify this deterioration. In the final two chapters of the book we move away from data fitting applications and concentrate more on investigating how SBF approximations can be used to solve PDEs on spheres. We will apply the mathematical frameworks developed in the previous chapters to solve two types of PDEs: elliptic and parabolic equations. For the elliptic PDE, since the error analysis is a direct consequence of previous chapters, we focus on a more practical aspect of the method: namely how to develop a preconditioner (based on the additive Schwarz method) to speed up an iterative solver. For the parabolic case, we consider the inhomogeneous heat equation on the sphere. A collocation method based on SBFs is developed and a full error analysis is given when the time variable is discretized using either the backward Euler method or the Crank-Nicolson method. The material contained in this book is heavily influenced by the authors’ shared research interests, however the reader should be aware of important works by other researchers in this area. In particular, pioneering work has been done by the Geomathematics group in Kaiserslautern under the leadership of Prof. Willi Freeden. The Kaiserlautern group focus on solving problems on the two dimensional sphere that are driven by real physical problems such as data fitting, inverse problems in the geosciences and boundary value problems in potential theory. They have contributed an enormous amount to the field and much of their work is covered in the two excellent textbooks [FGS98, FM04]. Another pioneering figure is the statistician Grace Wahba who, in the early-1980s, was one of the first researchers to consider the optimal interpolation problem set on the two dimensional sphere [Wah81]. Wahba’s work was extended a few years later by Freeden [Fre84] who established the basic theory for data fitting using so-called spherical splines. Wahba and Freeden’s contributions appeared long before RBFs (and consequently SBFs) were discovered, however there are many similarities between their spherical splines and what we call SBFs, and thus these original papers are certainly worth studying in conjunction with this book. In more recent times several important discoveries have been made. For instance, Thomas Hangelbroek [Han11] has improved error bounds for

8

1 Motivation and Background Functional Analysis

polyharmonic splines (a particular class of SBF). The approximation theory group at Texas A& M (led by Joe Ward and Fran Narcowich) have derived stability estimates [NSW98], a variety of error bounds, [NW02, NSW07, NSWW07] and, together with Hangelbroek, have also adapted the theoretical tools to apply to problems set on more general manifolds i.e., other smooth boundary free surfaces (of which the sphere is perhaps the most obvious example) [HNW10, HNW11, HNW12]. Similarly, the approximation theory group at Leicester University (led by Jeremy Levesley) have also made theoretical contributions on stability [LL98], on error bounds [BGL07, LS05, LL98] and also on the generalization to manifolds [Lev07, Lev01]. Ian Sloan and his co-workers have developed a multi-scale SBF method for analysing data on the two dimensional sphere [LSW10, LSW12]. This approach allows the finer details of the data to be accurately captured in the approximation. Finally, for those interested in more practical aspects, the following papers address the topic of fast evaluation methods [DH94, KKP09].

1.4 Hilbert Space Theory Let H be a vector space equipped with an inner product ·, ·H and denote the induced norm by 1

uH = ( u, uH ) 2 , for u ∈ H. We recall that H is said to be a Hilbert space if it is complete with respect to  · H . Of most interest to us are Hilbert function spaces by which we mean a Hilbert space of real valued functions on some domain Ω such that, for each x ∈ Ω the point evaluation functional δx : H → R is bounded. For such Hilbert spaces we have the following result. Lemma 1.1 (Reproducing Kernel Hilbert Spaces) Let (H, ·, ·H ) denote a Hilbert function space on a set Ω ⊂ Rd , then there exists a kernel K : Ω × Ω → R, called the reproducing kernel, with the following properties (i) K(x, ·) ∈ H,

(ii) f(x) = f, K(x, ·)H , (iii) K(x, y) = K(y, x),

for all f ∈ H and x, y ∈ Ω. Proof For any fixed x ∈ Ω the corresponding point evaluation functional δx : H → R is bounded, and so we can evoke a famous theorem of F. Riesz, see [Y88] Sect. 6.1, to deduce that there exists a unique kx ∈ H such that δx ( f ) = f (x) = f, kx H , for all f ∈ H. The element kx is called the “Riesz representor of δx ” and the reproducing kernel is defined to be the totality of all such elements, that is,

1.4 Hilbert Space Theory

9

K(x, y) = kx (y), for all x, y ∈ Ω. Properties (i) and (ii) of the lemma are immediate, and (iii) is true since K(x, y) = 

kx , ky H = ky , kx H = K(y, x). The reproducing kernel property of Hilbert function spaces is a very important one and, as the title of the above Lemma suggests, it is common to describe such objects as Reproducing Kernel Hilbert Spaces (RKHS). We now turn to a property that will feature heavily in this book, the notion of a symmetric positive definite kernel. Definition 1.1 (Positive Definite Kernel) A symmetric kernel K : Ω × Ω → R is N said to be positive definite on the domain Ω if for any set X = {xi }i=1 of N distinct N×N defined by points in Ω, the coefficient matrix AK ∈ R (AK )i, j = K(xi , x j ), for 1 ≤ i, j ≤ N,

(1.11)

is non-negative definite. Furthermore, if AK is positive definite then we say that K is strictly positive definite. The reproducing kernel K of any Hilbert function space H on Ω is positive definite. This is true because, for any x, y ∈ Ω, we have that K(x, y) = kx , ky H . Thus, the N resulting coefficient matrix AK (1.11) is simply the Gramian of N elements {kxi }i=1 of H, and hence non-negative definite. With this background material we are now able to tackle a standard problem in approximation theory, that of finding the optimal interpolant to data sampled from N , a set some f belonging to a Hilbert function space. Specifically, given X = {xi }i=1 of distinct points in Ω, our challenge is to solve: minimise {sH : subject to s ∈ H and s(xi ) = f (xi ), xi ∈ X} .

(1.12)

If we know in advance that the reproducing kernel K of H, is strictly positive definite, then the optimal interpolation problem (1.12) has a rather nice solution, as the following theorem illustrates. Theorem 1.1 Let (H, ·, ·H ) denote a Hilbert function space on Ω, and assume that its reproducing kernel K is strictly positive definite. In this case the optimal interpolation problem (1.12), has a unique solution which belongs to the subspace B := span{K(·, x1 ), . . . , K(·, xN )}. Proof For any f ∈ H, the solution space of the problem is defined to be V f = {s ∈ H : s(xi ) = f (xi ), xi ∈ X}.

(1.13)

10

1 Motivation and Background Functional Analysis

We note that V f is a non-empty, closed and convex subset of the Hilbert space H, and so it contains a unique element s ∗ of minimal norm. Let B ⊥ denote the orthogonal complement of B in H, then, for any, g ∈ B ⊥ we have g(xi ) = K(xi , ·), gH = 0, for 1 ≤ i ≤ n, and thus s ∗ + g ∈ V f . However, the minimal norm property of s ∗ implies that s ∗ H ≤ s ∗ + gH , for all g ∈ B ⊥ , thus we can deduce that s ∗ is orthogonal to B ⊥ . Hence s ∗ ∈ B, and so has the form s ∗ (x) =

N 

λ j K(x, x j ).

j=1

Applying the fact that s ∗ ∈ V f , we get the following system of linear equations N 

λ j K(xi , x j ) = f (xi ), for 1 ≤ i ≤ n,

j=1

which is uniquely solvable since the coefficient matrix is positive definite.



We remark that the reproducing kernel for a given Hilbert function space may not have a form that is amenable for numerical implementation. Indeed, the kernel itself may be difficult to compute. In view of this, we mention the following result [Sh71, Chap. 6] which is helpful in some special cases. Proposition 1.1 Let (H, ·, ·H ) denote a Hilbert function space on a domain Ω with reproducing kernel K. Let (H1 , ·, ·1 ) and (H2 , ·, ·2 ) be complementary orthogonal subspaces of H, that is, H = H1 ⊕ H2 and ·, ·H = ·, ·1 + ·, ·2 , then K = K1 + K2 where K1 and K2 are the reproducing kernels for H1 and H2 respectively.

1.5 Spherical Harmonics and Fourier Analysis In this section we present an introduction to spherical harmonics and spherical Fourier theory. Our aim here is only to provide the essential theory which will be used in Chap. 2 to establish the framework for the SBF method. Both [Mü6] and [AH12] are good references for this material. We begin our presentation in Rd where we have the following definition.

1.5 Spherical Harmonics and Fourier Analysis

11

Definition 1.2 A polynomial p : Rd → R of degree k ≥ 0, is said to be harmonic if it satisfies the Laplace equation  Δx p(x) =

∂2 ∂2 + ··· + 2 2 ∂x1 ∂xd

p(x) = 0,

(1.14)

and k-Homogeneous if it satisfies p(tx) = t k p(x) for any t > 0. We let HP k (Rd ) denote the space of all polynomials of degree k on Rd that are both harmonic and k−homogeneous. The restriction of this class to the unit sphere is of particular interest. Definition 1.3 (Spherical Harmonics of exact order) Let pk ∈ HP k (Rd ), then its restriction to the sphere, Yk = pk |Sd−1 , is a spherical harmonic of order k on S d−1 . We let Hk∗ (S d−1 ) denote the space of spherical harmonics of exact order k. Let pk ∈ HP k (Rd ) and p ∈ HP  (Rd ) then, applying Green’s identity we find  ( pk (x)Δx p (x) − p (x)Δx pk (x))dx    ∂ ∂f = p (r ξ) |r =1 − p (ξ) · pk (r ξ) |r =1 dωd−1 (ξ) pk (ξ) · ∂r ∂r S d−1  = (k − ) pk (ξ) · p (ξ)dωd−1 (ξ) d−1 S Yk (ξ) · Y (ξ)dωd−1 (ξ). = (k − )

0=

x≤1

S d−1

Thus, we have shown that spherical harmonics of different degrees are orthogonal with respect to the inner product 

f, gSd−1 :=

S d−1

f (ξ)g(ξ)dωd−1 (ξ).

(1.15)

In view of this we let Hk (S d−1 ) = ⊕kj=0 H∗j (S d−1 ) denote the space of spherical harmonics of order at most k. We now consider the spherical harmonic spaces from an alternative point of view. Let x ∈ Rd \ {0} then we can write x = r ξ, where r = x and ξ ∈ S d−1 . This observation allows us to rewrite Laplace operator as follows Δx =

1 ∂2 d −1 ∂ + 2 Δ∗ξ , for x ∈ Rd \ {0}, + ∂r 2 r ∂r r

(1.16)

12

1 Motivation and Background Functional Analysis

where Δ∗ξ , having no radial component, is commonly called the Laplace-Beltrami operator for S d−1 ; for a complete derivation see [Mü66]. Let pk ∈ HP k (Rd ) then, for x ∈ Rd \ {0}, we can write pk (x) = r k Yk (ξ) and apply (1.16) to deduce that 0 = Δx pk (x) = k(k + d − 2)r k−2 Yk (ξ) + r k−2 Δ∗ξ Yk (ξ) =⇒ Δ∗ξ Yk (ξ) + λk Yk (ξ) = 0, where λk = k(k + d − 2).

(1.17)

This means that Hk∗ (S d−1 ) is contained in the eigenspace of Δ∗ξ for the eigenvalue λk = k(k + d − 2). In fact, more is true as it can be shown (see [FGS98, Chap. 3]) that the space of spherical harmonics Hk∗ (S d−1 ) of degree k consist precisely of the infinitely differentiable functions that are eigenfunctions of Δ∗ξ corresponding to λk . Furthermore, it can also be shown (see [War83, Chap. 6]) that the direct sum of these eigenspaces is all of L2 (S d−1 ), the space of square integrable function on S d−1 , and so we can write ∗ d−1 ). L2 (S d−1 ) = ⊕∞ k=0 Hk (S

(1.18)

The space Hk∗ (S d−1 ) has dimension Nk,d which is given by the multiplicity of the eigenvalue λk in (1.17), these numbers are computable and are given by N0,d = 1, and Nk,d

  2k + d − 2 k + d − 3 , k ≥ 1. = k k−1

We note that there exist positive constants C1 and C2 independent of k such that C1 (1 + k)d−2 ≤ Nk,d ≤ C2 (1 + k)d−2 , k ≥ 0.

(1.19)

Given an orthonormal basis {Yk, :  = 1, . . . , Nk,d } for Hk∗ (S d−1 ) the collection {Yk, :  = 1, . . . , Nk,d , k ≥ 0}

(1.20)

forms an orthonormal basis for L2 (S d−1 ). In view of this we see how the spherical harmonics can be used to give a Fourier analysis on the sphere. In particular, every function f ∈ L2 (S d−1 ) has an associated spherical Fourier expansion f =

Nk,d ∞  

f k, Yk,

(1.21)

k=0 =1

where

f k, = f, Yk, Sd−1 =

 S d−1

f (ξ)Yk, (ξ)dωd−1 (ξ)

(1.22)

1.5 Spherical Harmonics and Fourier Analysis

13

are the spherical Fourier coefficients of f. In the case of the circle S 1 the dimension of the spherical harmonic spaces Hk∗ (S 1 ) are given by N0,2 = 1 and Nk,2 = 2 for k ≥ 1, and the spherical harmonics are defined by Y0,1 =

1 1 1 , Yk,1 = cos kφ, Yk,2 = sin kφ, φ ∈ [0, 2π], k ≥ 1. (1.23) 2π π π

Here our definition coincides with the usual Fourier series on S 1 . According to the celebrated Addition theorem [Mü66] the orthonormal basis {Yk, :  = 1, . . . , Nk,d } for Hk∗ (S d−1 ) satisfies the following identity: Nk,d 

Yk, (ξ)Yk, (η) =

=1

Nk,d Pk,d (ξ T η), ξ, η ∈ S d−1 , k ≥ 0, ωd−1

(1.24)

where Pk,d is the d−dimensional Legendre polynomial of degree k, defined on [−1, 1] and normalized such that Pk,d (1) = 1. A useful alternative definition is provided by Rodrigues formula [Mü66] Pk,d (t) =

(−1)k Γ ( d−1 2 ) 2k Γ (k +

d−1 2 )

(1 − t 2 )

3−d 2

d−3 dk (1 − t 2 )k+ 2 . k dt

(1.25)

For a given dimension d the family of d−dimensional Legendre polynomials (Pk,d )k≥0 satisfy the inequality |Pk,d (t)| ≤ 1, t ∈ [−1, 1], k ≥ 0.

(1.26)

They also satisfy the following orthogonality relation 

1 P j,d (t)Pk,d (t)(1 − t ) 2

d−3 2

dt =

−1

ωd−1 ωd−2 Nk,d

if j = k;

0

otherwise,

d≥2

j, k ≥ 0,

(1.27) which allows us to work with Legendre expansions of function. In particular, if ψ is continuous on the interval [−1, 1] such that 1 |ψ(t)|(1 − t 2 )

d−3 2

dt < ∞,

(1.28)

−1

then ψ has a Legendre series expansion ψ(t) =

∞  k=0

aψ (k)Pk,d (t),

(1.29)

14

1 Motivation and Background Functional Analysis

where the Legendre coefficients {aψ (k)}∞ k=0 are given by, ωd−2 Nk,d aψ (k) = ωd−1

1 Pk,d (t)ψ(t)(1 − t 2 )

d−3 2

dt.

(1.30)

−1

A useful trick, and one that we shall use in this book, is to allow a univariate ψ : [−1, 1] → R (which we assume satisfies (1.28)), to induce a zonal kernel Ψ : S d−1 × S d−1 → R via the formula Ψ (ξ, η) = ψ(ξ T η), where ξ, η ∈ S d−1 .

(1.31)

This kernel will possess a spherical Fourier series of the form, Ψ (ξ, η) =

Nk,d ∞  

k Yk, (ξ)Yk, (η), ψ

(1.32)

k=0 =1

and if we know the Legendre expansion coefficients of ψ then we can use (1.24) and (1.29) to deduce that Fourier coefficients of (1.32) are given by

k = aψ (k)ωd−1 = ωd−2 ψ Nk,d

1 Pk,d (t)ψ(t)(1 − t 2 )

d−3 2

dt, k ≥ 0.

(1.33)

−1

Remark 1.1 Using the elementary observation ξ T η = cos θ where θ is the smallest angle between ξ, η ∈ S d−1 , we can also view ψ as a function of [0, π] by considering ψ(cos θ). Throughout this book we will employ both representations and, rather than introduce additional notation, we will simply consider ψ(t) (as a function of the inner product) and ψ(θ) (as a function of the angle), where the context is clear.

1.6 Sobolev Spaces in Euclidean Space In this section we review the theory and properties of classical Sobolev spaces. Once again our aim is to be self-contained as we provide the reader with handpicked results which, as we shall see, play their role in developing the theory of spherical basis functions. A good reference for this material is [AD75]. We begin in the Euclidean setting where we have the following definitions. Definition 1.4 Given a domain Ω in Rd we say that a Lebesgue measurable function 1 (Ω), if f : Ω → R is locally integrable, and write f ∈ Lloc

1.6 Sobolev Spaces in Euclidean Space

15

 | f (x)|dx < ∞ for all compact subsets K ⊂ Ω. K 1 (Ω) has a weak derivative of order |α| if there Definition 1.5 A function f ∈ Lloc exists a locally integrable function (denoted by Dα f ) such that

 Ω

f (x)Dα γ(x)dx = (−1)|α|

 Ω

Dα f (x)γ(x)dx,

for all γ ∈ D(Ω).

With this notion of differentiability we define the integer order Sobolev spaces as W k (Ω) := { f ∈ L2 (Ω) : Dα f ∈ L2 (Ω), for all 0 ≤ |α| ≤ k},

(1.34)

where it is assumed that the derivatives Dα f are taken in the weak sense. The space is then equipped with the inner product

f, gW k (Ω) :=

 

Dα f (x)Dα g(x)dx,

0≤|α|≤k Ω

(1.35)

which induces the norm ⎛  f W k (Ω) = ( f,

1/2 f )W k (Ω)

=⎝



⎞1 2

D

α

f 2L2 (Ω) ⎠

.

(1.36)

0≤|α|≤k

One can easily show that the norm (1.36) makes W k (Ω) a Hilbert space, i.e., a complete inner product space. There are several equivalent ways of defining Sobolev spaces for a more general index s. Perhaps the most intuitive is to start with a parameter τ ∈ (0, 1) for which we define the space  | f (x) − f (y)|2 dxdy < ∞ . d+2τ Ω Ω x − y (1.37) This gives rise to the following definition of fractional order Sobolev spaces. W τ (Ω) :=



 

f ∈ L2 (Ω) : | f |2W τ (Ω) :=

Definition 1.6 Let Ω be a bounded domain with a suitably smooth boundary. Then, for s = k + τ we define   (1.38) W s (Ω) := f ∈ W k (Ω) : Dα f ∈ W τ (Ω), for all |α| = k . We equip this space with the inner product

16

1 Motivation and Background Functional Analysis

f, gW s (Ω) = f, gW k (Ω) +

   |α|=k Ω

Ω

| f (x) − f (y)| · |g(x) − g(y)| dxdy x − yd+2τ (1.39)

and this, in turn, induces a norm whose square is given by  f 2W s (Ω) :=  f 2W k (Ω) +

 |α|=k

|Dα f |2W τ (Ω) ,

(1.40)

which, as with the integer order case, makes W s (Ω) a Hilbert space. The most familiar Sobolev space is L2 (Ω) which corresponds to the index s = 0. Indeed we can think of Sobolev spaces for positive indices as subspaces of L2 (Ω). As the index increases, we require more derivatives to be square integrable, the smoothness requirements become stronger and the spaces become smaller. The Sobolev spaces are nested, that is,  · W s (Ω) ≤  · W t (Ω) whenever s ≤ t. The link between the increasing index and the continuity requirements is made explicit by the Sobolev embedding theorem [AD75, Chap. 5] which assets that if s > n + d/2 for some nonnegative integer n, then W s (Ω) is continuously embedded in C n (Ω). In particular, if s > d/2 then all functions in W s (Ω) will be continuous. For our purposes, it turns out that another construction of the fractional spaces is more convenient. This is obtained by applying the interpolation theory of Banach spaces and, for the convenience of the reader, we briefly compose the relevant material, however for a detailed account see [Tri78] or [BL76]. We begin with a definition. Definition 1.7 (Interpolation Couple) Given two Banach spaces (A0 ,  · A0 ) and (A1 ,  · A1 ) we say that {A0 , A1 } is an interpolation couple if A1 ⊂ A0 and a1 A0 ≤ ca1 A1 for all a1 ∈ A1 , where c > 0 does not depend on a1 . Let {A0 , A1 }, {B0 , B1 } denote two interpolation couples and let T be a continuous linear operator such that T : Ai → Bi , i ∈ {1, 2}. The main aim of the interpolation of these spaces is then, to construct, for τ ∈ (0, 1), Banach spaces Aτ and Bτ , (commonly called interpolation spaces), satisfying (i) A1 ⊂ Aτ ⊂ A0 , and B1 ⊂ Bτ ⊂ B0 ; (ii) the restriction of T to Aτ is a continuous linear operator from Aτ to Bτ . There are many different methods available for constructing interpolation spaces, and we choose a rather easy to handle approach due to Peetre, see [BS02]. Definition 1.8 (Peetre’s K-functional) Let {A0 , A1 } be an arbitrary interpolation couple. Then, for any f ∈ A0 and t > 0, we define the K-functional on {A0 , A1 } by

1.6 Sobolev Spaces in Euclidean Space

17

K(t, f ) = inf ( f − gA0 + tgA1 ).

(1.41)

g∈A1

For τ ∈ (0, 1), we use the K-functional to define a τ −dependent norm on A0 ,  f τ = K(t, f )t

−τ

 L2 ((0,∞), dt ) = t

∞  K(t, 0

f)

2

tτ

dt t

1/2 .

(1.42)

The set Aτ = (A0 , A1 )τ = { f ∈ A0 :  f τ < ∞}

(1.43)

then forms a Banach interpolation space for the interpolation couple {A0 , A1 }. Further, if A0 and A1 are Hilbert spaces then the norm  · τ satisfies the parallelogram law and therefore induces a Hilbert space structure. In order to demonstrate that this approach fulfils both (i) and (ii) above we consider the following result. Proposition 1.2 (Operator interpolation property) Let {A0 , A1 }, {B0 , B1 } denote two interpolation couples and, for τ ∈ (0, 1), let Aτ and Bτ denote the corresponding interpolation spaces, given by (1.43). Let T be a linear operator that maps Ai to Bi , such that T f Bi ≤ Ci ·  f Ai , for all f ∈ Ai , i ∈ {0, 1}. Then T maps Aτ to Bτ and T f Bτ ≤ C01−τ C1τ ·  f Aτ , for all f ∈ Aτ . Proof Let f ∈ Aτ , then for any g ∈ A1 we have K(t, T f ) ≤ T f − T gB0 + tT gB1 ≤ C0 ·  f − gA0 + tC1 · gA1   tC1 = C0 ·  f − gA0 + · gA1 . C0 Taking the infimum over g ∈ A1 , we have K(t, T f ) ≤ C0 · K(tC1 /C0 , f ). Setting s = tC1 /C0 we find that t −τ K(t, T f ) ≤ C01−τ C1τ · s −τ K(s, f ), and since dt/t = ds/s, we use (1.42) to deduce that,

(1.44)

18

1 Motivation and Background Functional Analysis

T f Bτ = K(t, T f )t −τ L2 ((0,∞), dt ) t

≤

C01−τ C1τ

· K(s, f )s

=

C01−τ C1τ

·  f Aτ .

−τ

L2 ((0,∞), ds ) s

This holds for all f ∈ Aτ and so completes the proof.



Let k be a non-negative integer and let Ω be a domain in Rd . The nesting property of the Sobolev spaces ensures that {W k (Ω), W k+1 (Ω)} is a valid interpolation couple. In particular, we have the following definition. Definition 1.9 (Sobolev space via interpolation) Let Ω be a bounded domain with a suitably smooth boundary. Then, for s = k + τ we define the “Sobolev interpolation space” W s (Ω) = (W k (Ω), W k+1 (Ω))τ := { f ∈ W k (Ω) :  f W k+τ (Ω) < ∞},

(1.45)

where, by (1.42), we have ⎛  f W k+τ (Ω) = ⎝

∞ 

K(t, f ) tτ

2

⎞1/2 dt ⎠ t

,

(1.46)

0

and K(t, f ) =

inf

( f − gW k (Ω) + tgW k+1 (Ω) ).

g∈W k+1 (Ω)

(1.47)

It can be shown, see [BS02, Chap. 9], that the two fractional Soboloev spaces given by Definitions 1.6 and 1.9 respectively, coincide as sets and their norms are equivalent. In view of this, and unless stated otherwise, we shall select the Sobolev interpolation space as our definitive fractional Sobolev space. In approximation theory Sobolev spaces serve as benchmark spaces of target functions that we wish to interpolate at a set of points X in some domain Ω. Any investigation into the accuracy of the interpolant will usually involve analysing the error function which itself belongs to the Sobolev space but vanishes at every point in X. In view of this it is helpful to explore the Sobolev subspaces where the functions are given zero conditions. In the integer order case we define consider  k (Ω) = W



 f ∈ W k (Ω) : f (x) = 0, x ∈ X .

(1.48)

For fractional subspaces, defined via interpolation, we propose the analogous definition    k+τ (Ω) = f ∈ W k+τ (Ω) : f (x) = 0, x ∈ X . (1.49) W

1.6 Sobolev Spaces in Euclidean Space

19

However, before we make this assertion some technical justification is required; we need to ensure that (1.49) is the same space that results from interpolating between the integer order subspaces. We establish this in the next few results which require some further background from Banach space theory. Definition 1.10 (Banach space projection) Let A be a Banach space. A continuous linear mapping P : A → A is called a projection on A if P 2 = P, that is, if P(P(a)) = Pa for every a ∈ A. Definition 1.11 (Complemented Banach subspace) A closed linear subspace  A of a Banach space A is said to be a complemented subspace of A if and only if there exists a continuous projection P on A with P(A) =  A. Let {A0 , A1 } be an arbitrary interpolation couple and let P denote a projection operator acting upon both A0 and A1 . Since a complemented subspace of a Banach A1 = P(A1 ) as Banach space is closed, we can consider the spaces  A0 = P(A0 ) and  spaces in their own right, with the inherited norms  · A0 and  · A1 respectively. Indeed, since A1 ⊂ A0 ⇒ P(A1 ) ⊂ P(A0 ) and a1 A0 ≤ ca1 A1 ⇒ Pa1 A0 ≤ cPa1 A1 we can deduce that { A0 ,  A1 } is a valid interpolation couple. The following result, concerning interpolation between complemented subspaces, is due to Triebel, see [Tri78]; Sect. 1.17. Theorem 1.2 Let {A0 , A1 } and { A0 ,  A1 } denote two interpolation couples, where  A0  and A1 are the complemented subspaces of A0 and A1 respectively, with common projection operator P. Then, for τ ∈ (0, 1), we have  A0 ,  A1 )τ = Aτ ∩  A0 = P(Aτ ), Aτ = (

(1.50)

that is,  Aτ is the complemented subspace of Aτ with the same projection P. We illustrate this result with the our Sobolev space application. N Theorem 1.3 Let X = {xi }i=1 denote a set of distinct points in a domain Ω ⊂ Rd . d  k (Ω) (1.48) is a Let k > 2 be a non-negative integer, then the Sobolev subspace W complemented subspace of W k (Ω).

 k (Ω) = PX (W k (Ω)), for some continuous projection Proof We need to show that W k PX on W (Ω). To accomplish this we can choose a set of N linearly independent cardinal functions

γi ∈ D(Ω) ⊂ W k (Ω) with the following properties 1.

γi (xi ) = 1, for i = 1, . . . , N, 2.

γi has compact support Ki ⊂ Ω and Ki ∩ K j = ∅ whenever i = j.

20

1 Motivation and Background Functional Analysis

Then, since k > d2 the Sobolev embedding theorem allows us to define a projection operator on W k (Ω) by N  QX : f → f (xi )

γi . (1.51) i=1

 k (Ω). Thus, setting PX := I − QX , We note that the null space of QX is precisely W where I denotes the identity, completes the proof.   k (Ω), W  k+1 (Ω)} is a valid interpolation For k > d2 , we can deduce that {W couple and thus, for τ ∈ (0, 1), we can define its interpolation space in the spirit of Definition 1.9. Furthermore, as a corollary of Theorem 1.2, we can formulate the following result. Theorem 1.4 (Equivalence of fractional subspaces) Let k be a non-negative integer and τ ∈ (0, 1). The Sobolev subspaces    k+1 (Ω)  k (Ω), W defined via interpolation W

 k+τ (Ω) (see (1.49)) and W

τ

are equivalent. Returning now to the global spaces, when Ω = Rd , there is an equivalent definition of W s (Rd ) in terms of the Fourier transform, defined for f ∈ L2 (Rd ) by

f (x) =



1 (2π)

d 2

Rd

f (y)e−ix y dy, x ∈ Rd .

For s ≥ 0 we can show that   W s (Rd ) = f ∈ L2 (Rd ) : ||| f | ||2W s (Rd ) :=

T

Rd

 (1 + x2 )s |

f (x)|2 d x < ∞ , (1.52)

where the norms |||·| ||W s (Rd ) and  · W s (Rd ) are equivalent. See [AD75] and the references therein. In order to generalise the various results established for Rd to the case of a bounded domain Ω, it is important to know whether there exists a continuous linear extension operator E : W k (Ω) → W k (Rd ), satisfying (E f )|Ω = f, for all f ∈ W k (Ω).

(1.53)

In [Ste70], Stein showed that provided the bounded domain Ω ⊂ Rd has a “minimally smooth boundary” then it is always possible to construct such an extension operator (1.53). The minimally smooth boundary condition imposed on Ω is rather technical, however, for our application it is sufficient to note that any open ball B(x, r )

1.6 Sobolev Spaces in Euclidean Space

21

is a valid candidate. With this in mind we present a specialized version of Stein’s remarkable theorem. Theorem 1.5 Let Ω = B(x, r ) be an open ball in Rd then there exists an extension operator (1.53) defined for all k ∈ N0 such that E f W k (Rd ) ≤ Cext ·  f W k (B(x,r )) wher e Cext is independent o f f. Furthermore, there exists 0 > 0 such that for any ∈ (0, 0 ) E can be chosen so that the support of E f is contained in B(x, (1 + )r ).

1.7 Sobolev Spaces on the Unit Sphere In this section we discuss Sobolev spaces on spheres. When the domain is the entire sphere S d−1 , the simplest way to describe W s (S d−1 ) [LM72] is somewhat analogous to definition (1.52) for Rd . We define W s (S d−1 ) for s ≥ 0 by ⎧ ⎨ ⎩

f ∈ L2 (S d−1 ) :  f 2W s (Sd−1 ) :=

Nk,d ∞   k=0 =1

⎫ ⎬   2 (1 + λk )β 

f k  < ∞ . ⎭

(1.54)

We note that the norm  · W s (Sd−1 ) is induced by the inner product

f, g =

Nk,d ∞   (1 + λk )β

gk, , f k,

(1.55)

k=0 =1

which we can show makes W s (S d−1 ) a Hilbert space. Recall that the numbers λk = k(k + d − 2) are the eigenvalues of the LaplaceBeltrami operator, which behave like k 2 for large k and

f k, are the Fourier coeffif (x) in (1.52) then cients of f defined by (1.22). We see that if we identify

f k, with

in some sense the Sobolev space W s (S d−1 ) is a periodic version of W s (Rd ). Clearly the Sobolev spaces are nested, that is  · W s (Sd−1 ) ≤  · W t (Sd−1 ) , whenever s ≤ t. The analogue of the Sobolev embedding theorem still holds on S d−1 . The theorem asserts that whenever s > n + (d − 1)/2, for some non-negative integer n, then W s (S d−1 ) is continuously embedded in C n (S d−1 ). The above definition is sufficient for functions that are defined globally on S d−1 . If we want to examine functions locally then we need to define Sobolev spaces for subsets of the sphere. We do this by considering the sphere to be a special instance of a differentiable manifold. We start with a review of some material from differential geometry [L95]. The notion of a differentiable manifold is necessary for extending the methods of calculus to spaces more general than Rd . Roughly speaking, a differentiable manifold M of dimension d − 1 is a topological space such that every

22

1 Motivation and Background Functional Analysis

point of M is in a neighbourhood that is homeomorphic to an open set in Rd−1 . All the manifolds considered will be compact. The manifold is said to be of class C ∞ if there exists a collection of open sets {U j }nj=1 in M that covers M such that each coordinate neighbourhood U j is homeomorphic to the unit ball B0 = B(0, 1) in Rd−1 , and the homeomorphism or coordinate map φ j : U j → B0 and its inverse φ−1 j : B0 → U j are infinitely differentiable mappings. The pair (U j , φ j ) is called a chart and the collection {U j , φ j }nj=1 is called an atlas for M. We need to ensure that  the charts are compatible, namely if Ui U j = ∅ then the change of coordinates φi ◦ φ−1 j : φ j (Ui ∩ U j ) → φi (Ui ∩ U j ) is an infinitely differentiable homeomorphism with positive Jacobian. In essence the existence of the covering {U j }nj=1 and associated maps {φ j }nj=1 means that we can treat the manifold as made up of patches that map down to the unit ball in Rd−1 . The advantage of this strategy is that many of the problems on differentiable manifolds are then easy to solve locally, i.e. on a suitable coordinate domain. In order to construct global solutions we require a partition of unity, which allows us to patch together the local solutions. Definition 1.12 (Partition of Unity) Let M be a compact manifold. A collection of n is said to be a partition of unity infinitely differentiable functions {χi : M → R}i=1 n subordinate to an open covering {Ui }i=1 of M if n (i) χi ≥ 0, (ii) χi has compact support in Ui , (iii) i=1 χi = 1. n We remark that, given any open cover {Ui }i=1 of a compact manifold M, one n can always find a partition of unity {χi }i=1 subordinate to it. The partition of unity allows us to express a function f : M → R, as

f =

n  (χi f ), where (χi f )(m) = χi (m) f (m), m ∈ M.

(1.56)

i=1

This gives us a decomposition of f in terms of local functions χi f, which are compactly supported in Ui . We define an associated function πi (χi f ) : Rd−1 → R, which we refer to as the (φi , χi )−projection of f onto Rd−1 . The map πi is defined for any f ∈ L2 (M) with compact support in Ui by  πi ( f )(x) =

f ◦ φi−1 (x), if x ∈ B0 ; 0 otherwise.

(1.57)

We note that for a general differentiable manifold we would have to define what we mean by L2 (M), however, for the sphere and its subsets, this is well understood. Therefore to find the value of πi (χi f ) at a point x ∈ B0 we take the inverse image of x under φi (the coordinate map associated to Ui ) and then use the value of χi f at that point. The function πi (χi f ) has compact support in B0 = φi (Ui ).

1.7 Sobolev Spaces on the Unit Sphere

23

After this preparation we can now define W s (M), for s ≥ 0 [Shi92, LM72] as   f ∈ L2 (M) : πi (χi f ) ∈ W s (Rd−1 ) for i = 1, . . . , n , (1.58) that is, we take the Sobolev space on M to be the set of functions in L2 (M) whose (φi , χi )−projection onto Rd−1 belongs to the Sobolev space W s (Rd−1 ) for i = 1, . . . , n. We define the norm on this space by  f W s (M) =

 n 

1 2

πi (χi f )2W s (Rd−1 )

.

(1.59)

i=1

We now illustrate the above definitions for the case of the unit sphere which is the classical example of a compact differentiable manifold. We begin by constructing a specific atlas for S d−1 . The same atlas for the sphere will be used throughout the book. We begin by defining a simple open cover and for this we set NP = (0, . . . , 0, 1) and SP = (0, . . . , 0, −1) to denote the north and south poles of S d−1 respectively. We then fix an angle π/2 < θ0 < 2π and define open sets U1 := G(NP, θ0 ) and U2 = G(SP, θ0 ).

(1.60)

If we choose θ0 close to π/2, then these open geodesic balls each cover a hemisphere and only overlap each other by a small amount. The sets U1 and U2 provide an open cover for S d−1 . We now need associated coordinate charts that map the covering sets to the unit ball. To do this we consider the stereographic projection map T1 of the punctured sphere S d−1 \ {NP} that takes a point ξ = (ξ1 , . . . , ξd ) ∈ S d−1 \ {NP} to the intersection of the the equatorial hyperplane {(x1 , . . . , xd−1 , 0) ∈ Rd } (which we identify as Rd−1 ) and the extended line that passes through ξ and NP. Explicitly we have   ξ1 ξd−1 . (1.61) ,..., T1 (ξ) = T1 (ξ1 , . . . , ξd ) = 1 − ξd 1 − ξd We remark that the stereographic projection T2 based on SP can be defined analogously. Specifically we have  T2 (ξ) = T2 (ξ1 , . . . , ξd ) =

ξ1 ξd−1 ,..., 1 + ξd 1 + ξd

 .

The stereographic projection map T1 (and similarly T2 ) has several nice properties. It is the restriction of a Möbius transformation therefore it will map a geodesic ball on S d−1 \ {NP} to a Euclidean ball in Rd−1 . For more details see [R06]. Let ξ, η ∈ U1 , then the image of U1 under T1 is an open Euclidean ball in Rd−1 centred at the origin. From [R06] the relationship

24

1 Motivation and Background Functional Analysis

ξ − η =

2T1 (ξ) − T1 (η) 1 + T1 (ξ)2 1 + T1 (η)2

,

(1.62)

holds. On S d−1 the Euclidean distance and the geodesic distance have the following elementary relationship   g(ξ, η) ξ, η ∈ S d−1 . (1.63) ξ − η = 2 sin 2 To determine the radius of T1 (U1 ) we set η = SP in (1.62), and suppose that ξ ∈ U1 then   T1 (ξ) g(ξ, SP) = sin 2 1 + T1 (ξ)2 from which we deduce that     θ0 g(ξ, SP) = T1 (ξ) < tan . tan 2 2 This implies that T1 (U1 )=B(0, tan(θ0 /2)). Similarly, T2 (U2 ) = B(0, tan(θ0 /2)). With this established we can define charts φi : Ui → B0 i = 1, 2, by the scaled stereographic projections φ1 =

1 1 · T1 and φ2 = · T2 , tan(θ0 /2) tan(θ0 /2)

(1.64)

2 and conclude that {Ui , φi }i=1 is a C ∞ atlas of covering coordinate charts for the 2 subordinate to these charts and, following sphere. We fix the partition of unity {χi }i=1 (1.58) and (1.59) we provide the following definition of a Sobolev space on S d−1 .

Definition 1.13 The Sobolev space W k (S d−1 ) of order k ∈ N0 is defined to be the set   f ∈ L2 (S d−1 ) : πi (χi f ) ∈ W k (Rd−1 ) for i = 1, 2 , (1.65) and is equipped with the norm  f W k (Sd−1 ) =

 2 

21 πi (χi f )2W k (Rd−1 )

.

(1.66)

i=1

This definition seems to depend on the choice of atlas used to define S d−1 . However, it can be shown that any two spaces defined using two different atlases coincide as sets, and the norms (1.66) are equivalent. Further, the definition also coincides with the Fourier theoretic one (1.54), see [LM72] for details.

1.7 Sobolev Spaces on the Unit Sphere

25

The charts have the useful property that two points in Ui that are a geodesic distance h apart on S d−1 are mapped to two points in Rd−1 whose Euclidean distance is equivalent to h. This is proved using (1.62) and (1.63) which imply  cos

2

θ0 2





g(ξ, η) Ti (ξ) − Ti (η) ≤ sin 2

 ≤ Ti (ξ) − Ti (η)

(1.67)

for ξ, η ∈ Ui , i, = 1, 2. The equivalence between sin θ and θ for small values of θ θ ≤ sin θ ≤ θ, θ ∈ [0, π/3] 2

(1.68)

combined with (1.64) and (1.67) shows that  sin θ0 φi (ξ) − φi (η ≤ g(ξ, η) ≤ 4 tan

θ0 2

 φi (ξ) − φi (η)

(1.69)

for all pairs of points ξ, η ∈ Ui such that g(ξ, η) ≤ 2π/3. With this in place we are now in position to define local Sobolev spaces on a geodesic ball G(z, θ) where z ∈ S d−1 . We begin by fixing an angle π/2 < θ0 < 2π/3 2 as described above, together and accordingly choose the global atlas A = {Ui , φi }i=1 2 with an associated partition of unity {χi }i=1 . We define open sets in Rd−1 by Ωi := φi (G(z, θ) ∩ Ui ) , i = 1, 2.

(1.70)

We define the Sobolev space W s (G(z, θ)) to be the set  f ∈ L2 (G(z, θ)) : (χi f ) ◦ φ

 

−1 

 ∈ W (Ωi ), for i = 1, 2, Ωi = ∅ (1.71) s

Ωi

and define the norm on W s (G(z, θ)) by  f 2W s (G(z,θ))

2 !  !  ! !2 −1  ! ! = !(χi f ) ◦ φ Ωi ! i=1

W s (Ωi )

,

(1.72)

where Ωi is defined by (1.70) and we adopt the convention that  · W s (Ωi ) = 0 if Ωi = ∅. We also note that if G(z, θ) lies outside of the support of χi (i.e., if supp(χi ) ∩ G(z, θ) = ∅) then χi f is identically zero and does not contribute to the norm. We close this section by presenting a closer inspection of the geometry behind the Sobolev space construction. We study the way the atlas and partition of unity work together mapping geodesic balls and defining functions and demonstrate how to ensure that a Sobolev space on a geodesic ball (with a sufficiently small radius) will always behave like a Sobolev space on a ball in Euclidean space.

26

1 Motivation and Background Functional Analysis

2 Remark 1.2 For a fixed angle θ0 ∈ (π/2, 2π/3) let A = {Ui , φi }i=1 denote the 2 global atlas as described above. We choose a partition of unity {χi }i=1 associated to A such that

supp(χi ) ⊂ Ui , i = 1, 2. In view of this we observe that there exists a positive constant CA , depending only on A and {χ1 , χ2 }, such that the geodesic distance from the set supp(χi ) to the boundary of Ui is strictly greater than CA , for i ∈ {1, 2}. For α ∈ (0, 1) we let Vi (α) denote the αCA −geodesic neighbourhood of supp(χi ). To visualise this we can think of Vi (α) as the set that covers supp(χi ) with a band of width αCA to spare, but does not cover Ui (the covering set of our atlas). Let us consider a geodesic ball G(z, θ) centred at some point z ∈ S d−1 with geodesic radius θ < CA /3. In view of (1.71) we can examine the cases for which functions f ∈ L 2 (S d−1 ) are eligible for membership of a local Sobolev space. 1. z ∈ / V2 (1/3) ⇒ G(z, θ) ⊂ V1 (2/3) ⊂ U1 , and supp{χ2 } ∩ G(z, θ) = ∅, 2. z ∈ / V1 (1/3) ⇒ G(z, θ) ⊂ V2 (2/3) ⊂ U2 , and supp{χ1 } ∩ G(z, θ) = ∅, 3. z ∈ V1 (1/3) ∩ V2 (1/3) ⇒ G(z, θ) ⊂ Vi (2/3) ⊂ Ui , for i ∈ {1, 2}, and (a) either supp{χ1 } ∩ G(z, θ) or supp{χ2 } ∩ G(z, θ) is nonempty, (b) both supp{χ1 } ∩ G(z, θ) and supp{χ2 } ∩ G(z, θ) are nonempty. The condition θ < CA /3 is sufficient to guarantee that closure of G(z, θ) is a subset of at least one of the open subsets U1 or U2 , defined by (1.60). Since we know that (1.64) map geodesic balls to Euclidean balls we can deduce that φi (G(z, θ)) = Ωi = B(xi , ri ) and φi (G(z, CA /3)) = B(xiA , riA ).

(1.73)

In general, concentric geodesic balls are not mapped to concentric Euclidean balls, i.e., we cannot conclude that xi coincides with xiA . Of the cases above 1 and 2 are the most straightforward. In these cases we have that G(z, θ) is not completely contained in one of the Ui and furthermore, since G(z, θ) does not intersect the support of one of the partitions of unity we’re guaranteed that only one of the terms in (1.72) is non-zero. Figure 1.1 illustrates all cases of the positioning of a geodesic ball. The more difficult case occurs when z ∈ S d−1 is positioned as in case 3 above. In this case we have the following strict inclusions G(z, θ) ⊂ G(z, CA /3) ⊂ Vi (2/3) ⊂ Ui , i ∈ {1, 2}, and thus both Ω1 and Ω2 are Euclidean balls. Indeed, taking the φi images and using (1.73) gives:

1.7 Sobolev Spaces on the Unit Sphere

27

Fig. 1.1 To illustrate the positioning of a geodesic ball of radius θ < CA /3

Ωi ⊂ φi (G(z, θ)) ⊂ φi (G(z, CA /3)) ⊂ φi (Vi (2/3)) ⊂ B0 (:= B(0, 1)), i ∈ {1, 2}. " #$ % " #$ % =B(xi ,ri )

=B(xiA ,riA )

Since these inclusions are strict there exists a positive constant eA such that B(xiA , riA ) ⊂ φi (Vi (2/3)) ⊂ B(0, 1 − eA ) ⊂ B0 , i ∈ {1, 2}. Thus, for any positive < eA , we have (see Fig. 1.2) Ωi = B(xi , ri ) ⊂ φi (G(z, θ)) ⊂ B(xi , ri + ) (1.74) ⊂

B(xiA , riA

+ ) ⊂ B0 , i ∈ {1, 2}.

In summary we have: Lemma 1.2 Let CA be the constant defined above. Then for any z ∈ S d−1 and θ < CA /3 we have 1. at least one of the open sets Ωi , i ∈ {1, 2} (1.70) is an open Euclidean ball; 2. there exists a positive constant eA , depending only on the atlas A, such that, if Ωi = B(xi , ri ), then B(xi , ri + ) ⊂ B0 for all 0 < < eA ; 3. if Ωi is not an open Euclidean ball and f ∈ W k (G(z, θ)) then (χi f )◦φi−1 |Ωi = 0. Lemma 1.2 allows us to take the view that a Sobolev space on a suitable geodesic ball in S d−1 essentially “behaves” like a Sobolev space on a Euclidean ball in Rd−1 .

28

1 Motivation and Background Functional Analysis

Fig. 1.2 Illustration of Lemma 1.2

The following result shows that the radii of the geodesic and Euclidean balls are comparable. Lemma 1.3 Assume that G(z, θ) ⊂ Ui , i ∈ {1, 2}, and let B(xi , ri ) be as in (1.73). Then there exist positive constants c0 and C0 such that c0 · θ ≤ ri ≤ C0 · θ, i ∈ {1, 2}.

(1.75)

Proof Inequality (1.69) gives sin θ0 φ1 (ξ) − φ1 (η) ≤ g(ξ, η) ≤ 2θ and g(ξ, η) ≤ 4 tan(θ0 /2)φ1 (ξ) − φ1 (η) ≤ 8 tan(θ0 /2)r1 . Since we have that 2r1 =

sup

ξ,η∈G(z,θ)

φ1 (ξ) − φ1 (η) and 2θ =

sup

ξ,η∈G(z,θ)

g(ξ, η),

we can complete the proof of the supremum on the left hand sides of the two inequalities above. We find that c0 = (4 tan(θ0 /2))−1 and C0 = (sin θ0 )−1 . 

Chapter 2

The Spherical Basis Function Method

2.1 Introduction Since the early 1990s the radial basis function (RBF) method has become a well established tool for reconstructing functions and for solving partial differential equations based on data prescribed at scattered locations throughout a domain in Rd . The method is extremely flexible, it has good approximation properties and does not become more elaborate as the space dimension increases. In the past decade or so approximation on spheres has become an area of growing interest with applications to physical geodesy, potential theory, geophysics, oceanography and meteorology. As more satellites are launched into space, the acquisition of global data is becoming more widespread and the demand for spherical data processing and solving problems of a global nature is increasing. In this chapter we will introduce the spherical basis function (SBF) method; the spherical analogue of the famous RBF method. In particular, we will use tools from Chap. 1 to construct a theoretical framework within which we can analyse the accuracy of the method. Specifically, we present two point-wise error bounds which both rely on the remarkable fact that, provided the data locations fill up the sphere sufficiently well, then it is possible to annihilate spherical harmonics of a certain order by using only a linear combination of point evaluations. The first error bound we encounter uses a global annihilation, i.e., every data location is used in the linear combination of point evaluations. In this case the relationship between the density of the data locations and the order of spherical harmonics to be annihilated is explicit and this is crucial to the error analysis that follows. The second error bound delivers a result of the same strength but uses a local annihilation of spherical harmonics and from this perspective one may consider this to be a stronger result. A drawback of this local approach is that, unlike the global case, there is no explicit connection between the density of the points and the degree of spherical harmonics to be annihilated and, as a result, much more additional work is required to establish the final result. The payoff for putting in this extra effort will be realized in the next chapter where the local error bound plays a key rôle in making significant improvements in the error bounds. © The Author(s) 2015 S. Hubbert et al., Spherical Radial Basis Functions, Theory and Applications, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-17939-1_2

29

30

2 The Spherical Basis Function Method

2.2 A Brief History of the RBF Method There already exists several excellent textbooks on the subject of radial basis functions [Fass07, Wen05, Buh03]. In view of this, the aim of this section is simply to establish the idea behind the method and also to highlight some of the crucial ideas and pioneering discoveries. This will set the scene for the rest of the chapter where we will demonstrate how these ideas can be recast onto the spherical setting. The problem of interpolating data measured at scattered locations in Euclidean space Rd (d > 1) arises in many areas of science and engineering. The importance of this problem is reflected in the literature, where a large number of different methods for its solution have been proposed. The problem itself is stated as follows. N of distinct data points in Rd and a target function Problem 1 Given a set X = {xi }i=1 d d f : R → R, find a function s : R → R that satisfies the interpolation conditions

s(xi ) = f (xi ), 1 ≤ i ≤ N.

(2.1)

The radial basis function (RBF) approach proposes a solution of the form s(x) =

N 

λ j φ(d(x, x j )), for λ j ∈ R, 1 ≤ j ≤ N,

(2.2)

j=1

where d(x, y) = x − y is usually the Euclidean metric and φ : [0, ∞) → R is the RBF. Applying the interpolation conditions (2.1) provides the linear system Aφ λ = f, where Aφ ∈ RN×N : Aφ,i j = φ(d(xi , x j )), 1 ≤ i, j ≤ N.

(2.3)

Thus a unique RBF interpolant exists if and only if the interpolation matrix Aφ is non-singular. One of the most attractive features of the RBF method is the fact that a unique interpolant is often guaranteed under rather mild conditions on the data points. In particular, if we choose our basis function to be any one of the following φ(r ) = e−cr

2

(Gaussian), 2 − 21

φ(r ) = (r + c ) 2

φ(r ) = (r 2 + c2 ) φ(r ) = r

1 2

(inverse multiquadric), (multiquadric), (linear),

r ≥ 0, c > 0,

(2.4)

then uniqueness is guaranteed provided that the N data points (N ≥ 2) are distinct, which is as simple a condition as one could wish for.

2.2 A Brief History of the RBF Method

31

Two other commonly used RBFs are φ(r ) = r 2 log r φ(r ) = r

(thin plate spline), (cubic),

3

r ≥ 0.

(2.5)

Now, in contrast to the previous candidates (2.4), there is no guarantee that the resulting interpolation matrix Aφ will be non-singular. Indeed, for the thin plate spline we can choose x2 , . . . , xN to be any distinct points on the unit sphere centred at x1 , in which case the first row and column of Aφ consists entirely of zeros, and hence is singular. In such cases, it is usual to add to s a polynomial of degree k ≥ 1, and so consider an interpolant of the form s(x) =

N 

λ j φ(d(x, x j )) +

j=1

M 

μ j p j (x),

(2.6)

j=1

where M = dim Πk (Rd ) and { p1 , . . . , pM } is a basis for Πk (Rd ). If we then impose the usual interpolation conditions N 

λ j φ(d(xi , x j )) +

j=1

M 

μ j p j (xi ) = f (xi ), 1 ≤ i ≤ N,

(2.7)

j=1

we observe that the addition of the polynomial introduces an extra M degrees of freedom. These are usually taken up by insisting that the RBF coefficients satisfy the following moment conditions: N 

λ j pi (x j ) = 0, 1 ≤ i ≤ M,

(2.8)

j=1

or equivalently, we have the linear system      Aφ P λ f = μ 0 PT 0

(2.9)

where Aφ is as in (2.3) and where P ∈ RN×M is given by Pi j = p j (xi ), for 1 ≤ i ≤ N and 1 ≤ j ≤ M.

(2.10)

Thus a unique augmented RBF interpolant exists if and only if the augmented interpolation matrix (2.9) is non-singular.

32

2 The Spherical Basis Function Method

One of the earliest examples of an RBF interpolant dates back to the late 1960s when the cubic spline method was developed for interpolating a univariate function f : [x1 , xN ] → R at distinct data points x1 < · · · < xN . The resulting interpolant s is composed of cubic polynomial pieces, that are joined so that the second derivative of s is continuous. A good account of the cubic spline method is given in [Pow81] where it is shown that in order to guarantee the uniqueness of the interpolant, it is necessary to impose suitable end conditions at x1 and xN . One useful condition is to set (2.11) s  (x1 ) = s  (xN ) = 0, in which case s is called the “natural” cubic-spline and has the form s(x) =

N 

λ j |x − x j |3 + a + bx, x ∈ R.

(2.12)

j=1

We  remark that the end conditions (2.11) are equivalent to the moment conditions λj = λ j x j = 0 which appear in (2.8), and thus we may regard the natural cubic-spline method as a special case of univariate RBF interpolation with φ(r ) = r 3 . The natural cubic spline is a good starting point because the method itself has several interesting theoretical properties. In particular, if we consider the following function space  1 2  2 H = { f ∈ L2 (R) : | f | := | f (x)| d x < ∞}, (2.13) R

then |·| is a semi-norm with null space Π1 (R) making H a semi-Hilbert space. Further, it is well known that (2.12) is the unique solution to the following variational problem minimise {|s| : s ∈ H and s(xi ) = f (xi ) 1 ≤ i ≤ N}.

(2.14)

For a detailed account of the many aspects of spline interpolation see [dB78]. The next appearance of an RBF came in 1971 when Rolland Hardy, a geoscientist, first suggested the use of the multiquadric basis function (see (2.4)) to interpolate scattered data in the plane. The discovery of this function arose from a purely heuristic approach to a problem in topography and the success of the resulting interpolation scheme, for solving 2D contour and mapping problems, is reported in [Ha71]. This appears to be the first application of the RBF method beyond the univariate setting. The next landmark discovery occurred in 1977 when Jean Duchon [Du77] approached the data fitting problem from the variational perspective. Duchon was one of the first mathematicians to generalise the notion of a natural cubic spline to higher dimensions and, to illustrate his contribution, consider a non-negative integer m > d/2 which indexes the following space of functions Hm,d = { f ∈ L2 (Rd ) : Dα f ∈ L2 (Rd ), for all |α| = m}.

(2.15)

2.2 A Brief History of the RBF Method

33

This space is then equipped with the semi-norm ⎛

| f |m,d

⎞1/2  m!  =⎝ |(Dα f )(x)|2 dx⎠ , α! Rd

(2.16)

|α|=m

with null space Πk−1 (Rd ). We note that, for d = 1 and m = 2, this space is the same as H above. More generally, Hm,d is closely related to the Sobolev space W m (Rd ) and shares many of its properties. In particular, since m > d/2, we know from the Sobolev embedding theorem that Hm,d is a semi-Hilbert space of continuous functions. Thus, following the cubic spline approach, for any f ∈ Hm,d the following variational problem was considered: minimise{|s|m,d : s ∈ Hm,d and s(xi ) = f (xi ) 1 ≤ i ≤ N}.

(2.17)

Using sophisticated techniques from distribution theory Duchon showed that the solution to (2.17), which he termed the Dm (Rd )−spline, has the form of an augmented RBF (2.6), where  φ(r ) :=

(−1)n r 2m−d log r (with n = m − d−2 2 ), d−1 n 2m−d (with n = m − 2 ), (−1) r

if d is even, if d is odd,

(2.18)

and where the augmented polynomial is of degree n −1. We note again that for d = 1 and m = 2 we recover the natural cubic spline. A year later Duchon [Du78] presented a study of the accuracy of Dm (Rd )−spline interpolation. To set the scene, it is assumed that we wish to interpolate a function N , located in a smooth, bounded f ∈ Hm,d over a set of distinct data points X = {xi }i=1 d domain Ω ⊂ R . The density of the set X ⊂ Ω is measured by using the mesh-norm h := h(X, Ω) := sup min y − xi , y∈Ω xi ∈X

(2.19)

and our aim is to investigate how the Dm (Rd )−spline interpolant s f approximates f as the data points become dense in Ω, that is, as h → 0. Definition 2.1 (The Duchon strategy) The Duchon strategy for delivering error bounds for RBF interpolation consists of the following steps: 1. construct a scalable quasi-uniform mesh for the domain, that is a collection of points on Ω so that it can be covered by a union of small open balls Bi (centred at each of the mesh points) that have uniformly bounded overlap: 2. estimate the local interpolation error using data prescribed on each Bi ; 3. by way of a suitable extension operator create the gluing result which combines the local error estimates to provide a final estimate for Ω.

34

2 The Spherical Basis Function Method

The justification for each of the three steps above is provided in [Du78], where geometric arguments and techniques from Sobolev space theory play a prominent role. Moreover, employing the strategy yields error bounds of the form s f − f L p (Ω) = O(h

m− d2 + dp

),

p ∈ [2, ∞].

(2.20)

In 1982, Richard Franke [F82] published the results of his survey on scattered data interpolation methods for R2 . In his report, over 30 different methods are tested including Hardy’s multiquadric and Duchon’s thin plate spline. Each method was assessed over a range of criterion including accuracy, ease of implementation and visual smoothness. Of all the methods tested Hardy’s multiquadric scheme performed the best and Duchon’s thin plate spline was also highly rated. These findings were particularly intriguing since, at the time, there was no mathematical basis to justify the use of multiquadric interpolation. In view of this Franke proposed the conjecture that the interpolation matrix (2.3) corresponding to the multiquadric basis function is non-singular. The invertibility of the interpolation matrices associated with the common RBFs was proven in two stages. First, Schoenberg [Sch38] in 1938 proved the unique solvability of (2.3) for a small class of RBFs. Then, in 1986, Micchelli [Mi86] extended Schoenberg’s result and established a larger class of RBFs for which (2.9) is uniquely solvable. In particular, Micchelli showed how this extension could be used to settle Franke’s conjecture on the multiquadric basis function. For the convenience of the reader, we present a brief account of the Schoenberg-Micchelli theory. Definition 2.2 (Positive Definite Functions) A continuous function φ : [0, ∞) → R N of is said to be positive definite (φ ∈ PD) if, for any d ≥ 1 and any set X = {xi }i=1 d distinct points in R , the quadratic form λT Aφ λ =

N  N 

λ j λk φ(d(x j , xk ))

(2.21)

j=1 k=1

is non-negative for all λ ∈ RN . Furthermore, if (2.21) is positive for all λ ∈ RN \{0} then we say that φ is strictly positive definite (φ ∈ SPD). In addition, we consider the following interesting class of functions first studied by Bernstein in the early 1930s. Definition 2.3 (Completely Monotone Functions) A continuous function f : [0, ∞) → R is said to be completely monotone on (0, ∞) if f ∈ C ∞ (0, ∞) and (−1)l f (l) (r ) ≥ 0, for all r > 0 and l = 0, 1, 2 . . . . In [Sch38] Schoenberg provided the following important theorem

(2.22)

2.2 A Brief History of the RBF Method

35

Theorem 2.1 (Schoenberg) A continuous function φ : [0, ∞) → R, belongs to PD √ if and only if the function f = φ( ·) is completely monotone on (0, ∞). Moreover, if, in addition, f is not a constant then φ belongs to SPD. Using Schoenberg’s theorem we can immediately deduce that the Gaussian and inverse multiquadric basis functions belong to SPD. This observation establishes the solvability of the RBF method for these cases. Following on from Schoenberg’s work we consider the following, more general, class of functions. Definition 2.4 (Conditionally Positive Definite Functions) A continuous function φ : [0, ∞) → R is said to be conditionally positive definite of order m ∈ N N of distinct points in Rd , (φ ∈ CPD(m)) if, for any d ≥ 1 and any set X = {xi }i=1 the quadratic form (2.21) is non-negative on the subspace Vm−1 = {λ ∈ RN :

N 

λi p(xi ) = 0 for all p ∈ Πm−1 (Rd )}.

(2.23)

i=1

Furthermore, if (2.21) is positive for all λ ∈ Vm−1 \{0} then we say that φ is conditionally strictly positive definite of order m (φ ∈ CSPD(m)). For augmented RBF interpolation (2.6) it is usual to insist that the geometry of the locations satisfy the following mild property. Definition 2.5 (Unisolvency) Let m be a positive integer and let M= dim Πm−1 (Rd ). A set of distinct points {x1 , . . . , xM } is said to be unisolvent with respect to Πm−1 (Rd ) if the only element of Πm−1 (Rd ) to vanish at each xi is the zero polynomial. The following theorem establishes a unique solution to the augmented interpolation problem (2.9) in the case where φ ∈ CSPD(m). Theorem 2.2 Let φ ∈ CSPD(m) and X = {x1 , . . . , xN } denote a set of N distinct data points in Rd such that (i) N ≥ M = dim Πm−1 (Rd ), (ii) X contains a subset that is unisolvent with respect to Πm−1 (Rd ). Then the augmented interpolation problem (2.9) has a unique solution. Proof It is sufficient to show that, if λ ∈ RN and μ ∈ RM satisfy the homogeneous linear system      Aφ P λ 0 = (2.24) μ 0 PT 0 then they are zero vectors. We note that PT λ = 0 implies that λ ∈ Vm−1 , and thus

36

2 The Spherical Basis Function Method

0 = λT (Aφ λ + Pμ) = λT Aφ λ + (μT (PT λ))T = λT Aφ λ. Since Aφ induces a positive definite form on Vm−1 , this implies that λ = 0. d Let { p1 , . . . , p M } denote the basis of Πm−1 (R ) used to define the matrix P (2.10) M ∗ and let p (x) = j=1 μ j p j (x). Now, since λ = 0 we have that Pμ = ( p ∗ (x1 ), . . . , p ∗ (xN ))T = 0, and so, using the unisolvency of X, we conclude that μ = 0.



Inspired by Schoenberg’s characterisation theorem, Micchelli proved the following important extension. Theorem 2.3 (Micchelli) A continuous function φ : [0, ∞) → R, belongs to √ CPD(m) if the function f = φ( ·) is such that (−1)m f (m) is completely monotone on (0, ∞). Moreover, if, in addition, f ∈ C m−1 [0, ∞) and is not a polynomial of degree at most m, then φ belongs to CSPD(m). We note in passing that, as Micchelli suspected, the converse of this theorem is also true and this was settled in 1993 by Guo et al. [GHS93]. As it stands, Micchelli’s theorem serves as an important source of applicable RBFs. The popular choices are the generalised Duchon splines φ(r ) = (−1)k+1 r 2k log r φ(r ) = (−1)

β 2β

r

∈ CSPD(k + 1), k ∈ N, ∈ CSPD( β + 1), β > 0 and β ∈ / N,

(2.25)

and the generalised multiquadrics φ(r ) = (−1) β +1 (r 2 + c2 )β ∈ CSPD( β + 1), β > 0 and β ∈ / N, φ(r ) = (r 2 + c2 )β

∈ SPD,

β < 0.

(2.26)

In addition, Micchelli also proved the following important theorem concerning CSPD(1) functions. Theorem 2.4 Let −φ be CSPD(1) with φ(0) ≥ 0 then the corresponding interpolation matrix Aφ given by (2.3) is non-singular. Proof By definition the matrix Aφ induces a positive definite form on the N − 1 dimensional hyperplane, given by V0 = {λ = (λ1 , . . . λN )T ∈ RN :

N 

λi = 0}.

i=1

Thus Aφ has at least N − 1 positive eigenvalues. However trace(Aφ ) ≤ 0, and so the remaining eigenvalue must therefore be negative. 

2.2 A Brief History of the RBF Method

37

In view of Franke’s numerical findings this theorem is particularly important, for it establishes the solvability of Hardy’s original multiquadric interpolation scheme. Indeed, Micchelli’s overall contribution has encouraged a large communities of mathematicians to study the properties of RBFs. One of the most important of the post-Micchelli discoveries is the so-called variational approach, i.e., that every RBF interpolant can be viewed as the solution to a minimal norm interpolation problem in some reproducing kernel Hilbert space (commonly called the Native space). This approach, which can be viewed as a generalization of Duchon’s work, was developed by Madych and Nelson in the early 1980 s and finally published in 1990 [MN90]. Over the years many researchers have studied the original Madych-Nelson approach and, as a result, a sound theoretical framework for RBF interpolation has emerged where error estimates can easily be delivered. To give a flavour of the Madych-Nelson theory, we consider the following definition. Definition 2.6 Let Φ ∈ C(Rd ) be of polynomial growth, i.e., there exists k ∈ N0 , : Rd \{0} → R such that |Φ(x)| = O(xk ) as x → ∞. A continuous function Φ is said to be the generalised Fourier transform of Φ if there exists m ∈ N0 , such that 

 Rd

Φ(x) γ (x)dx =

Rd

Φ(ω)γ(ω)dω

(2.27)

holds for all functions γ from the subspace  S m−1 (Rd ) = {γ ∈ S(Rd ) :

Rd

γ(x) p(x)dx = 0 for all p ∈ Πm−1 (Rd )}.

Furthermore, the minimal choice of m is called the order of Φ. We now quote a specialisation of a result due to Iske, which can be found in [I95]. Theorem 2.5 Let φ ∈ C[0, ∞) and assume that Φ(x) = φ(x) is of polynomial growth, then the following are equivalent (i) φ ∈ CPD(m); of order m which is non(ii) Φ possesses a generalised Fourier transform Φ d negative and not identically zero on R \{0}. It turns out that the generalised Fourier transforms of the most commonly used basis functions φ ∈ CSPD(m) are positive on Rd \{0} [SW01]. This fact allows us to define the so-called native space by  Hφ = { f ∈ L2 (Rd ) : | f |2φ =

1 | f (ω)|2 dω < ∞}, Rd Φ(ω)

(2.28)

where | · |φ is a semi-norm whose kernel is Πm−1 (Rd ). For a given φ ∈ CSPD(m), the nature of Hφ is largely determined by the decay rate of Φ(ω). Specifically, if Φ(ω) has a polynomial rate of decay then Hφ is closely

38

2 The Spherical Basis Function Method

related to a certain Sobolev space. On the other hand, if Φ(ω) decays exponentially quickly then Hφ is a smaller space of C ∞ functions. Madych and Nelson were the first to illustrate the importance of the native spaces. Specifically, they showed that given any f ∈ Hφ , the solution to the following Duchon-like variational problem minimise{|s|φ : s ∈ Hφ and s(xi ) = f (xi ) 1 ≤ i ≤ N},

(2.29)

is precisely the unique φ−based RBF interpolant. Micchelli’s non-singularity results and the Madych-Nelson variational framework are the fundamental starting points from which a whole host of theoretical and practical advances have been made; the reader is encouraged to consult the textbooks highlighted earlier to discover more. All of the candidate RBFs that we have encountered so far are globally supported. When implementing algorithms on large data sets the global support can be a drawback; the associated dense interpolation matrices can be poorly conditioned and also the evaluation of resulting interpolants can be expensive. In the mid 1990 s several researchers set about overcoming these issues by constructing tailor made strictly positive definite RBFs that have compact support, for these examples the interpolation matrices are sparse and better conditioned and the evaluation of their interpolants is simpler as it requires relatively few evaluations of the RBF. Unlike their global counterparts the compactly supported RBFs are dimension dependent so Michelli’s theorem does not apply. Instead we appeal to a famous and more general result of Bochner which tells us that a candidate RBF φ is strictly positive definite on Rd (where d is fixed) whenever the d−dimensional Fourier transform of its induced kernel Φ(x) = φ(x) (x ∈ Rd ) is positive, i.e., whenever Φ(ω) > 0, for all ω ∈ Rd . One of the most commonly used families of compactly supported RBFs examples are the Wendland functions (named after their discoverer). In order to introduce these functions we begin by investigating the following family of parameterised basis functions defined by:

φμ,α (r ) :=

1 α−1 2 Γ (α)



1

α−1

(1 − t)μ t t 2 − r 2 dt for r ∈ [0, 1],

(2.30)

r

where μ > −1, α > 0. It well known (see [Gne02]) that if α = k ∈ {0, 1, 2, . . .} then the function φμ,k generates a strictly positive definite function on Rd if and only if μ ≥ d+1 2 + k. In [Wen95] Wendland considers the case where μ =  :=

  d + k + 1, 2

(2.31)

2.2 A Brief History of the RBF Method

39

i.e., the smallest allowable integer that still allows positive definiteness. In practical cases it is usual to introduce a support parameter  > 0 and define () φ,k (r )

1 = k−1 2 (k − 1)!



1

r

  α−1

1 2 2 (1 − t) t t − (r ) dt for r ∈ 0, .  

()

It is known that the functions φ,k are polynomial of degree 2k +  on 0 ≤ r ≤ 1/, and furthermore it is shown in [Hub12] that they are given explicitly by

() φ,k (r )

= (−1) 2 ! k k

  k+ 2

i=0

  Γ i + 21 (r )2i   Γ i − k + 21 ( + 2k − 2i)!(2i)!

−1 2

−

 i=0

+ i)!(r )2k+2i+1



(2.32)

(k . i!( − 2i − 1)!(2k + 2i + 1)!

By construction each Wendland function induces a radial kernel on the appropriate Rd whose d−dimensional Fourier transform is positive. Furthermore, it can also be shown that there exists positive constants C1 and C2 such that C1

2k+1 2k+1  () ≤ Φ (ω) ≤ C . 2 ,k ωd+1+2k ωd+1+2k

(2.33)

The Madych-Nelson variational theory applies equally to the more general dimension dependent RBFs. In particular, in the case of the Wendland functions we note that the polynomial decay rates of their Fourier transforms (2.33) ensures that their corresponding Native spaces (2.28) are norm equivalent to certain Sobolev spaces.

2.3 The Spherical Basis Function Method The global spherical interpolation problem is as follows: N Problem 2 Given a set Ξ = {ξi }i=1 of distinct data points on S d−1 and a target d−1 → R, find a function s : S d−1 → R that satisfies the interpolation function f : S conditions (2.34) s(ξi ) = f (ξi ), 1 ≤ i ≤ N.

In this setting we can consider specializing the RBF method to the sphere by considering an interpolant of the form s(ξ) =

N  j=1

αi ψ(g(ξ, ξ j )), ξ ∈ S d−1 ,

(2.35)

40

2 The Spherical Basis Function Method

where g denotes the geodesic metric on S d−1 g(ξ, η) = cos−1 (ξ T η), ξ, η ∈ S d−1 .

(2.36)

and where ψ : [0, π] → R is a continuous function which we will call the spherical basis function (SBF). Applying the interpolation conditions (2.34) provides the linear system Aψ α = f, where Aψ ∈ RN×N : Aψ,i j = ψ(g(ξi , ξ j )), 1 ≤ i, j ≤ N. (2.37) Thus a unique SBF interpolant exists if and only if the interpolation matrix Aψ is non-singular. Just as polynomial reproduction is important in Euclidean data fitting problems it is also common, in the spherical setting, that one requires that an interpolant should reproduce the low order spherical harmonics. Following the RBF approach we can conveniently add to s (2.35) a spherical harmonic of order k, which gives the form s(ξ) =

N 

α j ψ(g(ξ, ξ j )) +

M 

j=1

β j Y j (ξ), ξ ∈ S d−1 ,

(2.38)

j=1

where M = dim Hk (S d−1 ), and {Y1 , . . . , YM } is a basis for Hk (S d−1 ). The interpolation conditions (2.34) now provide N linear equations in N + M unknowns, and so, following RBF theory, it is usual to impose M linear constraints N 

α j Yi (ξ j ) = 0, 1 ≤ i ≤ M,

(2.39)

j=1

which leads to the augmented linear system 

Aψ Y YT 0



α β

 =

  f 0

(2.40)

where Aψ is as in (2.37) and Y ∈ RN×M is given by Yi j = Y j (ξi ), where 1 ≤ i ≤ N, and 1 ≤ j ≤ M.

(2.41)

Thus a unique augmented SBF interpolant exists if and only if the augmented interpolation matrix in (2.40) is non-singular. In order to make a transfer of the RBF machinery to the spherical setting we require the spherical analogue of Michelli’s discovery, i.e., a notion and characterization of positive and conditionally positive definite functions on spheres. Recasting from the Euclidean setting we have the following definitions.

2.3 The Spherical Basis Function Method

41

Definition 2.7 (SPD functions on spheres) A continuous function ψ : [0, π] → R is said to be strictly positive definite on S d−1 (ψ ∈ SPD(S d−1 )) if, for any set N of distinct points on S d−1 , the quadratic form Ξ = {ξi }i=1 αT Aψ α =

N  N 

α j αk ψ(g(ξ j ξ k ))

(2.42)

j=1 k=1

is positive on RN \{0}. Definition 2.8 (CSPD functions on spheres) Let m be a positive integer. A continuous function ψ : [0, π] → R is said to be conditionally strictly positive definite of N order m on S d−1 (ψ ∈ CSPDm (S d−1 )) if, for any set Ξ = {ξi }i=1 of distinct points d−1 on S , the quadratic form (2.42) is positive on the subspace Wm−1 = {α ∈ RN \{0 :

N 

αi Y(ξi ) = 0 for all Y ∈ Hm−1 (S d−1 )}.

(2.43)

i=1

Following the story of the RBF method it is clear that any ψ ∈ SPD(S d−1 ) function can be used to provide a unique interpolant of the form (2.35). Furthermore, it is straightforward to show that ψ ∈ CSPDm (S d−1 ) can be used to provide a unique augmented interpolant (2.38) provided that the following spherical unisolvency condition holds: Definition 2.9 (Unisolvency on the sphere) Let m be a positive integer and let M = M is said to be unisolvent with dim Hm−1 (S d−1 ). A set of distinct points Ξ = {ξi }i=1 d−1 respect to Hm−1 (S ) if the only element of Hm−1 (S d−1 ) to vanish at each ξi is the zero spherical harmonic. So far we have a method in theory. The following partial characterization theorem is the result which allows us to practically implement this method and to perform in depth analysis of its properties, it can be viewed as a modification/extension of Schoenberg’s pioneering work from the early 1940s. Theorem 2.6 If ψ ∈ CSPDm (S d−1 ), then it has the following form ψ(θ) =

∞ 

aψ (k)Pk,d (cos θ),

(2.44)

k=0

such that aψ (k) ≥ 0 for k ≥ m and

∞ 

aψ (k) < ∞,

k=0

where {Pk,d } denote the d−dimensional Legendre polynomials (1.24).

(2.45)

42

2 The Spherical Basis Function Method

The above theorem gives rise to several remarks and observations. Firstly, given that cos(g(ξ, η)) = ξ T η we may view our SBF ψ as a function of the inner product. Secondly, we note that SPD functions are contained in Theorem 2.6 by considering the case m = 0; in view of this we shall take ψ ∈ CSPD0 (S d−1 ) to mean ψ ∈ SPD(S d−1 ). Thirdly, and finally, we remark that a complete characterization of functions of the form (2.44) satisfying (2.45) has been established for d ≥ 3 by Chen et al. [CMS03] who show that, in this case, a necessary and sufficient condition is that the set {k ∈ N0 \{0, 1, . . . , m − 1} : ak > 0} must contain infinitely many odd and infinitely many even integers. The case of d = 2 remains an open problem. For our purposes we will only consider SBFs whose Legendre coefficients satisfy the sufficient condition that they are all positive for k ≥ m. We now turn to developing a variational setting for SBF interpolants. This begins with an application of the addition formula (1.24) which shows that for every ψ ∈ SCPDm (S d−1 ) we can associate a zonal kernel Ψ (ξ, η) = ψ(ξ T η) which has a unique spherical Fourier expansion, given by Ψ (ξ, η) =

Nk,d ∞  

k Yk, (ξ)Yk, (η), ψ

(2.46)

k=0 =1

k )k≥0 denote the spherical Fourier coefficients (1.33) of Ψ. We say that the where (ψ coefficients decay at a polynomial rate as k → ∞ if there exist positive constants A1 , A2 and α independent of k such that k ≤ A2 (1 + k)−(d−1+α) , A1 (1 + k)−(d−1+α) ≤ ψ

k ≥ m,

(2.47)

otherwise they decay exponentially quickly. We remark that if (2.47) holds for the spherical Fourier coefficients then, using formula (1.33) together with (1.19), we can deduce that there exists constant A1 and A2 (again independent of k) such that A1 (1 + k)−(1+α) ≤ aψ (k) ≤ A2 (1 + k)−(1+α) , α > 0 k ≥ m.

(2.48)

With this insight we define the so-called Native space for the SBF ψ as follows. k }k≥m Definition 2.10 (Native space of the SBF) Let ψ ∈ SCPDm (S d−1 ) and let {ψ denote the spherical Fourier coefficients of its associated zonal kernel (2.46). We define the native space of ψ to be Hψ,m := { f ∈ L2 (S d−1 ) : | f |2ψ,m =

Nk,d ∞   | f k, |2 < ∞}, k ψ

k=m =1

where | · |ψ,m is a (semi-)norm induced via the (semi-)inner product

(2.49)

2.3 The Spherical Basis Function Method

( f, g)ψ,m

43 Nk,d ∞   gk, f k, = . ψk

(2.50)

k=m =1

We note that this definition is analogous to the RBF native spaces (2.28) where k )k≥0 and Φ(ω), ω ∈ Rd is a counterpart to (ψ f (ω) corresponds to f k, . The Sobolev s d−1 spaces W (S ), s > 0 (see Definition 1.54) are a special instance of native spaces k = (1 + λk )−s . One can generated by the kernel whose Fourier coefficients are ψ easily show that there exists constants A1 and A2 such that A1 (1 + k)−2s ≤ (1 + λk )−s ≤ A2 (1 + k)−2s , k ≥ 0, and, in view of this, we can see that the native space of an SBF ψ ∈ SPD(S d−1 ) for k )k≥0 (the Fourier coefficients of its induced kernel) decay at a polynomial which (ψ rate (2.47) is norm equivalent to the Sobolev space W s (S d−1 ), s = (d −1+α)/2. The Sobolev embedding theorem guarantees  that this is a space of continuous functions or, in the language of Chap. 1, the pair Hψ,0 , (·, ·)ψ,0 is a RKHS. When m > 0 the native space is a semi-Hilbert space with the spherical harmonics Hm−1 (S d−1 ) being the null space of the induced semi-norm. In this case, in order to make use of the Hilbert space theory as presented in Chap. 1, it is common to modify ( f, g)ψ,m so that it becomes a genuine inner product; we do this by defining an appropriate inner product for the null space and add this to the semi-inner product. One common approach is to select a set {ξ 1 , . . . , ξ M } which is unisolvent with respect to Hm−1 (S d−1 ) and use this to define the following inner product  f, gHm−1 (Sd−1 ) =

M 

f (ξi )g(ξi ),

f, g ∈ Hm−1 (S d−1 ).

(2.51)

i=1

We now propose the following modified native space. Definition 2.11 (Native Hilbert space of the SBF) Let ψ ∈ SCPDm (S d−1 ) and let k }k≥m denote the spherical Fourier coefficients of its associated zonal kernel (2.46). {ψ We define the native Hilbert space of ψ to be Hψ :=



f ∈ L2 (S d−1 ) :  f 2ψ =

M   i=1

f (ξi )

2

+

Nk,d ∞    | f k, |2 < ∞ , (2.52) k ψ

k=m =1

where  · ψ is the norm induced via the inner product  f, gψ =  f, gHm−1 (Sd−1 ) + ( f, g)ψ,m .

(2.53)

We remark that as all norms are equivalent on finite dimensional spaces, we can use the same arguments as above (for the m = 0 case) to deduce that if the spherical Fourier coefficients exhibit the polynomial decay rate (2.47) then Hψ is

44

2 The Spherical Basis Function Method

norm equivalent to W s (S d−1 ), with s = (d − 1 + α)/2. This means both spaces coincide as sets and there exist constants 0 < keq < Keq , such that keq  · W s (Sd−1 ) ≤  · ψ ≤ Keq  · W s (Sd−1 ) .

(2.54)

When the coefficients decay exponentially quickly then Hψ is a much smaller subspace of infinitely differentiable functions. With this preparation we now investigate the following variational problem: N denote Proposition 2.1 (Optimal interpolation in the native space) Let Ξ = {ξi }i=1 d−1 d−1 and ψ ∈ CSPDm (S ). Assume further that a set of distinct points on S {ξ1 , . . . ξ M } ⊂ Ξ is unisolvent with respect to Hm−1 (S d−1 ), and let Hψ , ·, ·ψ denote the native Hilbert space of ψ (2.52). Then, for any f ∈ Hψ , the solution to the variational problem:

  minimise sψ : subject to s ∈ Hψ and s(ξi ) = f (ξi ), ξi ∈ Ξ ,

(2.55)

is the unique ψ−based SBF interpolant to f at Ξ.   Proof Given that Hψ , ·, ·ψ is a Hilbert function space we know, from Sect. 1.3, that the solution has the form s f (ξ) :=

N 

λk K(ξ, ξ k ),

(2.56)

k=1

where K is the reproducing kernel of Hψ . Thus, we need to compute K. We begin this process by defining a projection operator from Hψ onto Hm−1 (S d−1 ). Specifically we use the subset of Ξ that is unisolvent with respect to Hm−1 (S d−1 ) to define the M } for Hm−1 (S d−1 ) satisfying 1 , . . . , Y unique “Lagrange” basis {Y  1, Yi (ξ j ) = 0,

if i = j, if i = j.

(2.57)

The “Lagrange” projection P : Hψ → Hm−1 (S d−1 ) is given by (P f )(ξ) =

M 

j (ξ) f (ξ j ), where ξ ∈ S d−1 . Y

(2.58)

j=1

This operator maps each f ∈ Hψ to its unique spherical harmonic interpolant based d−1 ), and we have on {ξ j }M j=1 . In particular, (PY)(ξ) = Y(ξ) for all Y ∈ Hm−1 (S the following decomposition ψ ⊕ Hm−1 (S d−1 ), Hψ := H

(2.59)

2.3 The Spherical Basis Function Method

45

where ψ = (I − P)Hψ = { f ∈ Hψ : f (ξ j ) = 0 for j = 1, . . . , M}. H

(2.60)

ψ , (·, ·)ψ,m ) and (Hm−1 (S d−1 ), ·, ·H (Sd−1 ) ) are complementary We note that (H m−1 orthogonal subspaces of (Hψ , ·, ·ψ ) in the sense of Proposition 1.1. The reproducing ψ was computed in [S99], and we quote kernel K1 of H K1 (ξ, η) := (I − P)ξ (I − P)η Ψ (ξ, η),

(2.61)

where Ψ is the zonal kernel associated with ψ, and where the subscript denotes the variable to which the operator applies. Furthermore, it is easily verified that K2 (ξ, η) =

M 

k (ξ)Y k (η) Y

(2.62)

k=1

is the reproducing kernel of Hm−1 (S d−1 ). Using Proposition 1.1, the reproducing kernel of (Hψ , ·, ·ψ ) is given by K = K1 + K2 , that is, K(ξ, η) = Ψ (ξ, η) −

M 

⎛ j (η)Ψ (ξ, ξ j ) + ⎝−Pξ (I − P)η Ψ (ξ, η) + Y

j=1

M 

⎞ k (ξ)Y k (η)⎠ . Y

k=1

We observe that the term in the brackets, as a function of ξ, is simply an element of Hm−1 (S d−1 ) whose coefficients depend upon η, we denote this as Yη (ξ) and rewrite the reproducing kernel as K(ξ, η) = Ψ (ξ, η) −

M 

Y(η)Ψ (ξ, ξ j ) + Yη (ξ).

(2.63)

j=1

It is known that K belongs to SPD(S d−1 ) (see [S99, Sect. 6]) and thus (2.55) has a unique solution of the form s f (ξ) =

N  k=1

λk K(ξ, ξ k ) =

N 

⎛ λk ⎝Ψ (ξ, ξ k ) −

k=1

M 

⎞ Y j (ξ k )Ψ (ξ, ξ j )⎠ +

j=1

N 

λi Yξi (ξ).

i=1

The final sum in the above expression is an element of Hm−1 (S d−1 ) which we can express in terms of the Lagrange basis. This observation, together with a little further manipulation, allows us to write s f (ξ) =

N  j=1

α j Ψ (ξ, ξ j ) +

M  j=1

j (ξ), βjY

(2.64)

46

2 The Spherical Basis Function Method

where

⎧ N ⎪ ⎨λ −  λ Y j k j (ξ k ), αj = k=1 ⎪ ⎩λ , j

if 1 ≤ j ≤ M,

(2.65)

if M + 1 ≤ j ≤ N.

It is easy to check that the α j satisfy the SBF side conditions given by (2.39), specifically let Y ∈ Hm−1 (S d−1 ) then N 

α j Y(ξ j ) =

j=1

N 

λ j Y(ξ j ) −

j=1

=

N  j=1

N  k=1

λ j Y(ξ j ) −

N 

λk

M 

j (ξ k )Y(ξ j ) Y

j=1

λk (PY)(ξk ) = 0.

k=1

Thus, s f ∗ is precisely the unique ψ−based SBF interpolant to f at Ξ. We close this section with a result that provides two important properties of the optimal SBF interpolants. Lemma 2.1 For a given f ∈ Hψ let s f denote its optimal ψ−based SBF interpolant, then we have (i)  f −s f 2ψ =  f, f −s f ψ

(ii)  f −s f ψ ≤  f ψ

(iii) s f ψ ≤  f ψ

Proof To prove (i) consider  f − s f 2ψ =  f, f − s f ψ − s f , f − s f ψ . Using (2.56) and the reproducing kernel property, we have N N   λk K(·, ξ k ), f − s f ψ = λk ( f − s f )(ξ k ) = 0, s f , f − s f ψ =  k=1

k=1

and thus (i) follows. To prove (ii) and (iii) we use (i) to provide: s f 2ψ +  f − s f 2ψ = s f 2ψ +  f, f − s f  = s f 2ψ +  f 2ψ −  f, s f ψ =  f 2ψ , where the final equality comes from s f , f − s f ψ = 0; inequalities (ii) and (iii) follow from this. 

2.4 Framework for Pointwise Error Estimates

47

2.4 Framework for Pointwise Error Estimates The undoubted appeal of the variational approach to SBF interpolation is that it provides a physical interpretation of the construction process. The surfaces generated minimize a certain energy measure (bending energy in the case of cubic and thin plate splines) and this gives a reassuring sense that they will be sensibly shaped and wellbehaved. In addition, the variational approach also provides a rather nice framework for delivering error bounds and the following development show how easily it is to access such bounds. We begin by choosing ψ ∈ CSPDm (S d−1 ) and let s f denote the unique ψ−based SBF interpolant to a given target function f ∈ Hψ . We observe ψ , | · |ψ,m ) that the error function f − s f belongs to the Hilbert function space (H (2.60) and so, for any ξ ∈ S d−1 , we have | f (ξ) − s f (ξ)| = |(K1 (ξ, ·), f − s f )ψ,m |, ψ given by (2.61). Applying the Cauchywhere K1 is the reproducing kernel of H Schwarz inequality we have | f (ξ) − s f (ξ)| ≤ |K1 (ξ, ·)|ψ,m | f − s f |ψ,m = |K1 (ξ, ·)|ψ,m  f − s f ψ .

(2.66)

The factor |K1 (ξ, ·)|ψ,m is called the Lagrange power function for ψ, and we write Lψ (ξ) = |K1 (ξ, ·)|ψ,m . The square of this function can be computed explicitly since |K1 (ξ, ·)|2ψ,m = (K1 (ξ, ·), K1 (ξ, ·))ψ,m = K1 (ξ, ξ) = Lψ2 (ξ). Thus, employing (2.61), we have Lψ2 (ξ) :=

M  M 

i (ξ)Y j (ξ)ψ(ξ T ξ j ) − 2 Y i

i=1 j=1

M 

i (ξ)ψ(ξ T ξi ) + ψ(1), Y

(2.67)

i=1

M } is the Lagrange basis for Hm−1 (S d−1 ). This ensures spherical 1 , . . . , Y where {Y harmonic reproduction via Y(ξ) =

N 

i (ξ)Y(ξi ), for all Y ∈ Hm−1 (S d−1 ). Y

(2.68)

i=1

In particular, in view of (2.66) we have | f (ξ) − s f (ξ)| ≤ Lψ (ξ) f − s f ψ , ξ ∈ S d−1 .

(2.69)

The Lagrange power function clearly provides a bound on the pointwise interpolation error. However, it only makes use of information based on the subset of Ξ that is unisolvent with respect to Hm−1 (S d−1 ). Intuitively, we would expect an improvement

48

2 The Spherical Basis Function Method

if the function were allowed to depend upon the whole of Ξ. In view of this, we fix N so that ξ ∈ S d−1 and generalize (2.68) by selecting N real coefficients {γi }i=1 Y(ξ) =

N 

γi Y(ξi ), for all Y ∈ Hm−1 (S d−1 ).

(2.70)

i=1

In addition, we define a bounded linear functional on Hψ by Λξ ( f ) = (δξ −

N 

γi δξi )( f ), for all f ∈ Hψ .

(2.71)

i=1

Using Lemma 1.1, the Riesz representor of Λξ in Hψ , is given by kΛξ (·) = K(ξ, ·) −

N 

γi K(ξi , ·),

(2.72)

i=1

where K is the reproducing kernel of Hψ , see (2.63). Now, applying the same analysis as before we find | f (ξ) − s f (ξ)| = |Λξ ( f − s f )| = |kΛξ , f − s f ψ | ≤ kΛξ ψ  f − s f ψ . We can evoke, again, the Riesz representation theorem, to deduce that kΛξ ψ = Λξ ψ∗ = δξ −

N 

γi δξi ψ∗ ,

(2.73)

i=1

where  · ψ∗ denotes the usual dual space norm given by T ψ∗ = sup{|T f | :  f ψ ≤ 1}.

(2.74)

The factor kΛξ ψ is said to be a power function for ψ at ξ, and we write Pψ,γ (ξ) = kΛξ ψ . The square of this function can be computed explicitly since 2 (ξ), kΛξ 2ψ = kΛξ , kΛξ ψ = Λξ (kΛξ ) = Pψ,γ

indeed this calculation was made in [LLRS99] and, again, we quote ⎛ Pψ,γ (ξ) = ⎝

N  N  i=1 j=1

γi γ j ψ(ξiT ξ j ) − 2

N  i=1

⎞1/2 γi ψ(ξT ξi ) + ψ(1)⎠

.

2.4 Framework for Pointwise Error Estimates

49

To improve the presentation, we set ξ 0 = ξ and γ0 = −1, this enables us to write the power function in the more compact form ⎛ Pψ,γ (ξ) = ⎝

⎞1/2

N  N 

γi γ j ψ(ξiT ξ j )⎠

N     = γi δξi  ∗ .

i=0 j=0

ψ

i=0

(2.75)

N satisfying (2.70), For a given ξ ∈ S d−1 , each selection of coefficients {γi }i=1 gives rise to its own power function Pψ,γ , which, in turn, provides the following error bound (2.76) | f (ξ) − s f (ξ)| ≤ Pψ,γ (ξ) ·  f − s f ψ .

Stated in this way, it is clear that a close investigation of Pψ,γ , and especially the choice of coefficients, ought to provide an insight into the accuracy of the SBF interpolation method. In particular, in [WuS93], Wu and Schaback solve the linearly N which constrained optimisation problem of choosing the optimal coefficients {γi∗ }i=1 minimises (2.75) subject to (2.70). In view of this we define N      γi δξi  Pψ,γ ∗ (ξ) = min  i=0

ψ∗

 N : {γi }i=1 satisfy condition (2.70) ,

(2.77)

to be the optimal power function for ψ at ξ. We remark that the error bound (2.76) may be viewed as a specific instance of the following more general result for functions with vanishing conditions. N Proposition 2.2 Let ψ ∈ CSPDm (S d−1 ) and Ξ = {ξi }i=1 denote a set of distinct d−1 d−1 we have the following bound points on S . For any ξ ∈ S

| f (ξ)| ≤ Pψ,γ ∗ (ξ) ·  f ψ , where f ∈ Hψ and f (ξi ) = 0, i = 1, . . . , N. (2.78) N Proof Let {γi }i=1 denote a choice of real coefficients satisfying (2.70). Set ξ 0 = ξ and γ0 = −1, then, for any f ∈ Hψ that satisfies f (ξi ) = 0, i = 1, . . . , N, we have N N         γi δξi ( f ) ≤  γi δξi  ∗ ·  f ψ , | f (ξ)| = |δξ ( f )| =  i=0

i=0

ψ

N completes the proof. taking the infimum over all such choices of {γi }i=1



50

2 The Spherical Basis Function Method

2.5 Pointwise Error Estimate I Let ψ ∈ CSPDm (S d−1 ) denote an SBF whose Legendre coefficients decay like (2.48), i.e., whose native Hilbert space Hψ is equivalent to the Sobolev space W s (S d−1 ), s = (d − 1 + α)/2. In this section we aim to estimate the uniform rate at which the interpolant converges to its target function as the interpolation points fill the surface of the sphere. To measure the relative density of the point set Ξ in S d−1 we use the so-called mesh norm h := h(Ξ, S d−1 ) := sup min{g(η, ξi ) = cos−1 (η T ξi ) : ξi ∈ Ξ },

(2.79)

η∈S d−1

and our specific aim is then to estimate the value p such | f (ξ) − s f (ξ)| = O(h p ), for all f ∈ Hψ and ξ ∈ S d−1 .

(2.80)

The strategy we employ is again borrowed from RBF theory where we attempt to bound the optimal power function of ψ by a function of h. We begin our analysis with the following remarkable result from [JSW99]. N denote a set of distinct Lemma 2.2 (Jetter, Stöckler and Ward) Let Ξ = {ξi }i=1 d−1 with mesh-norm h, and let K be the positive integer satisfying data points on S

1 1 ≤ 2h ≤ . K +1 K

(2.81)

N Let ξ ∈ S d−1 , then there exist coefficients {γi }i=1 such that

Y(ξ) =

N 

γi Y(ξi ), for all Y ∈ HK (S d−1 ),

(2.82)

i=1

and such that

N 

|γi | ≤ 2.

(2.83)

i=1

This result has played an important role in advancing our understanding of the SBF interpolation method. Specifically, it enables us to deliver our first error estimate. Theorem 2.7 Let ψ ∈ CSPDm (S d−1 ) be an SBF whose Legendre coefficients decay N denote a set distinct data points on like (2.48) for some α > 0. Let Ξ = {ξi }i=1 d−1 S whose mesh-norm h satisfies (2.81) for some positive integer K ≥ m − 1. Let f ∈ Hψ and s f denote its unique SBF interpolant. Then, for any ξ ∈ S d−1 , we have α

| f (ξ) − s f (ξ)| ≤ C · h 2  f − s f ψ , where C is a positive constant independent of h.

(2.84)

2.5 Pointwise Error Estimate I

51

N Proof The assumption K ≥ m − 1 allows us to deduce that the coefficients {γi }i=1 from Lemma 2.2 also satisfy condition (2.70), and so, in view of (2.77), they can be used to bound the optimal power function of ψ :

P

ψ,γ ∗

N     (ξ) ≤  γi δξi 

= ≤ ≤

=

ψ∗

i=0

sup

f ∈Hψ :  f ψ ≤1

i=0

Nk,d N      γi f k, Yk, (ξi ) 

sup

f ∈Hψ :  f ψ ≤1 i=0 N 

sup

f ∈Hψ :  f ψ ≤1 i=0

k>K =1 Nk,d 

    |γi | ·  f k, Yk, (ξi )

 N 

sup

f ∈Hψ :  f ψ ≤1

≤ 3·

N     γi δξi )( f ) (

k>K =1



|γi | ·

i=0

sup

max

Nk,d      max  f k, Yk, (ξi )

i∈{0,...,N}

|

Nk,d 

f ∈Hψ :  f ψ ≤1 i∈{0,...,N} k>K =1

k>K =1

f k, Yk, (ξi )|.

(2.85)

To bound the maximum value we can employ the Cauchy-Schwarz inequality together with the addition formula (1.24). Specifically, for any ξ ∈ S d−1 , we have ⎛ ⎝

Nk,d  k>K =1

⎞2

⎛

⎞ ⎞ ⎛ Nk,d 2 Nk,d   f k, k Y 2 (ξ)⎠ ⎠·⎝ f k, Yk, (ξ)⎠ ≤ ⎝ ψ k, k ψ k>K =1

≤  f 2ψ ·

k>K =1

 Nk,d ψ  Nk,d ψ  k k ≤ = aψ (k), ωd−1 ωd−1

k>K

k>K

k>K

where, by (1.33), the aψ (k) denotes the k th Legendre expansion coefficient of ψ. We can continue bounding from above by using (2.48) and (2.81) to give 

aψ (k) ≤ Ca

k>K

=

 k>K

1 ≤ Ca (1 + k)α+1

∞ K

dx (1 + x)α+1

1 Ca 2α Ca · . ≤ C · h α , where C = α α (1 + K) α

(2.86)

52

2 The Spherical Basis Function Method

Now, linking (2.85) and (2.86) together gives Pψ,γ ∗ (ξ) ≤ C · h α/2 , ξ ∈ S d−1 . Thus, for any ξ ∈ S d−1 , we can use (2.76) to deduce that | f (ξ) − s f (ξ)| ≤ C · h α/2 ·  f − s f ψ , for all f ∈ Hψ , where C is a positive constant independent of h.



2.6 Pointwise Error Estimate II In this section we will examine a similar but local approach to bounding the pointwise interpolation error. This development was initiated by Golitschek and Light [GL01] and later refined in [M01]. As usual the basic idea is to bound the optimal power function of ψ at ξ ∈ S d−1 ; however, the new approach shows that this can be done by using only those data points ξi ∈ Ξ which lie within a certain neighbourhood of ξ. This is in contrast to Theorem 2.7 where every location in Ξ is used to bound the power function. In our previous result we have used the mesh-norm h to measure the relative N in S d−1 . Geometrically speaking, h density of a set of data points Ξ = {ξi }i=1 represents the radius of the largest spherical cap (open geodesic ball) which can be placed on S d−1 without covering any ξi . In [GL01], von Golitschek and Light use the height h d of the maximal spherical cap as an alternative mesh-norm; that is, they define h d to be the smallest number such that inf max{η T ξi : ξi ∈ Ξ } > 1 − h d ,

η∈S d−1

(2.87)

is satisfied. We shall call h d the “dot product” mesh norm of Ξ. Using some elementary trigonometry we can show that h d = 2 sin2 (h/2). Furthermore, if h ∈ (0, 2π/3) then we can apply the small angle result for sin(h/2) to give h2 h2 ≤ hd ≤ 8 2

(2.88)

that is, h d is equivalent to h 2 . The idea of using the dot product as an alternative measure of distance will prove to be a useful one. Definition 2.12 (Dot product neighbourhood) For every ξ ∈ S d−1 we define an associated a dot product distance function dξ : S d−1 → [−1, 1], given by dξ (η) = ξ T η.

2.6 Pointwise Error Estimate II

53

Furthermore, we can define a dot product neighbourhood of ξ by N(ξ, rd ) = {η ∈ S d−1 : dξ (η) > 1 − rd }, where rd ∈ (0, 1).

(2.89)

Proposition 2.3 Let ξ ∈ S d−1 be a fixed point and let rd ∈ (0, 1), then η 1 , η 2 ∈ N(ξ, rd ) =⇒ 1 − η T1 η 2 < 4rd . Proof For any η 1 , η 2 ∈ S d−1 we have the following useful relation 2 − 2η T1 η 2 = η 1 − η 2 2 . Furthermore, if η 1 , η 2 ∈ N(ξ, rd ) then we also know that 2 − 2η iT ξ < 2rd , i ∈ {1, 2}, which allows us to deduce that 2 − 2η T1 η 2 = (η 1 − ξ) + (ξ − η 2 ) ≤ η 1 − ξ + ξ − η 2  =

2 − 2η T1 ξ +

! 2 − 2η T2 ξ < 2 2rd , 

and the proof is complete. The following crucial result is quoted from [GL01].

Lemma 2.3 (von Golitschek and Light) Let ξ ∈ S d−1 and let J be a fixed positive integer. Let Ξ = {ξ 1 , . . . , ξ N } denote a set of N distinct data points on S d−1 with dot product mesh-norm h d . There is a number h 0 ∈ (0, 1) such that if h d < h 0 , then N such that there exist coefficients {γi }i=1 1. Y(ξ) =

N 

γi Y(ξi ), for all Y ∈ HJ−1 (S d−1 ),

i=1

2. there exists a constant K1 (independent of ξ and h d ) such that if ξi ∈ / N(ξ, K1 h d ), then γi = 0, and N  |γi | ≤ K2 . 3. there exists a constant K2 (independent of ξ and h d ) such that i=1

It is pertinent to mention that Lemma 2.3 is similar in spirit to Lemma 2.2. To illustrate this, we provide the following useful comparison list. C1. For a sufficiently dense set of data-points, both Lemmata supply coefficients N  N which satisfy condition (2.70) and, in both cases, the quantity |γi | {γi }i=1 i=1

is suitably bounded.

54

2 The Spherical Basis Function Method

N C2. For a given ξ ∈ S d−1 , the coefficients {γi }i=1 arising from Lemma 2.3 are said to be “local” since γi = 0 if and only if ξi ∈ N(ξ, K1 h d ).

C3. Lemma 2.3 is stated for a “fixed” positive integer J, whereas Lemma 2.2 is stated for an integer K which depends upon the mesh-norm h of Ξ. In both cases the result for m, and hence condition (2.70), follows if we assume that m − 1 ≤ max{K, J}. The main aim, again, is to provide a suitable bound on the optimal power function of ψ and hence, using (2.76), deduce error estimate results for SBF interpolation. However, in contrast to the previous attempt, we will pursue a different approach which relies heavily on Taylor series analysis; for the convenience of the reader we briefly compose the key arguments.  Analysis via Taylor series. Let ψ ∈ CSPDm−1 (S d−1 ) be an SBF whose Legendre coefficients decay like (2.48) for some α > 0. Let Ψ denote the zonal kernel induced by ψ. For a fixed ξ ∈ S d−1 , we consider the function Fξ : S d−1 → R given by Fξ (η) = Ψ (ξ, η). In particular, we can write Fξ (η) =

∞ 

ak Pk,d (ξ T η).

(2.90)

k=m

Our aim is to investigate the behaviour of Fξ in a local neighbourhood N(ξ, rd ) and, in view of (2.90), we can do this by studying the local behaviour of the d−dimensional Legendre polynomials. Specifically, we choose a suitable positive integer J and consider the Taylor expansion Pk,d (t) =

J−1 (r )  Pk,d (1) r =0

r!

(1 − t)r + RJ (k, t), t ∈ (1 − rd , 1],

(2.91)

where the remainder term RJ (k, t) satisfies |RJ (k, t)| ≤

(1 − t)J J!

sup t∈(1−r,1]

(J)

|Pk,d (t)|, for k ≥ J.

For all d ≥ 2, we can use (1.26) together with Markov’s inequality for algebraic polynomials ([DL93, Chap. 4], to deduce that |RJ (k, t)| ≤

(1 − t)J 2J k , for k ≥ J. J!

(2.92)

As a final remark we note that, for each d ≥ 2, the polynomials {Pk,d }rk=0 form a basis for the space of univariate polynomials on [−1, 1] of degree at most r. In particular, for a given r ∈ N0 , there exists real coefficients {αr s }rs=0 such that

2.6 Pointwise Error Estimate II

55

(1 − t)r =

r 

αr s Ps,d (t).

(2.93)

s=0

With this preparation we are now in position to prove our next error estimate. Theorem 2.8 Let ψ ∈ CSPDm (S d−1 ) be an SBF whose Legendre coefficients decay N denote a set distinct points on S d−1 like (2.48) for some α > 0. Let Ξ = {ξi }i=1 with mesh-norm h. Set " # α + 1 J = max m, , (2.94) 2 where x denotes the smallest integer ≥ x, and assume that the dot product meshnorm h d (2.87) of Ξ satisfies 1 1 ≤ hd < 2 , 2 (K + 1) K

(2.95)

where K > J is a positive integer. Let f ∈ Hψ and s f denote its unique SBF interpolant. Then, for any ξ ∈ S d−1 , we have | f (ξ) − s f (ξ)| ≤ C · h α/2 ·  f − s f ψ ,

(2.96)

where C is a positive constant independent of h. Proof The choice of integer J (2.94), allows us to evoke Lemma 2.3 to provide, for any ξ ∈ S d−1 , a neighbourhood N(ξ, K1 h d ) and a set of local coefficients {γi }i∈Ioc , where Ioc := {i : ξi ∈ Ξ ∩ N(ξ, K1 h d )}, which satisfy condition (2.70). Furthermore, these coefficients can be used to define a local power function which, in turn, provides a bound on the optimal power function of ψ : 

2 2 Pψ,γ ∗ (ξ) ≤ Pψ,oc (ξ) =

γi γ j ψ(ξiT ξ j ) =

i, j∈Ioc ∪{0}

∞ 



aψ (k)

γi γ j Pk,d (ξiT ξ j ),

i, j∈Ioc ∪{0}

k=m

where we have employed the Legendre expansion of ψ. For our investigation it is useful to split the above sum into two parts; that is, we shall consider K 



aψ (k)

k=m

$

∞ 

γi γ j Pk,d (ξiT ξ j ) +

i, j∈Ioc ∪{0}

%&

'

aψ (k)

k=K+1

$

sum 1



γi γ j Pk,d (ξiT ξ j ) .

i, j∈Ioc ∪{0}

%&

'

sum 2

(2.97) We begin by considering “sum 1” of (2.97). In particular, substituting in the Taylor expansion (2.91) of the Legendre polynomials yields K  k=m

⎛ aψ (k)

(r ) J−1  Pk,d (1) ⎝ r =0

r!

 i, j∈Ioc ∪{0}

γi γ j (1 − ξiT ξ j )r +

 i, j∈Ioc ∪{0}

⎞ γi γ j RJ (k, ξiT ξ j )⎠ .

56

2 The Spherical Basis Function Method

We continue our development by analysing the first sum appearing in the brackets. Using identity (2.93) followed by an application of addition formula (1.24), gives 

γi γ j (1 − ξiT ξ j )r =

i, j∈Ioc ∪{0}

r 

r 

αr s

s=0

=

r 

γi γ j Ps,d (t)

i, j∈Ioc ∪{0}

s=0

=



αr s

αr s

s=0

ωd−1 Ns,d ωd−1 Ns,d



γi γ j

i, j∈Ioc ∪{0} Ns,d 

⎛ ⎝

Ns,d 

Ys, (ξi )Yk, (ξ j )

=0



⎞2

γi Ys, (ξi )⎠ .

i∈Ioc ∪{0}

=0

This expression vanishes for 0 ≤ r ≤ J − 1, by part (i) of Lemma 2.3, and thus, the bound reduces to K 

aψ (k)



K 

γi γ j RJ (k, ξiT ξ j ) =

i, j∈Ioc ∪{0}

k=m

aψ (k)



γi γ j RJ (k, ξiT ξ j ),

i, j∈Ioc ∪{0}

k=J

since the remainder RJ (k, t) is zero for k ≤ J − 1. Now, for any i, j ∈ Ioc we have ξi , ξ j ∈ N(ξ, K1 h d ), and so, by Proposition 2.3, it follows that (1 − ξiT ξ j )J < (4K1 h d )J . We can use this fact, together with (2.92), to deduce K  k=J



aψ (k)

i, j∈Ioc ∪{0}

γi γ j RJ (k, ξiT ξ j ) ≤

K  k=J

K  k=J



aψ (k)

aψ (k) ⎝



i∈Ioc ∪{0}

|γi γ j |

i, j∈Ioc ∪{0}

|γi γ j |

i, j∈Ioc ∪{0}

⎛



aψ (k)

k=J

≤

=

K 

⎞2 |γi |⎠

(1 − ξiT ξ j )J 2J k J!

(4K1 h d )J 2J k J! (4K1 h d )J 2J k . J!

We can now apply part (iii) of Lemma 2.3 to give K  k=m

aψ (k)

 i, j∈Ioc ∪{0}

γi γ j Pk,d (ξiT ξ j ) ≤ (1 + K2 )2

K (4K1 h d )J  aψ (k)k 2J . (2.98) J! k=J

2.6 Pointwise Error Estimate II

57

Using the assumed decay rate of the Legendre coefficients we can write K 

aψ (k)k 2J ≤ Ca

k=J

K  k=J

 k 2J ≤ C k 2J−(α+1) . a (1 + k)α+1 K

k=J

The definition of J guarantees that the function x → x 2J−(α+1) is non-decreasing on [0, ∞) and hence we have the bound K 

 aψ (k)k

2J

K

≤ Ca J

k=J

−J+ α2

x 2J−(α+1) d x ≤ Ca · K 2J−α ≤ Ca · h d

,

(2.99)

where the final inequality follows from (2.95). Linking (2.98) and (2.99) together gives us our final bound for “sum 1” that is, K 



aψ (k)

α

γi γ j Pk,d (ξiT ξ j ) ≤ Csum1 · h d2 ,

(2.100)

i, j∈Ioc ∪{0}

k=m

where Csum1 = Ca · (1 + K2 )2 ·

(4K1 )J J!

(2.101)

is independent of h d . We now turn to “sum 2” of (2.97) which is easier to bound. Specifically, we use (1.26) followed by part (iii) of Lemma 2.3 to yield ∞ 

aψ (k)

γi γ j Pk,d (ξiT ξ j ) ≤

i, j∈Ioc ∪{0}

k=K+1

=



∞  k=K+1

⎛ aψ (k) ⎝

∞ 

⎞2

∞ 

|γi |⎠ ≤ (K2 + 1)2

i∈Ioc ∪{0}

|γi γ j |

i, j∈Ioc ∪{0}

k=K+1





aψ (k)

aψ (k).

(2.102)

k=K+1

Again we use the decay of the Legendre coefficients together (2.95) to deduce that ∞ 

aψ (k) ≤ Ca

k=K+1

∞  k=K+1

=

1 ≤ Ca (1 + k)α+1



α 1 Ca Ca 2 · h ≤ d. α (K + 1)α α

∞

K

dx (1 + x)α+1 (2.103)

58

2 The Spherical Basis Function Method

Linking (2.102) and (2.103) together provides the following bound for “sum 2” ∞  k=K+1

aψ (k)



α

γi γ j Pk,d (ξiT ξ j ) ≤ Csum2 · h d2 ,

(2.104)

i, j∈Ioc ∪{0}

where Csum2 = (1 + K2 )2 ·

Ca , α

(2.105)

is independent of h d . We are now in a position to provide a more meaningful bound on the optimal power function. In particular, in view of (2.100), (2.104) and the mesh-norm equivalence relation (2.88), we choose to set C = 2−α/2 max{Csum1 , Csum2 } and deduce α

2 2 2 ≤ C · hα, Pψ,γ ∗ (ξ) ≤ Pψ,oc (ξ) ≤ max{Csum1 , Csum2 } · h d

the proof is then completed by employing this bound in (2.76).

(2.106) 

Chapter 3

Error Bounds via Duchon’s Technique

The goal of this chapter is to investigate the SBF interpolation error when measured in the more general L p −norms, where p ∈ [1, ∞]. To do this we once again revisit RBF theory and, in particular, we focus on Duchon’s strategy (see Definition 2.1) developed for D m −splines in Euclidean space. The chapter itself falls into two parts. The first part carefully sets up spherical versions of the crucial results used in Duchon’s approach in Rd and in the second part we demonstrate how these results can be used, together with the variational framework for SBF interpolation, to provide the desired interpolation error estimates in the L p -norms.

3.1 Duchon’s Recipe for the Sphere We recall that the success of Duchon’s approach (in the Euclidean setting) hinges on three crucial ingredients: 1. A scalable quasi-uniform mesh which is needed to collect local pieces of approximation order. 2. A gluing result that shows how the sum of all local norms of a function (restricted to local regions) can be bounded by its global norm. 3. A suitable extension operator which faithfully maps a locally defined function to a global one defined on the whole Euclidean space. In this section we shall establish that these results have a spherical analogue. We begin by constructing the scalable quasi-uniform mesh. In Euclidean space Duchon made use of the scaled integer lattice to provide a regular mesh with a specified spacing. While we do not have such a regular mesh for the sphere, we can find quasiuniform meshes that will satisfy our requirements. This is the subject of our next lemma. For any set G ⊂ S d−1 we introduce the notation  FG (ξ) =

1 0

if ξ ∈ G; otherwise.

© The Author(s) 2015 S. Hubbert et al., Spherical Radial Basis Functions, Theory and Applications, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-17939-1_3

59

60

3 Error Bounds via Duchon’s Technique

Lemma 3.1 For a given constant M > 0 there exists h 0 > 0 such that, for all h ∈ (0, h 0 ), there is a set of points Z h ⊂ S d−1 such that S d−1 =



G(z, M∗ h), M∗ =

√ d − 1,

z∈Z h

and for which there exists an integer Q independent of h such that 

FG(z,Mh) ≤ Q, where M = M + M∗ .

(3.1)

z∈Z h

Further, the cardinality of Z h is bounded by Cq h −(d−1) , Cq independent of h. Proof We begin by fixing the condition of the density measure h (needed for the subsequent surface area argument). Let M be a given positive constant and set h0 =

π 6



1 M∗ + M

 .

(3.2)

Choose h ∈ (0, h 0 ). Let C d denote the d−dimensional cube inscribed into the unit √ sphere S d−1 . This cube will have side 2/ d. Let n h ≥ 2 denote the integer such that 1 1 . ≤h< nh nh − 1

(3.3)

On each face of the inscribed cube we √ place a regular mesh of dimension d − 1 such that each subcube has side 2/(n h d). That is, we place a lattice of points isomorphic to  1 2 1 d−1 √ · Zd−1 ∩ − √ , √ nh d d d on each side of the cube. We now define Z h for the sphere to be the radial projection of these points on the cube onto S d−1 . Suppose that two points x1 and x2 lie on one side of the inscribed cube and are a distance b apart. Let ξ 1 and ξ 2 be the radial projections onto S d−1 of x1 and x2 respectively. Consider Fig. 3.1. The point x is the closest point to the origin from the (extended) line connecting x1 and x2 . Further, r1 and r2 represent the respective x (with the convention that r2 ≥ 0). We observe distances from x1 and x2 relative to that the distance a, from x to the origin, satisfies 1 √ ≤ a < 1. d

(3.4)

3.1 Duchon’s Recipe for the Sphere

61

Fig. 3.1 Illustration of quasi-uniform mesh proof

The geodesic distance g(ξ 1 , ξ 2 ) is given by

r1 r2



g(ξ 1 , ξ 2 ) = tan−1 − tan−1 . a a Employing the mean value theorem we can deduce that  1 1 b g(ξ1 , ξ 2 ) = · , for some t ∈ − a 1 + t2 a



d −1 1 , d a



d − 1 . d

Maximising the RHS of (3.5) and using (3.4) gives g(ξ 1 , ξ 2 ) ≤ Similarly, minimising the RHS gives g(ξ 1 , ξ 2 ) ≥ Thus, we have shown

b a

(3.5) √ < b d.

ad 2 b 1 b b · > · > . 2 a d(a + 1) − 1 a 2d − 1 2d √ b ≤ g(ξ1 , ξ 2 ) ≤ b d. 2d

(3.6)

For any ξ ∈ S d−1 , let zξ denote its closest point from Z h . Let C d denote the d-dimensional cube inscribed into S d−1 , then we can view the radial projection as a mapping Proj : C d → S d−1 . In particular, let x ∈ C d be such that Proj (x) = ξ.

62

3 Error Bounds via Duchon’s Technique

The point x necessarily lies within a (d − 1)-dimensional subcube, which contains thelattice point x. The furthest that x can be away from x is bounded above by half the diameter of the subcube. Hence, by (3.6) and (3.3), any ξ ∈ S d−1 is such that √ √ d −1 min g(z, ξ) = g(zξ , ξ) < ≤ h d − 1. z∈Z h nh 1 nh

d−1 d ,

This proves the first part of the theorem. The minimum separation distance of the lattice points on the surface of the √ inscribed cube is 2/(n h d). Therefore by (3.6) and (3.3) min g(z, z ) > 

z,z ∈Z h

1 1 1 1 2 1 1 h 1 > > = M1 h, √ 3/2 3/2 2 nh d d 2 nh − 1 d 2d 2

(3.7)

where we’ve defined M1 = d −3/2 . We now show that the second part of the theorem holds by an elementary surface area argument. Let ξ ∈ S d−1 , M > 0 and suppose that 

FG(z,Mh) (ξ) = N , where M = M∗ + M,

z∈Z h

that is, g(ξ, zi ) < Mh, for zi ∈ Z h , i = 1, . . . , N . Before continuing with the proof we note that for any geodesic ball G(ξ, θ) of radius θ < π/3 there exists positive constants C1a and C2a , depending only on d, such that its surface area is bounded by  C1a · θd−1 ≤

G(ξ,θ)

dωd−1 ≤ C2a · θd−1 , ξ ∈ S d−1 .

(3.8)

We make two observations. Firstly, as consequence of (3.7), the geodesic balls G(zi , M1 h), i = 1, . . . , N , are disjoint. Secondly the larger ball G(z, Mh) must contain all of the smaller balls G(zi , M1 h), i = 1, . . . , N . In order to accommodate all of the balls the area of G(z, Mh + M1 h), which, by (3.8), is bounded above by C2a ((M + M1 )h)d−1 must be at least N C1a (M1 h)d−1 . Therefore, there exists an integer Q that is independent of ξ and h such that Ca N ≤ 2a C1



d−1 M +1 ≤ Q. M1

3.1 Duchon’s Recipe for the Sphere

63

To finish the proof we let |Z h | denote the cardinality of the mesh. Then we have |Z h | · C1a (Mh)d−1 =



C1a (Mh)d−1

z∈Z h

≤



z∈Z h



=

dωd−1 (ξ) = G(z,Mh)

 S d−1 z∈Z h



That is,

 |Z h | ≤

 z∈Z h

S d−1

FG(z,Mh) (η)dωd−1 (ξ)

FG(z,Mh) (η) dωd−1 (ξ) ≤ Qωd−1 . 

≤Q





Qωd−1 d−1 C1a M

h −(d−1) = C Q h −(d−1) , 

where C Q is independent of h.

Now we are equipped with the quasi-uniform mesh for the sphere we can accomplish our second task and deliver the following gluing result. Theorem 3.1 (Duchon’s Inequality for the Sphere) Let M be a given constant and h 0 be given by (3.2). Let h ∈ (0, h 0 ) and Z h and Q be given by Lemma 3.1. Then, for s > 0 we have that  z∈Z h

f 2W s (G(z,Mh)) ≤ Q f 2W s (S d−1 ) ,

f ∈ W s (S d−1 ).

(3.9)

2 Proof Following Sect. 1.6, let {χi }i=1 be a partition of unity subordinate to atlas 2 A = {Ui , φi } j=1 for the sphere. For z ∈ Z h and i ∈ {1, 2} we shall set

Ωi (z) = φi (G(z, Mh) ∩ Ui ) ⊂ B0 (= B(0, 1)).

(3.10)

We shall prove the result in the integer case, the fractional case follows in the same fashion. Thus, for any f ∈ W k (S d−1 ), k a non-negative integer, we use (1.72) and consider 2    2    f k πi (χi f )|Ω (z) 2 k = . (3.11) i W (G(z,Mh)) W (Ω (z)) i

z∈Z h

i=1 z∈Z h

64

3 Error Bounds via Duchon’s Technique

Let g ∈ W k (Rd−1 ), then using (3.10) and Lemma 3.1 we can write   g|Ω (z) 2 i

W k (Ωi (z))

=

    D α g|Ω (z) 2 i

z∈Z h |α|≤k

z∈Z h

=

 

|α|≤k z∈Z h

=



Ωi (z)



d−1 |α|≤k R z∈Z h

≤Q

   D α g 2

|α|≤k

L 2 (Ωi (z))

 α 2 (D g|Ωi (z) (x)dx

(3.12)

FΩi (z) (x)(D α g(x))2 dx

L 2 (Rd−1 )

 2 = Q g W k (Rd−1 ) .

Applying these arguments to g = πi (χi f ) ∈ W k (Rd−1 ), and substituting into (3.11) provides the result for the integer order spaces.  We now turn to our final task which is to demonstrate that functions defined in a local Sobolev space can be continuously extended to the global space. Specifically, the aim is to construct a continuous extension operator E G(z,θ) : W k (G(z, θ)) → W k (S d−1 ), with the property that (E G(z,θ) f )|G(z,θ) = f for all f ∈ W k (G(z, θ)). The development here will focus on integer order spaces however, as we shall see, results of the same strength can easily be recast for fractional order spaces using interpolation arguments. Much of the groundwork in terms of the notation and basic concepts has already been established in Sect. 1.6, and the reader may want to briefly review this material before progressing. In view of (1.56) we start by extending the local functions (χi f ) ◦ φi−1 |Ωi ∈ W k (Ωi ) to W k (Rd−1 ) for i ∈ {1, 2}. Remark 3.1 Let CA be the constant from Remark 1.2 and assume that θ < CA /3. Then, by Lemma 1.2, we can restrict attention to the case where Ωi is an open Euclidean ball, since otherwise (χi f ) ◦ φi−1 |Ωi is the zero function which has a trivial extension. For the unit ball B0 and for  > 0 sufficiently small, we can appeal to Theorem 1.5 for a continuous extension operator E B0 : W k (B0 ) → W k (Rd−1 ), where supp(E B0 f ) ⊂ B(0, 1 + ) for all f ∈ W k (B0 ). To define an extension operator on B(x, r ) we use the coordinate transform

and set

σ(y) = r y + x, for r > 0 and y ∈ Rd−1 ,

(3.13)



E B(x,r ) f (y) = E B0 ( f ◦ σ) ◦ σ −1 (y) y ∈ Rd−1 .

(3.14)

3.1 Duchon’s Recipe for the Sphere

65

In addition, we have that, supp(E B(x,r ) f ) ⊂ B(x, r (1 + )) for all f ∈ W k (B(x, r )).

(3.15)

Remark 3.2 Let z ∈ S d−1 , θ < CA /3, and assume that Ωi = B(xi , ri ) ⊂ B0 . We can use Lemma 1.2 (2) to choose an  < eA such that supp(E B(xi ,ri ) f ) ⊂ B(xi , ri (1 + )) ⊂ B(xi , ri + ) ⊂ B0 , i ∈ {1, 2}, i.e., we can ensure that E B(xi ,ri ) f is compactly supported in B0 , for all f ∈ W k (B(xi , ri )), i ∈ {1, 2}. Theorem 3.2 (Sobolev extension theorem for the sphere) Let k be a non-negative integer, τ ∈ [0, 1) and CA be the constant defined in Remark 1.2. If z ∈ S d−1 and θ < CA /3, then there exists an extension operator E G(z,θ) : W k+τ (G(z, θ)) → W k+τ (S d−1 ) satisfying: 1. (E G(z,θ) f )|G(z,θ) = f, for every f ∈ W k+τ (G(z, θ)), 2. E G(z,θ) f W k+τ (S d−1 ) ≤ K f W k+τ (G(z,θ)) , where K is independent of f and z. Proof We begin with the integer order case where τ = 0. Let f ∈ W k (G(z, θ)) and p ∈ S d−1 , then we define a candidate extension operator E G(z,θ) : W k (G(z, θ)) → W k (S d−1 ) by E G(z,θ) f (p) =

2 



E Ωi (χi f ) ◦ φi−1 |Ωi (φi (p)) · FUi (p), p ∈ S d−1 .

(3.16)

i=1

Remark 3.1 tells us that, due to the bound on θ, we need only focus on the case where Ωi is a Euclidean ball. Thus, we shall assume that z is located as in case (3b) of Fig. 1.1. That is, z ∈ U1 ∩ U2 and Ωi = B(xi , ri ), for i ∈ {1, 2}. In this case we have 2 

 E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) (φi (p)), (3.17) E G(z,θ) f (p) = i=1

where E B(xi ,ri ) : W k (B(xi , ri )) → W k (Rd−1 ) is given by (3.14). In addition, we also choose  > 0 as in Remark 3.1 to ensure that 

  (3.18) supp E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) ⊂ B0 , for i ∈ {1, 2}. To prove part 1 we assume that p ∈ G(z, θ), that is, φi (p) ∈ B(xi , ri ), for i ∈ {1, 2}. Then, by Theorem 1.5, we have 

E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) (φi (p)) = (χi f ) ◦ φi−1 (φi (p)) = (χi f )(p),

66

3 Error Bounds via Duchon’s Technique

hence E G(z,θ) f (p) =

2  (χi f )(p) = f (p),

as required.

i=1

To prove part 2 we use (1.66) and consider E G(z,θ) f 2W k (S d−1 ) =

2 

π j (χ j E G(z,θ) f ) 2W k (Rd−1 )

j=1

=

2 

π j (χ j )π j (E G(z,θ) f ) 2W k (Rd−1 ) .

j=1

We note that π j (χ j ) ∈ D(Rd−1 ) (1.5) and so there exists a constant Kχ depending only on A and the partition of unity {χ j }2j=1 such that E G(z,θ) f 2W k (S d−1 ) ≤ Kχ

2 

π j (E G(z,θ) f ) 2W k (Rd−1 )

j=1

= Kχ

2  2 

2 

 E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) (φi ◦ φ−1 ) j 

by (3.17)

2  2 2 

   E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) (φi ◦ φ−1 )  j 

by (3.18)

 

W k (Rd−1 )

j=1 i=1

= Kχ

W k (B0 )

j=1 i=1

≤ 2Kχ

2  2  2 

   )  E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) (φi ◦ φ−1 j 

W k (B0 )

j=1 i=1

.

2 is an atlas for S d−1 , the coordinate changes, φi ◦ φ−1 Since A = {Ui , φi }i=1 j : φ j (Ui ∩ U j ) → φi (Ui ∩ U j ), for i = j ∈ {1, 2}, are infinitely differentiable. Therefore there exists a constant KA , depending only on A, such that

E G(z,θ) f 2W k (S d−1 ) ≤ 2Kχ KA

2  2  

2    E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri ) 

W k (B0 )

j=1 i=1

≤ 4Kχ KA

2  

2    E B(xi ,ri ) (χi f ) ◦ φi−1 | B(xi ,ri )  i=1

W k (Rd−1 )

.

(3.19)

The function (χi f ) ◦ φi−1 | B(xi ,ri ) belongs to W k (B(xi , ri )), for i ∈ {1, 2}. Thus, we can appeal to Theorem 1.5 to deduce the existence of a constant Cext , independent of (χi f ) ◦ φi−1 | B(xi ,ri ) (and therefore of f ), such that

3.1 Duchon’s Recipe for the Sphere

2 E G(z,θ) f 2W k (S d−1 ) ≤ 4Kχ KA Cext

67 2  2    (χi f ) ◦ φi−1 | B(xi ,ri )  i=1

W k (B(xi ,ri ))

.

Taking square roots gives  E G(z,θ) f W k (S d−1 ) ≤ K f W k (G(z,θ)) , where K = 2Cext Kχ KA ,

(3.20)

and this completes the proof in the case where z is positioned as (3b) Fig. 1.1; the proof for the other cases follows in a similar, but simpler, fashion. For the fractional case, τ ∈ (0, 1), we use the fact that the same extension operator works for all integer order Sobolev spaces. In particular, E G(z,θ) maps W k+i (G(z, θ)) to W k+i (S d−1 ), for i = 0, 1, such that E G(z,θ) f W k+i (S d−1 ) ≤ K f W k+i (G(z,θ)) , i = 0, 1. Given that the fractional Sobolev spaces are defined via interpolation we can evoke the operator interpolation property (see Proposition 1.2) to deduce that E G(z,θ) is a bounded linear map from W k+τ (G(z, θ)) to W k+τ (S d−1 ) and E G(z,θ) f W k+τ (S d−1 ) ≤ K1−τ Kτ f W k+τ (G(z,θ)) = K f W k+τ (G(z,θ)) . The first property follows from the fact that W k+τ (S d−1 ) ⊂ W k (S d−1 ) and W k+τ (G(z, θ)) ⊂ W k (G(z, θ)).  We now turn to the extension constant K (3.20) of the operator E G(z,θ) . In particular, we shall investigate its dependence upon the geodesic radius θ. By inspection, it is clear that the factors Kχ and KA are both independent of θ. However, the factor Cext may depend upon the radii r1 or r2 and these are both related to θ by (1.75). Indeed, this dependence is established as follows. Lemma 3.2 (Extension constant’s dependence on radius) Let k > d−1 2 be a nonnegative integer and let E B(x,r ) : W k (B(x, r )) → W k (Rd−1 ) be a linear extension operator such that E B(x,r ) f W k (Rd−1 ) ≤ Cext f W k (B(x,r )) ,

f ∈ W k (B(x, r )),

(3.21)

then the extension constant Cext > 0 depends upon the radius r of the ball. k d−1 ) is a Proof Since k > d−1 2 the Sobolev embedding theorem tells us that W (R space of continuous functions. Hence, there exists a constant c > 0 such that

c · | f (y)| ≤ f W k (Rd−1 ) , for all f ∈ W k (Rd−1 ) and y ∈ Rd−1 .

(3.22)

68

3 Error Bounds via Duchon’s Technique

Let E denote a continuous linear extension operator and choose f ∈ W k (B(x, r )) to be f = 1. A combination of (3.21) and (3.22) yields c ≤ E B(x,r ) f W k (Rd−1 ) ≤ Cext f W k (B(x,r )) .

(3.23)

Now since f W k (B(x,r )) → 0 as r → 0 we can deduce, from (3.23), that the constant  Cext must grow to ∞ as r → 0. Hence Cext depends on the radius of the ball. Unfortunately, we require that the extension constant of our operator must be independent of the geodesic radius. It may be possible to achieve this from an indepth study of the variety of rather technical constructions of extension operators but this remains an open question. For our purposes, rather than delving so deeply into this theory, we shall instead restrict the function in W s (B(x, r )) that we will extend. For this restricted class we are able to show that the extension constant Cext is independent of the radius r, and hence the spherical extension operator constant K is independent of θ. The restriction we place on the functions that we extend is that they should vanish on a suitable finite collection of points. As our interest is in developing error bounds for the difference between a function and its interpolant, which by definition will vanish at a finite collection of points, we see that, in the circumstances, this is a reasonable restriction to make. To set the scene we begin by providing some background material. First of all, we shall represent a set of distinct points {w1 , . . . , wn } in Rd−1 as the n−tuple (w1 , . . . , wn ). Furthermore, following Definition 2.5, we shall say that (w1 , . . . , wn ) is a unisolvent n−tuple (with respect to Πk−1 (Rd−1 )) whenever n > dim Πk−1 (Rd−1 ) and the only polynomial p ∈ Πk−1 (Rd−1 ) that satisfies p(wi ) = 0, i = 1, . . . , n is the zero polynomial. Let W = W1 ×· · ·× Wn be a product of compact subsets in Rd−1 such that (w1 , . . . , wn ) is unisolvent with respect to Πk−1 (Rd−1 ) for any choice of wi ∈ Wi , i = 1, . . . , n. For example, on the real line we can choose each Wi to be a bounded interval such that Wi ∩ W j = ∅ if i = j. In this case, provided n > k = dim Πk−1 (R) each n−tuple (w1 , . . . , wn ) is a set of n distinct points in R and can thus be described as a unisolvent n−tuple for Πk−1 (R). We now state a lemma that dates back to Duchon which shows we can bound the derivatives of a function f (in the L 2 norm) by higher derivatives, whenever f vanishes on a suitable unisolvent set of points. Lemma 3.3 (Duchon’s Lemma) Let k be a non-negative integer. Let Ω be a domain in Rd−1 with a smooth boundary. Let W = W1 × · · · × Wn be a product of compact subsets in Rd−1 such that (w1 , . . . , wn ) is unisolvent with respect to Πk−1 (Rd−1 ) for any choice of wi ∈ Wi , i = 1, . . . , n. Suppose that i ≤ k, then there exists a positive constant CW (i) depending on Ω, W, k and i such that  |α|=i

D α f 2L 2 (Ω) ≤ CW (i) ·

 |α|=k

D α f 2L 2 (Ω)

(3.24)

for all f ∈ W k (B0 ) such that f (w j ) = 0, (1 ≤ j ≤ n), for some (w1 , . . . , wn ) ∈ W.

3.1 Duchon’s Recipe for the Sphere

69

We have not been explicit about the smoothness conditions on the boundary of Ω in Lemma 3.3 because we will only apply this result in the case when Ω is a Euclidean ball, whose boundary satisfies the required smoothness condition. We will now set about demonstrating how our candidate extension operator (3.14) continuously extends functions in W k (B(x∗ , r )) that have certain vanishing properties in such a way that the extension constant in independent of r. A crucial ingredient to our proof is the following change of variables result. Lemma 3.4 Let α ∈ Nd0 be a multi-index with |α| ≤ k. Then, for any f ∈ W k (B(w∗ , r )), we have D α f 2L 2 (B(x∗ ,r )) = r (d−1)−2|α| D α ( f ◦ σ) 2L 2 (B0 ) ,

(3.25)

and similarly, for any f ∈ W k (Rd−1 ), we have D α f 2L (Rd−1 ) = r (d−1)−2|α| D α ( f ◦ σ) 2L (Rd−1 ) . 2 2

(3.26)

Proof Let f ∈ W k (B(x∗ , r )), then we have D

α

 f 2L 2 (B(x∗ ,r ))

=

B(x∗ ,r )

(D α f (y))2 dy



=r

 =r

((D α f ) ◦ σ)(y))2 dy

d−1 B0

(D α ( f ◦ σ)(y))2 r −2|α| dy

d−1 B0

= r (d−1)−2|α| D α ( f ◦ σ) 2L 2 (B0 ) . This proves (3.25), and the proof of (3.26) is identical.



We can now combine Duchon’s lemma and the change of variable result to prove our stated goal. Proposition 3.1 (Euclidean extension for functions with zeros I) Let k > d−1 2 be a non-negative integer. Let W = W1 × · · · × Wn be a product of compact subsets in B0 such that (w1 , . . . , wn ) is unisolvent with respect to Πk−1 (Rd−1 ) for any choice of wi ∈ Wi . Suppose that r ∈ (0, 1] and x∗ ∈ Rd−1 . Let σ be given by (3.13) and suppose that f ∈ W k (B(x∗ , r )) is such that f ◦ σ(wi ) = 0, i = 1, . . . , n, for some (w1 , . . . , wn ) ∈ W. Let E B(x∗ ,r ) be the extension operator given by (3.14). Then there exists a constant CW independent of x∗ , r and f (but depending on W) such that (3.27) E B(x∗ ,r ) f W k (Rd−1 ) ≤ CW f W k (B(x∗ ,r )) .

70

3 Error Bounds via Duchon’s Technique

Proof Applying (1.36), (3.26), and (3.14) respectively, we have 

E B(x∗ ,r ) f 2W k (Rd−1 ) =

 

2   r (d−1)−2|α| D α E B0 ( f ◦ σ) 

L 2 (Rd−1 )

0≤|α|≤k

.

Furthermore, since r ∈ (0, 1) we can deduce that E B(x∗ ,r ) f 2W k (Rd−1 ) ≤ r (d−1)−2k · E B0 ( f ◦ σ) 2W k (Rd−1 ) . By Theorem 1.5 there exists a constant C B0 , independent of x∗ and r, such that E B(x∗ ,r ) f 2W k (Rd−1 ) ≤ C B0 · r (d−1)−2k · f ◦ σ 2W k (B ) . 0

(3.28)

We can now apply Duchon’s Lemma 3.3 to the function f ◦ σ, which satisfies f ◦ σ(wi ) = 0, i = 1, . . . , n, for some (w1 , . . . , wn ) ∈ W to eliminate the terms involving derivatives of order less than k. Thus, setting AW = max{CW (i) : i = 0, . . . , k}, we can employ (1.36), (3.24) and (3.25) to deduce that f ◦ σ 2W k (B ) ≤ AW r −(d−1)+2k 0

W −(d−1)+2k

≤A r

 |α|=k

D α f 2L 2 (B(x∗ ,r ))



D α f 2L 2 (B(x∗ ,r )) .

0≤|α|≤k

That is, we have f ◦ σ 2W k (B ) ≤ AW · r −(d−1)+2k f 2W k (B(x∗ ,r )) . 0

(3.29)

Substituting (3.29) into (3.28) and taking square roots yields, E B(x∗ ,r ) f W k (Rd−1 ) ≤

 C B0 · AW · f W k (B(x∗ ,r )) .

We note that the constant AW is independent of x∗ and r, and hence, setting CW =  C B0 · AW completes the proof.  For a given k ≥ 0, we can easily construct products W that have the unisolvency properties required by Proposition 3.1. Take any set of points {x1 , . . . , xn } in B0 that are known to be unisolvent with respect to Πk−1 (Rd−1 ). It is possible to define Lagrange cardinal functions in Πk−1 (Rd−1 ) associated with these points. Any small perturbation of the points will cause a small perturbation in these cardinal functions. For a small enough perturbation, the cardinal functions will still be cardinal function for the new points. Therefore, there exists a δk > 0 such that if the points are each moved by a distance less than or equal to δk then the perturbed points will still be unisolvent with respect to Πk−1 (Rd−1 ). If we set

3.1 Duchon’s Recipe for the Sphere

71

Wi = B(xi , δk ) ⊂ B0 , i = 1, . . . , n, then W := W1 × · · · × Wn is a product of sets in B0 such that for any choice wi ∈ Wi the set {w1 , . . . , wn } is unisolvent with respect to Πk−1 (Rd−1 ). In summary, we have shown the following: Lemma 3.5 For any k ≥ 0, there exists a δk > 0 and points xi ∈ B0 , i = 1, . . . , n, such that if we set Wi = B(xi , δk ) ⊂ B0 , then W := W1 × · · · × Wn is such that {w1 , . . . , wn } is unisolvent with respect to Πk−1 (Rd−1 ) for any choice of wi ∈ Wi , i = 1, . . . , n. xn } say, in B0 that are unisolvent For a given k, we fix a set of points { x1 , . . . , with respect to Πk−1 (Rd−1 ). Then, in view of Lemma 3.5, this gives a fixed product of compact sets x1 , δk ) × · · · × B( xn , δk ) (3.30) W := B( with the unisolvency property required by Proposition 3.1. The next step is to show that if a function f ∈ W k (Rd−1 ) where k > d−1 2 vanishes at a sufficiently dense set of point X in B(x∗ , r ) then the translated and scaled function f ◦ σ on B0 will be zero on some set of points (w1 , . . . , wn ) ∈ W. We measure the density of X in B(x∗ , r ) by assigning the local Euclidean mesh norm ρ :=

sup

y∈B(x∗ ,r )

min{ x − y : x ∈ X }.

(3.31)

The image of X under σ −1 , where σ −1 (y) := (y − x∗ )/r, x ∈ Rd−1 , has a local Euclidean mesh norm ρ/r in B0 . Thus if ρ satisfies ρ/r < δk then for each point xi ∈ B0 , i = 1, . . . , n associated with W (3.30), we have min

y∈σ −1 (X )

y − xi ≤ ρ/r < δk .

This implies that there exists a point, call it wi , in Wi ∩(σ −1 X ). Now since f (x) = 0, for x ∈ X we have f ◦ σ(wi ) = 0, i = 1, . . . , n. Therefore we can apply Proposition 3.1 to f to give the following: Proposition 3.2 (Euclidean extension for functions with zeros II) Let  > 0 and k > d−1 2 be a non-negative integer. Let W and δk be given by Lemma 3.5. Let X denote a set of distinct points in B(x∗ , r ) where x∗ ∈ Rd−1 and r ∈ (0, 1]. Let f belong to the following Sobolev subspace  k (B(x∗ , r )) = W



 f ∈ W k (B(x∗ , r )) : f (x) = 0, x ∈ X .

(3.32)

Let ρ denote the mesh norm of X measured by (3.31) and let E B(x∗ ,r ) be the extension operator given by (3.14) which has support in B(x∗ , r (1 + )). If ρ/r < δk then there exists a constant CW independent of x∗ , r, f and X such that

72

3 Error Bounds via Duchon’s Technique

 k (B(x∗ , r )). E B(x∗ ,r ) f W k (Rd−1 ) ≤ CW f W k (B(x∗ ,r )) for all f ∈ W

(3.33)

Corollary 3.1 The result of Proposition 3.2 also holds for fractional order Sobolev spaces. Proof The proof of this case requires that we apply Lemma 3.5 for k + 1 rather than k, i.e., we ensure n > dim(Πk (Rd−1 )) and define W := W1 × · · · × Wn such that {w1 , . . . , wn } is unisolvent with respect to Πk (Rd−1 ) for any choice of wi ∈ Wi , i = 1, . . . , n. Clearly, any such set {w1 , . . . , wn } will also contain a subset that is unisolvent with respect to Πk−1 (Rd−1 ), thus we can appeal to Lemma 3.3 for the spaces of order k and k + 1 simultaneously. We then recall, from Theorem 1.4,  k+τ (B(x ∗ , r )) is precisely the interpolation space of the integer that the subspace W k  k+1 (B(x∗ , r ))} (see Theorem 1.4). The fractional  (B(x∗ , r )), W order couple {W result then follows by applying the operator interpolation property in the same way as in Theorem 3.2.  Proposition 3.2 establishes that we can extend certain functions in W k (B(x∗ , r )) to functions in W k (Rd−1 ), such that the extension constant is independent of r. We now turn attention to proving a similar result on S d−1 . This is done using the 2 and partition of unity functions {χ }2 to fixed atlas of coordinate charts {Ui , φi }i=1 i i=1 project functions (with certain zero properties) in local Sobolev spaces on geodesic balls on S d−1 onto local Sobolev spaces on Euclidean balls in Rd−1 . Proposition 3.2 can be used to extend these projected functions to W k (Rd−1 ). By mapping back up to the sphere we are able to prove an analogue of Proposition 3.2 for S d−1 . To begin the analysis, we return to Theorem 3.2 where we demonstrated a construction of the Sobolev extension operator for the sphere. For the integer order case we trace the proof of this result to Eq. (3.19), where we have E G(z,θ) f 2W k (S d−1 ) ≤ 4Kχ KA

2 

E B(xi ,ri ) f i 2W k (Rd−1 )

(3.34)

i=1

where

f i = (χi f ) ◦ φi−1 | B(xi ,ri ) for i ∈ {1, 2},

and where the constants Kχ and KA are independent of z and θ. Suppose now that Ξ is a set of distinct points on S d−1 with mesh norm h (2.79). We shall restrict attention to those local functions that vanish on Ξ ∩ G(z, θ), i.e., for k > (d − 1)/2 we consider applying our extension operator to the subspace 2k (G(z, θ)) = W



 f ∈ W k (G(z, θ)) : f (ξ) = 0, ξ ∈ Ξ ∩ G(z, θ) .

(3.35)

To measure the density of Ξ locally, i.e., on some geodesic ball G(z, θ) we assign the local mesh norm

3.1 Duchon’s Recipe for the Sphere

hL =

sup

η∈G(z,θ)

73

  min g(η, ξ) : ξ ∈ Ξ ∩ G(z, θ) .

(3.36)

 k (G(z, θ)), then the projected functions f i = (χi f ) ◦ Observe that if f ∈ W 2 k ∈ W (B(xi , ri )) i = 1, 2, vanish on the transformed set of points given

φi−1 | B(xi ,ri ) by

  (i) X θ = φi (ξ) : ξ ∈ Ξ ∩ G(z, θ) ⊂ B(xi , ri ).

(3.37)

In summary, we conclude that  k (B(xi , ri )), for i = 1, 2,  k (G(z, θ)) ⇒ f i = (χi f ) ◦ φ−1 | B(xi ,ri ) ∈ W f ∈W i where 2k (B(xi , ri )) = W



(i)

f ∈ W k (B(xi , ri )) : f (x) = 0, for x ∈ X θ



.

(3.38)

(i)

We measure the density of X θ , which we assume to be non-empty, by assigning the local Euclidean mesh norm   (3.39) ρi = sup min x − φi (ξ) : ξ ∈ Ξ ∩ G(z, θ) . x∈B(xi ,ri )

Remark 3.3 Let k > d−1 2 be a positive integer and let δk be as in Lemma 3.5. Let  k (G(z, θ)) and assume that the local Euclidean mesh norms (3.39) satisfy f ∈W ρi < δk , for i ∈ {1, 2}. ri

(3.40)

Then, using Theorem 3.2, there exist a constant Cext , independent of xi and ri , i ∈ {1, 2}, such that 2 E G(z,θ) f 2W k (S d−1 ) ≤ 4Kχ KA Cext

2 

2 f 2 k f i 2W k (B(x ,r )) = K , W (G(x,θ)) i

i

i=1

= where K

 4Kχ KA Cext is independent of z and θ.

Remark 3.3 provides the route to a restricted extension theorem for the sphere. In particular, what is needed is a deeper understanding of two geometric issues: 1. How does the global density measure h relate to the local measure h L ? 2. How does h relate to the local Euclidean density measures ρi (i = 1, 2) when the local points are mapped to Euclidean balls? The following result provides a relationship between h and h L .

74

3 Error Bounds via Duchon’s Technique

Lemma 3.6 Let Ξ be a set of points on S d−1 with mesh norm h ∈ (0, π/6). Let z ∈ S d−1 , θ ≥ 3h and let h L denote the local mesh norm of Ξ ∩ G(z, θ). Then h L ≤ 4h.

(3.41)

Proof Let η ∈ G(z, θ) and let ξ be a closest point to η from Ξ. Then, by (2.79), we have g(η, ξ) ≤ h. We prove the lemma by splitting into two cases based on the position of ξ. (a) If ξ ∈ G(z, θ), then min{g(η, ξ) : ξ ∈ Ξ ∩ G(z, θ)} ≤ h < 4h. (b) If ξ ∈ / G(z, θ), then g(ξ, z) ≥ θ ≥ 3h. Thus, there exists a point η  ∈ G(z, θ), lying on the intersection between the boundary of G(ξ, 2h) and the geodesic arc connecting z and ξ, (see Fig. 3.2). That is, η  satisfies g(z, ξ) = g(z, η  ) + g(η  , ξ) = g(z, η  ) + 2h. Furthermore, there must exist a ξ  ∈ Ξ, such that g(η  , ξ  ) ≤ h. The triangle inequality allows us to deduce g(z, ξ  ) ≤ g(z, η  ) + g(η  , ξ  ) = g(z, ξ) − 2h + g(η  , ξ  ) ≤ g(z, η) + g(η, ξ) − h < θ + h − h = θ. Thus, ξ  ∈ G(z, θ), and this implies g(η, ξ  ) ≤ g(η, ξ) + g(ξ, η  ) + g(η  , ξ  ) ≤ h + 2h + h = 4h. Hence min{g(η, ξ) : ξ ∈ Ξ ∩ G(z, θ)} ≤ 4h. These arguments hold for any η ∈ G(z, θ) and so, by (3.36), the proof is complete.  The next result shows how the geodesic mesh norm of Ξ relates to the local (i) Euclidean mesh norms of the X θ , for i ∈ {1, 2}.

Fig. 3.2 Illustration of Lemma 3.6

3.1 Duchon’s Recipe for the Sphere

75

Lemma 3.7 Let Ξ be a set of points on S d−1 with mesh norm h ∈ (0, π/6). Let z ∈ S d−1 , θ ≥ 3h and assume that G(z, θ) ⊂ Ui , i ∈ {1, 2}. Let ρi denote the (i) Euclidean mesh norm of X θ = φi (Ξ ∩ G(z, θ)) given by (3.39), then ρi ≤ 4C0 h, for i ∈ {1, 2},

(3.42)

where C0 is as in Lemma 1.3. Proof Let x ∈ B(xi , ri ), then η = φi−1 (x) ∈ G(z, θ). By Lemma 3.6, there exists a point ξ ∈ Ξ such that g(η, ξ) ≤ 4h. We note that 4h ≤ 2π/3 and so we can use Lemma 1.3 to deduce that min

ξ∈Ξ ∩G(z,θ)

φi (η) − φi (ξ) =

min

ξ∈Ξ ∩G(z,θ)

x − φi (ξ) ≤ 4C0 h.

This result holds for all x ∈ B(xi , ri ) and so proves the lemma.



We now demonstrate how the extension constant for the operator from Theorem 3.2 is independent of the radius of the geodesic ball when we restrict the class of functions we wish to extend to those satisfying certain vanishing properties. Theorem 3.3 (Sobolev extension theorem for the sphere II) Let k > d−1 2 be a non-negative integer, τ ∈ [0, 1) and CA be the constant defined in Remark 1.2. Let z ∈ S d−1 , M be a given constant and Ξ be a set of points on S d−1 with mesh norm h ∈ (0, π/6) given by (2.79). There exists a constant M0 (independent of h) such that if (3.43) M0 ≤ M and Mh < CA /3 then, for any f belonging to the Sobolev subspace  k+τ (G(z, Mh)) = W



f ∈ W k+τ (G(z, Mh)) : f (ξ) = 0, ξ ∈ Ξ



the linear extension map E G(z,Mh) given in Theorem 3.2 satisfies  f k+τ E G(z,Mh) f W k+τ (S d−1 ) ≤ K W (G(z,Mh)) ,

(3.44)

 is independent of h. where the extension constant K Proof We begin by considering the integer order case where τ = 0. In this setting we can evoke Lemma 3.7, to deduce that if M ≥ 3, then ρi ≤ 4C0 · h, for i ∈ {1, 2}. Furthermore, using Lemma 1.3, there exists a constant c0 > 0, such that ρi 4C0 · h 4C0 · h 4C0 ≤ ≤ = , for i ∈ {1, 2}. ri ri c0 Mh c0 M

(3.45)

76

3 Error Bounds via Duchon’s Technique

Thus, if M ≥ 3 and M > 4C0 /c0 δk then condition (3.40) holds. In view of this, by setting   4C0 , (3.46) M0 = max 3, c0 δk the result follows by the observation in Remark 3.3. The fractional case follows by setting   4C0 M0 = max 3, , c0 δk+1

(3.47)

since this condition ensures that the point set is sufficiently dense for the interpolation arguments set out in Corollary 3.1 to hold. 

3.2 Global Error Bounds for SBF Interpolation We recall, from Chap. 2, that the accuracy of the SBF method depends upon several factors, including the smoothness of the basis function ψ, the smoothness of the target function to be interpolated, and the mesh norm h (2.79) of the interpolation points N ⊂ S d−1 . In particular, we demonstrated that if ψ ∈ C S P D (S d−1 ) Ξ = {ξi }i=1 m whose Legrendre coefficients decay like (2.48) for some α > 0 and the target function f belongs to the native space Hψ of ψ (which is norm equivalent to the Sobolev space W s (S d−1 ) with s = (d − 1 + α)/2) then f − s f L ∞ (S d−1 ) = O(h α/2 ). In this section we will show how our newly established Duchon framework can be used to supply L p −error estimates. Specifically, under the same assumptions as above, we will prove results of the form α

f − s f L p (S d−1 ) = O(h 2

+ d−1 p

),

p ∈ [2, ∞].

Furthermore, we show that if a stronger smoothness condition is imposed upon the target function then we can improve this result to f − s f L p (S d−1 ) = O(h

d−1 α+ d−1 2 + p

),

p ∈ [2, ∞].

The case of p ∈ [1, 2) will also be included. The Duchon framework for the sphere that we have established in the previous section can be viewed as providing the key ingredients for a recipe which yields error bounds.

3.2 Global Error Bounds for SBF Interpolation

77

Our first task is to use the covering of S d−1 (see Lemma 3.1) to write 

p

f − s f L

p (S

d−1 )

=

|( f − s f )(ξ)| p d S(ξ)  √ |( f − s f )(ξ)| p d S(ξ), for M∗ = d − 1. ≤ S d−1

z∈Z h

G(z,M∗ h)

This step gives us the advantage that we can consider the error locally. In particular, the function f − s f is continuous on G(z, M∗ h) and, as this is a compact subset of S d−1 , there exists a point ξ z ∈ G(z, M∗ h) at which f − s f attains its maximum. This observation allows us to write   p f − s f L (S d−1 ) ≤ |( f − s f )(ξz )| p d S(ξ) p

G(z,M∗ h)

z∈Z h

≤ Cd · h d−1



|( f − s f )(ξ z )| p ,

(3.48)

z∈Z h

where Cd is a constant depending only on d which satisfies

 Area G(z, M∗ h) ≤ Cd h d−1 .

(3.49)

We know, from the variational theory, that f − s f ∈ Hψ . Furthermore, due to the decay of the Legendre coefficients, the native space Hψ is norm equivalent to the Sobolev space W s (S d−1 ), where s = (d − 1 + α)/2, (see the paragraph following Definition 2.11). Now, rather than consider f − s f , we choose instead to consider the restriction f − s f |G(z,Mh) where M = M∗ + M1 , for some M1 ≥ 0 whose value, or more precisely whose range of values, is not yet determined. In choosing a suitable value for M1 , and hence M, we must take into account the following conditions. (a) In order to employ Theorem 2.8 to provide pointwise error estimates, we require that each G(z, Mh) must contain the dot product neighbourhood N (ξ z , K 1 h d ). (b) In order to apply the Sobolev extension operator to f − s f |G(z,Mh) ∈ W s (G(z, Mh)), we require that Mh ∈ (R0 h, CA /3), see Theorem 3.3. If we let R0 and CA denote the constants from Theorem 3.3 corresponding to √ , then the condition M1 > max{R0 − d − 1, 0} together with the s = α+d−1 2 assumption that the geodesic mesh norm of Ξ should satisfy 0 < h < CA /(3M) are sufficient to guarantee that condition (b) is satisfied, (see Theorem 3.3).

(3.50)

78

3 Error Bounds via Duchon’s Technique

We now turn to condition (a). Let K 1 denote the neighbourhood constant from Theorem 2.8. For any ξ z , the neighbourhood N (ξ z , K 1 h d ) can also be viewed, in more familiar terms, as an open geodesic ball G(ξ z , θ), where θ satisfies sin2 (θ/2) = K 1 h d /2. If we assume that the dot product mesh norm (2.87) satisfies (dot)

hd < h0

= 3/(2K 1 ),

(3.51)

then we have that θ ∈ (0, 2π/3), thus we can apply the small angle result for sin(θ/2), followed by the mesh-norm equivalence relation (2.88) to deduce that   h θ √ ≤ K1hd ≤ K1 · √ . 2 2 2 In particular, if M1 is chosen according to M1 > max{R0 −

√

 d − 1, 2 K 1 },

(3.52)

then this shows that  N (ξ z , K 1 h d ) = G(ξ z , θ) ⊂ G(ξ z , 2 K 1 h)  ⊂ G(z, (M + 2 K 1 )h) ⊂ G(z, (M + M1 )h) = G(z, Mh) ⊂ G(z, CA /3), and so both condition (a) and condition (b) are simultaneously satisfied. Let vz = f − s f |G(z,Mh) then, using the Sobolev extension operator, we have E1. E G(z,Mh) vz ∈ W s (S d−1 ). E2. E G(z,Mh) vz (ξ) = 0 for all ξ ∈ Ξ ∩ G(z, Mh). E3. Using Theorem 3.3, there exists a constant K, independent of h and z such that E G(z,Mh) vz W s (S d−1 ) ≤ K · vz W s (G(z,Mh)) . Since condition (a) is satisfied, we have that Ξξz = Ξ ∩ N (ξ z , K 1 h d ) ⊂ Ξ ∩ G(z, Mh) = Ξz . Thus, the optimal power function of ψ, based upon Ξz and evaluated at the point ξ z , can be bounded above by the local power function Pψ,oc (ξ z ). Moreover, if we let J denote the integer from Theorem 2.8 and we assume that the geodesic mesh norm of Ξ satisfies

3.2 Global Error Bounds for SBF Interpolation (geod)

h ∈ (0, h 0

79 (geod)

) where h 0

= min

1 , 3M K

C

A

,

(3.53)

where K > J is a positive integer, then we can subsequently use the local error bound (2.96) together with (2.54) and E3 respectively, to yield |( f − s f )(ξ z )| = |E G(z,Mh) vz (ξ z )| ≤ Pψ,oc (ξ z ) E G(z,Mh) vz ψ ≤ Pψ,oc (ξ z )K eq E G(z,Mh) vz W s (S d−1 ) α

≤ C K eq · h 2 E G(z,Mh) vz W s (S d−1 ) α

≤ KC K eq · h 2 vz W s (G(z,Mh)) . Substituting this into (3.48) gives p

f − s f L

p

(S d−1 )

≤ Cd (KC K eq ) p · h (

αp 2 +d−1)



f − s f |G(z,Mh)

z∈Z h

For p ≥ 2 we use Jensen’s inequality by Theorem 3.1, and (2.54) to give

N

p i=1 ai

≤(

p . W s (G(z,Mh))

N

2 p/2 [BS02], followed i=1 ai )

p

f − s f L ⎛ ≤ Cd (KC K eq ) p · h ≤ Cd (KC K eq ≤ Cd (KC K eq

 

( α2p +d−1)

·⎝



z∈Z h

Q) p · h

( α2p +d−1)

−1 p Qkeq ) · h(

p (S

d−1 )

⎞ p/2 ( f − s f )|G(z,Mh) 2W s (G(z,Mh)) ⎠

p/2  · f − s f 2W s (S d−1 )

αp 2 +d−1)

p

· f − s f ψ .

Finally, taking the pth root gives α

f − s f L p (S d−1 ) ≤ C · h 2

+ d−1 p

f − s f ψ ,

√ −1 1/ p where C = Cd (KC K eq Qkeq ) is independent of f and h. For p ∈ [1, 2) we execute the same arguments as above, however we replace 

p/2 N p p N 2 ai ≤ N 1− 2 [BS02]. Further, we use Jensen’s inequality with i=1 i=1 ai the fact that the cardinality of Z h is bounded by C Q h −(d−1) , see Lemma 3.1, to deduce that

80

3 Error Bounds via Duchon’s Technique p

f − s f L ⎛ ≤ Cd (KC K eq ) p · h ≤ Cd (KC K eq = Cd (KC K eq

 

( α2p +d−1)

·⎝



p (S

d−1 )

⎞ p/2 ( f − s f )|G(z,Mh) 2

Q) p C Q · h p(

α+d−1 ) 2

−1 p Qkeq ) C Q · h p(

β

W2 (G(z,Mh))

z∈Z h

⎠

p/2  · f − s f 2W s (S d−1 )

α+d−1 ) 2

p

· f − s f ψ .

Finally, taking the pth root provides f − s f L p (S d−1 ) ≤ C · h

α+d−1 2

f − s f ψ ,

√ −1 where C = (Cd C Q )1/ p (KC K eq Qkeq ) is independent of f and h. In summary we have proved the following theorem. Theorem 3.4 Assume that ψ ∈ C S P Dm (S d−1 ) whose Legendre coefficient decay like (2.48) for some α > 0. Let Ξ denote a set of distinct data points on S d−1 with geodesic mesh-norm h (2.79). There exists a positive number h 0 such that, if h ∈ (0, h 0 ) then the ψ−based SBF interpolant s f to any target function f ∈ Hψ satisfies

and

f − s f L p (S d−1 ) ≤ C · h 2

α

+ d−1 p

f − s f ψ , for p ∈ [2, ∞)

(3.54)

α

d−1 2

f − s f ψ , for p ∈ [1, 2),

(3.55)

f − s f L p (S d−1 ) ≤ C · h 2 +

where the generic constant C is independent of f and h. Proof Let M1 be chosen according to (3.52) and define 

 (geod) (dot) , , 2h 0 h 0 = min h 0 (geod)

(dot)

where h 0 and h 0 are defined by (3.53) and (3.51) respectively. If h ∈ (0, h 0 ), then the two density conditions (3.53) and (3.51) are satisfied. Thus, the arguments set out in the analysis leading up to the theorem can be employed to provide the desired results.  Remark 3.4 The constants appearing in the error bounds (3.54) and (3.55) depend 1/ p on the value of p. For p ∈ [2, ∞] the dependence is due to the factor Cd where Cd is given by (3.49). For p ∈ [1, 2) the dependence is due to the factor (Cd C Q )1/ p where C Q is taken from Lemma 3.1, In both cases we note that the constants do not grow excessively large as p varies.

3.2 Global Error Bounds for SBF Interpolation

81

At first glance it is tempting to “tidy up” the error results from Theorem 3.4 by employing the optimality bound f − s f ψ ≤ f ψ , from Lemma 2.1 (ii). This is a perfectly valid procedure, however we will show that an improved bound is available, provided that f belongs to a certain subspace of Hψ , which we shall denote as Hψ∗ψ . Once this improved bound is established we will use it to improve the L p −convergence order in (3.54) for target functions f ∈ Hψ∗ψ . Definition 3.1 Assume that ψ ∈ C S P Dm (S d−1 ) whose Legendre coefficient decay like (2.48) for some α > 0, and let Ψ denote its corresponding zonal kernel. We define the convolution kernel of Ψ by  (Ψ ∗ Ψ )(ξ, η) :=

S d−1

Ψ (ξ, ν)Ψ (ν, η)dωd−1 (ν), ξ, η ∈ S d−1 .

It is more revealing to work in terms of Fourier expansions since we have Ψ (ξ, η) =

Nk,d ∞  

ψk Yk, (ξ)Yk, (η) ⇒ (Ψ ∗ Ψ )(ξ, η) =

Nk,d ∞  

k=0 =1

ψk2 Yk, (ξ)Yk, (η).

k=0 =1

This observation allows us to define a convolution native space by ⎛

 Hψ∗ψ,m =

f ∈ L 2 (S d−1 ) : | f |ψ∗ψ,m

⎞1/2  Nk,d ∞  2  | f | k, ⎝ ⎠ = d/2. Before we examine how this family can be restricted to the sphere, we briefly review what is known about D m (Rd )−splines in the Euclidean setting. N is a finite set of distinct points in Rd Firstly, φ ∈ C S P D(n) and thus if X = {xi }i=1 containing a subset that is unisolvent with respect to Πn−1 (Rd ), and f : Rd → R is an arbitrary function, then the unique D m (Rd )−spline interpolant to f at X, denoted by s f , has the form (2.6)–(2.8). The generalized Fourier transforms of Φ(y) = φ(y) is computed in [SW01] and we quote:

 := Φ(y)

⎧ d−2 d 2m ⎪ 2 +1)Γ (m) ⎪ 2 π 2 Γ (m− , ⎪ ⎨ y2m ⎪ d ⎪ ⎪ ⎩ 22m−1 π 2 Γ (m− d2 +1)Γ (m) y2m

if d is odd, (4.3)

,

if d is even.

Thus, from the interpolation theory of RBFs one can show [LW92] that the native space of φ is (following a simple modification) equivalent to the Sobolev space W m (Rd ). For most practical applications we assume that we have a bounded open domain Ω ⊂ Rd for which X ⊂ closure(Ω). We measure the density of X in Ω by assigning the mesh norm (2.19). Let p ∈ [1, ∞] and let f : Ω → R be chosen from a sufficiently smooth class of functions. The D m (Rd )—spline interpolant is said to provide an L p —approximation order of λ > 0 if  f − s f  L p (Ω) = O(h λ ), as h → 0.

(4.4)

The largest possible value of λ such that (4.4) holds is called the optimal approximation order. In the late 1980s, Buhmann [Buh90] investigated the special case of D m (Rd )— spline interpolation where Ω = Rd on the scaled integer lattice X = h · Zd . In this framework it is shown that, given a function f ∈ C 2m (Rd ), its unique D m (Rd )— spline interpolant s f at h · Zd satisfies s f − f  L p (Rd ) = O(h 2m ), p ∈ [1, ∞].

(4.5)

Furthermore, 2m is the optimal approximation order. At the present time, the optimal L p —approximation orders ( p ∈ [1, ∞]) are not known for the case of D m (Rd )—spline interpolation on a smooth bounded domain Ω ⊂ Rd . Indeed, numerical evidence suggests that, in such cases, the accuracy is badly affected by the presence of a boundary. There has been a great deal of research into understanding (i.e., quantifying) the boundary effect, most notably by

4.1 Duchon Splines for the Sphere

87

Michael Johnson. Indeed, a combination of Johnson’s results from [Joh98, Joh00, Joh04] demonstrates that, for a suitably smooth domain Ω ⊂ Rd , the surface spline interpolant to a smooth enough target function f over a data set X ⊂ Ω with mesh norm h satisfies  λ ∈ [m − d2 + dp + 21 , m + 1p ], if p ∈ [2, ∞],  f −s f  L p (Ω) = O(h λ ), where if p ∈ [1, 2]. λ = m + 1p , It is shown in [Joh04] that m +

1 p

is optimal for p ∈ [1, 2] and, in the same paper,

numerical evidence is presented which strongly suggests that m + 1p is also the optimal L p —approximation order for p > 2; a theoretical explanation of these results remains an open problem. Let us suppose we wish to use the D m (Rd )—spline basis functions to interpolate data which are known to lie on a smooth compact (d −1)—dimensional differentiable manifold M ⊂ Rd . One possible scheme would be to use a modified surface spline to account for the dimension of the manifold, that is, we choose m, d ∈ N such that m > d−1 2 , and consider 

n 2m−d−1 log r (with n = m − d−3 ), 2

) := (−1) r φ(r (with n = m − d−2 (−1)n r 2m−d−1 2 ),

if d is odd, if d is even.

In the special case of the unit sphere M = S d−1 we can use (4.1) and completely specialise this family to the sphere to give ψ : [−1, 1] → R as  ψ(t) =

(−1)m−

d−3 2

(−1)m−

d−2 2

1 m− d−1 2 2 (2 − 2t) m− d−1 2

(2 − 2t)

log(2 − 2t),

,

if d is odd,

(4.6)

if d is even.

We can use (1.33) as a starting point to compute the Fourier coefficients of the induced zonal kernel Ψ (ξ, η) = ψ(ξ T η). Specifically, substituting the Rodrigues representation (1.25) for the Pk,d (k ≥ 0), and integrating by parts k times gives k = κd ψ

1 ψ

(k)

2 k+ d−3 2

(t)(1 − t )

dt, where κd =

−1

π

d−1 2

2k−1 Γ (k +

d−1 2 )

.

(4.7)

We begin with the case where d is even and consider ψ(t) = (2 − 2t)m− Substituting ψ (k) (t) =

(−1)k 2m−

d−1 2

Γ (m −

Γ (m −

d−1 2

d−1 2

− k + 1)

+ 1)

(1 − t)m−

d−1 2 −k

,

d−1 2

.

88

4 Radial Basis Functions for the Sphere

into (4.7) yields (even) ψ k

=

(−1)k 2m−

d−1 2

Γ (m −

Γ (m −

d−1 2

d−1 2

+ 1)

− k + 1)

1 κd

(1 − t)m−

d−1 2 −k

(1 − t 2 )k+

d−3 2

dt.

−1

Letting 2u = 1 + t allows us to write (even) = ψ k

(−1)k 22m+k−1 Γ (m − Γ (m −

d−1 2

d−1 2

+ 1)

− k + 1)

1 κd

(1 − u)m−1 u k+

d−3 2

du.

0

The integral in the above expression is the beta function 1 B(x, y) =

(1 − u) y−1 u x−1 du = 0

evaluated at y = m and x = k + (even) = ψ k

d−1 2 .

Thus we have that

(−1)k 22m+k−1 Γ (m − Γ (m −

d−1 2

Γ (x)Γ (y) , Γ (x + y)

d−1 2

+ 1)Γ (m)Γ (k +

− k + 1)Γ (m + k +

d−1 2 )

d−1 2 )

κd .

(4.8)

This can be simplified further by using the reflection formula ([AS64], 6.1.17) Γ (z)Γ (1 − z) = In particular setting z = m − Γ (m −

d−1 2

(4.9)

− k + 1 yields

d−2 d −1 d −1 − k + 1)Γ (−m + k + ) = (−1)k−m+ 2 π, 2 2

furthermore, expressing Γ (m + k + (m + k +

π . sin πz

d−1 2 )

as

d −1 d −1 d −1 − 1) · · · (m + k + − 2m)Γ (−m + k + ) 2 2 2

allows us to rewrite (4.8) as (even) = ψ k

(−1)m−

d−2 2

22m+k−1 Γ (m −

(m + k +

d−1 2

d−1 2

+ 1)Γ (m)Γ (k +

− 1) · · · (m + k +

d−1 2

− 2m)π

d−1 2 )

κd .

4.1 Duchon Splines for the Sphere

89

Substituting the value for κd yields (even) = ψ k

(−1)m−

d−2 2

(m + k +

22m π

d−1 2

d−3 2

Γ (m −

d−1 2

− 1) · · · (m + k +

+ 1)Γ (m) d−1 2

− 2m)

.

(4.10)

We now turn attention to the case where d is odd and so we consider ψ(t) = log(2 − 2t). This function can be represented as follows

1 m− d−1 2 2 (2 − 2t)

ψ(t) =

1 ∂ (2 − 2t)β . β=m− d−1 2 ∂β 2

(4.11)

The Fourier coefficients of the more general function t → (2 − 2t)β can be computed in the same way as above. Indeed, they are given by  ck (β) = (−1)k π where h(β) =

d−1 2

2d−1 h(β)

(4.12)

22β Γ (β + 1)Γ (β + d−1 2 ) . Γ (k + β + d − 1)Γ (−k + β + 1)

(4.13)

(odd)

 The Fourier coefficients {ψ }k≥0 of the zonal kernel induced by (4.11) are k given by (odd) = 1 ∂ cˆk (β) ψ k β=m− d−1 2 ∂β 2 . d−3

Since the function (−1)m− 2 ψ is conditionally positive definite of order m − d−1 2 (odd) where k > m − d−1 . In order to we shall restrict attention to the coefficients ψ k 2 differentiate h(β) it is useful to express it as h(β) = 22β Γ (β +

v(β) d − 1 β(β − 1) · · · (β − k + 1) ) = u(β) . 2 Γ (β + k + d − 1) w(β)

Differentiating with respect to β gives h (β) =

w(β){u (β)v(β) + v (β)u(β)} − u(β)v(β)w (β) . w(β)2

We observe that v(m − d−1 2 ) = 0 whenever k > m − above, evaluated at β = m − d−1 2 simplifies to h (m −

d−1 2

v (m − d−1 d −1 2 )u(m − )= 2 w(m − d−1 2 )

and so the expression

d−1 2 )

.

90

4 Radial Basis Functions for the Sphere

Furthermore, v (m −

d−3 d −1 d −1 d −1 ) = (−1)k−m+ 2 Γ (m − + 1)Γ (k − m + ), 2 2 2

from which we can see h (m − (−1)k+m−

d−3 2

d−1 2 )

equals

22m−(d−1) Γ (m)Γ (m −

Γ (k − m + d −1 + 1) 2 Γ (k + m +

d−1 2 ) . d−1 2 )

Using the fact that Γ (k − m + Γ (k + m +

d−1 2 ) d−1 2 )

=

1 (m + k +

we can deduce that, for k > m − (odd) = ψ k

(−1)m−

d−1 2 ,

d−3 2

(m + k +

d−1 2

d−1 2

− 2m)

,

we have

22m−1 π

d−1 2

− 1) · · · (m + k +

d−1 2

Γ (m −

− 1) · · · (m

d−1 2 + 1)Γ (m) . + k + d−1 2 − 2m)

(4.14)

In summary we have proved the following result. Lemma 4.1 The spherical Fourier coefficients of the D m (Rd )—splines restricted to S d−1 (4.6) are given by

k = ψ

⎧ d−3 22m π 2 Γ (m− d−1 ⎪ 2 +1)Γ (m) ⎪ , if d is even, ⎪ d−1 ⎪ ⎨ (m+k+ d−1 2 −1)···(m+k+ 2 −2m) ⎪ ⎪ ⎪ ⎪ ⎩

d−1

22m−1 π 2 Γ (m− d−1 2 +1)Γ (m) , d−1 (m+k+ d−1 −1)···(m+k+ 2 2 −2m)

if d is odd and k > m − (d − 1)/2. (4.15)

We make two observations. Firstly, the numerators appearing in the expressions for the spherical coefficients (4.15) have the same form as the numerators appearing in the expressions for the d—dimensional Fourier transform (4.3), where d is replaced by d − 1 in the spherical setting. Secondly, the Fourier coefficients exhibit the same decay rates as the generalised Fourier transform, i.e., we have that k = O(k −2m ).  Φ(y) = O(y−2m ) and ψ This is not a coincidence as the following more general result demonstrates (see [zCF05, NSW07]).

4.1 Duchon Splines for the Sphere

91

Theorem 4.1 Let Φ(x) = φ(x) denote a radial function induced by φ ∈ C S P Dm for some m ≥ 0. Assume that the (generalized) d—dimensional Fourier transform of  Φ has polynomial decay, i.e., Φ(y) = O(yd+α ) for some α > 0. The restriction of Φ to the sphere

 Ψ (ξ, η) = φ

 2 − 2ξ η , ξ, η ∈ S d−1 , T

has a representation as a spherical Fourier series Ψ (ξ, η) =

Nk,d ∞  

k Yk, (ξ)Yk, (η) ψ

k=0 =1

k = k have the analogous decay rate, i.e., ψ whose spherical Fourier coefficients ψ O(k −(d−1)+α ). With the decay rate of the spherical Fourier coefficients established, we can now appeal to the error bounds from Chap. 3 (specifically Theorem 3.5) to formulate the following result. Theorem 4.2 Let m be a non-negative integer such that m > d−1 2 . Let ψ denote d−1 m d of Duchon’s family of D (R )—splines (2.18). Let f ∈ the restriction to S W 2m (S d−1 ) and let s f denote the unique ψ—based SBF interpolant to f over a set Ξ ⊂ S d−1 of distinct data points with mesh-norm h. Then, we have  s f − f  L p (S d−1 ) =

d−1 2m− d−1 2 + p

O(h O(h 2m ),

), if p ∈ [2, ∞], as h → 0. if p ∈ [1, 2],

(4.16)

4.2 Numerical Investigation In this section we investigate the numerical performance of employing surface splines to fit scattered data on the sphere. All the code used in these investigations was written in MATLAB. In the testing environment attention is devoted to the circle S 1 , where we shall use the linear and cubic splines, that is, ψ(t) = −(2 − 2t)1/2 and ψ(t) = (2 − 2t)3/2 ,

(4.17)

and the 2-sphere S 2 , where we shall use the thin plate spline ψ(t) = (1 − t) log(2 − 2t).

(4.18)

For a thorough investigation it is important that we have some control on the distribution of the data points on either S 1 or S 2 . The important quantity to consider is the

92

4 Radial Basis Functions for the Sphere

mesh norm h given by (2.79). In view of this we perform numerical testing on S 1 with equally spaced points, where doubling the number of points causes h to halve. For S 2 , we use the spiral points of Saff and Kuijlaars [SK97]. Here the data points uniformly fill up the sphere by tracing out an imaginary spiral from the south pole ξ 1 to the north pole ξ N . Numerical experiments suggest that doubling √ the number of spiral points causes h to decrease by a factor of approximately 1/ 2. For testing purposes we choose to interpolate the following infinitely smooth target functions 3

f (x, y) = 1 + x 8 + e2y , for ξ = (x, y) ∈ S 1 ,

(4.19)

f (x, y, z) = sin x · sin y · sin z, for ξ = (x, y, z) ∈ S 2 .

(4.20)

and

In order to measure the interpolation error we generate 10,000 points (equally spaced for S 1 and randomly distributed for S 2 ) and approximate as follows, s − f  L ∞ (S d−1 ) ≈ max{|s(ξi ) − f (ξi )| : 1 ≤ i ≤ 10, 000}, and p

s − f  L

p (S

d−1 )

≈

10,000  1 |s(ξi ) − f (ξi )| p , 10, 000

p ∈ [1, ∞).

i=1

With this test environment in place, we ask the following question. Question: How close are the theoretical L p −convergence orders, given by Theorem 4.2 to those observed by experiment? Let p,N denote the L p —error measure achieved using N data points with mesh norm h N . The aim of our experiment is to examine how the error measure changes as the interpolation nodes double. The theory predicts that  d−1 2m− d−1 2 + p ,

p,N (h N / h 2N ) ≈

p,2N (h N / h 2N )2m ,

if p ∈ [2, ∞], if p ∈ [1, 2].

Using our√data point distributions we know that h N / h 2N = 2 for the circle, and h N / h 2N ≈ 2 for the 2-sphere. Thus we can use our numerical results to predict the optimal convergence orders. The results are displayed in Tables 4.1, 4.2 and 4.3. The numerical results suggest that, just as in the Euclidean case, the optimal L p (S d−1 ) convergence order ( p ∈ [1, ∞]), for restricted D m (Rd )—spline interpolation is 2m. This implies that Theorem 4.2 predicts the optimal orders for p ∈ [1, 2] and, for p ∈ [2, ∞] it suggests the following conjecture: Proposition 4.1 (Conjecture on global accuracy) Let m be a non-negative integer d−1 of Duchon’s family of such that m > d−1 2 . Let ψ denote the restriction to S D m (Rd )—splines (2.18). Let f ∈ W 2m (S d−1 ) and let s f denote the unique ψ—based

4.2 Numerical Investigation

93

Table 4.1 Accuracy of linear spline interpolation on S 1 with m = 1 Linear Error measure Convergence order Nodes 64 128 256 512 1024

L1 4.24−02 1.07−02 2.68−03 6.69−04 1.67−04

L2 2.73−02 6.84−03 1.71−03 4.28−04 1.07−04

L∞ 4.91−02 1.26−02 3.17−03 7.94−04 1.99−04

L1 − 1.99 2.00 2.00 2.00

L2 − 2.00 2.00 2.00 2.00

L∞ − 1.96 1.99 2.00 2.00

Table 4.2 Accuracy of cubic spline interpolation on S 1 with m = 2 Cubic Error measure Convergence order Nodes 64 128 256 512 1024

L1 2.00−05 1.14−06 7.05−08 4.38−09 2.73−10

L2 4.30−05 2.45−06 1.49−07 9.29−09 5.80−10

L∞ 2.57−04 1.54−05 9.50−07 5.92−08 3.70−09

L1 − 4.12 4.03 4.01 4.00

L2 − 4.13 4.03 4.01 4.00

L∞ − 4.06 4.02 4.00 4.00

Table 4.3 Accuracy of thin plate spline interpolation on S 2 with m = 2 TPS Error measure Convergence order Nodes 128 256 512 1024 2048

L1 3.39−04 7.99−05 1.90−05 4.68−06 1.14−06

L2 4.38−04 1.03−04 2.46−05 6.08−06 1.49−06

L∞ 1.27−03 2.72−04 6.73−05 1.67−05 4.22−06

L1 − 4.17 4.15 4.04 4.07

L2 − 4.17 4.14 4.04 4.05

L∞ − 4.45 4.02 4.02 3.97

SBF interpolant to f over a set Ξ ⊂ S d−1 of distinct data points with mesh-norm h. Then, we have s f − f  L p (S d−1 ) = O(h 2m ) as h → 0, for all p ∈ [1, ∞]. Moreover, the number 2m cannot be improved. We now begin to investigate the impact of a boundary upon the accuracy of surface spline interpolation. Specifically we consider the following question. Question: For S 1 and S 2 consider the interpolation problem set on the semi-circle and hemisphere respectively. How does the presence of the boundary affect the convergence of the interpolation scheme?

94

4 Radial Basis Functions for the Sphere

Table 4.4 Accuracy of linear spline interpolation on the semi-circle with m = 1 linear Error measure Nodes 128 256 512 1024 128/256 256/512 512/1024

L1 L2 2.06−03 2.24−03 5.16−04 8.30−04 1.29−04 2.88−04 3.23−05 1.03−04 Convergence orders 2.00 1.56 2.00 1.53 2.00 1.52

L3 3.42−03 1.35−03 5.32−04 2.11−04

L4 4.39−03 1.83−03 7.68−04 3.23−04

L∞ 9.34−03 4.64−03 2.31−03 1.15−03

1.35 1.34 1.34

1.26 1.25 1.25

1.01 1.01 1.00

Table 4.5 Accuracy of cubic spline interpolation on the semi-circle with m = 2 cubic Error measure Nodes 128 256 512 1024 128/256 256/512 512/1024

L1 L2 3.76−06 3.51−05 4.71−07 6.21−06 5.88−08 1.10−06 7.36−09 1.94−07 Convergence orders 3.00 2.50 3.00 2.50 3.00 2.50

L3 7.78−05 1.54−05 3.06−06 6.08−07

L4 1.17−04 2.45−05 5.14−06 1.08−06

L∞ 3.93−04 9.80−05 2.45−05 6.12−06

2.33 2.33 2.33

2.25 2.25 2.25

2.00 2.00 2.00

Table 4.6 Accuracy for thin plate spline interpolation on the hemisphere with m = 1 ψ3 Error measure Nodes 128 256 512 1024 128/256 256/512 512/1024

L1 L2 1.13−03 3.46−03 4.53−04 1.68−03 1.60−04 6.89−04 5.50−05 2.86−04 Convergence orders 2.64 2.09 3.00 2.57 3.08 2.54

L3 6.26−03 3.17−03 1.39−03 6.14−04

L4 8.90−03 4.58−03 2.07−03 9.46−04

L∞ 3.87−02 2.09−02 1.02−02 5.05−03

1.96 2.39 2.35

1.92 2.29 2.26

1.78 2.07 2.02

We tackle this question in the same way as for the global analysis of the previous question. The results are displayed in Tables 4.4, 4.5 and 4.6. The results clearly show that the optimal orders are not achieved for interpolation on the semi-circle and hemisphere and this corroborates with the conjecture in the

4.2 Numerical Investigation

95

Euclidean setting regarding how the accuracy of the RBF method deteriorates in the presence of a boundary. What is most interesting is that the deterioration can be quantified and we express this as a conjecture. Proposition 4.2 (Conjecture on the hemisphere) Let m be a non-negative integer d−1 of Duchon’s family of such that m > d−1 2 . Let ψ denote the restriction to S D m (Rd )−splines (2.18). Let f ∈ C 2m (S d−1 ) and let s f denote the unique ψ−based SBF interpolant to f over a set Ξ ⊂ H d−1 of distinct data points with mesh-norm h. Then, we have s f − f  L p (S d−1 ) = O(h

m+ 1p

) for all p ∈ [1, ∞].

Chapter 5

Fast Iterative Solvers for PDEs on Spheres

5.1 Introduction In the final two chapters of the book, we will describe the SBF method for constructing numerical solutions to PDEs on the unit sphere, which have many applications in physical geodesy, potential theory, oceanography, and meteorology [FGS98, FM04, Sve83, Sve84]. Evolution equations on spherical geometry such as the shallow water equations have been studied in weather forecasting services [Gö99, WDH+92]. The geometry of the sphere is a major obstacle in constructing the approximation space for the solution of the PDEs. One way to overcome this obstacle is to use the SBFs to construct the approximate solutions via a collocation method [MN02] or a Galerkin method [Le 04]. Whether a collocation or a Galerkin method is used a problem common to both is that we encounter very ill-conditioned linear systems. This is due to the fact that the condition numbers of the collocation or Galerkin matrices depend on the separation radius of the scattered data [LLS99], which can be very small for a large set of scattered data. In this chapter, we propose a remedy to this problem via the construction of a preconditioner based on the additive Schwarz method. To this end, we consider an elliptic PDE on the unit sphere of the form − Δ∗ u + ω2 u = f,

(5.1)

where Δ∗ is the Laplace-Beltrami operator and ω is some non-zero real constant. The equation arises, for example, when one discretizes in time the diffusion equation on the sphere. We propose a way of partitioning a given scattered data set into smaller subsets. This method of partitioning was inspired by scattered data which come from the MAGSAT satellite which measures the magnetic field of the earth. The data sites along the track of the satellite form a sequence of discrete points, see Fig. 5.1, and this sequence covers the globe (except for two small polar caps, in this case of

© The Author(s) 2015 S. Hubbert et al., Spherical Radial Basis Functions, Theory and Applications, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-17939-1_5

97

98

5 Fast Iterative Solvers for PDEs on Spheres

Fig. 5.1 Global scattered MAGSAT satellite data

radius about 0.1 radian) over a period of time. This happens because the earth is rotating around its own axis, while the satellite traverses from near the north pole to near the south pole then back to the north pole in an elliptical path. Using the overlapping subsets of scattered data, we define an additive Schwarz operator for solving (5.1). We then prove a theorem which gives a bound on the condition number of the Schwarz operator. We conclude by illustrating the method using numerical experiments on relatively large scattered point sets taken from MAGSAT satellite data (see [LH98]). It is noted that, for the interpolation problem in Rd using radial basis functions, the idea of dividing the scattered data set into smaller subsets for the purpose of defining the Schwarz alternating algorithm has been proposed in [BLB00]. However, in that paper it is proved only that the Schwarz alternating method is a contraction. Moreover, the method is not used there as a preconditioner, and the problem to which it is applied, namely interpolation with thin plate splines in Rd , is different from that studied here. In addition, work on applying the multiplicative Schwarz alternating algorithm using spherical splines has also been carried out in [Hes02], but in that work the data points are not scattered, and again the Schwarz method is not used as a preconditioner.

5.2 The Weak Formulation of the PDE

99

5.2 The Weak Formulation of the PDE The weak formulation for the Eq. (5.1) is obtained by multiplying both sides by a test function v ∈ W 1 (S d−1 ): 

−Δ∗ u + ω2 u, v

 S d−1

=  f, v S d−1 .

We introduce the bilinear form 

∗

a(u, v) := −Δ u + ω u, v 2

 S d−1

=

Nk,d ∞  

(λk + ω2 ) uk, vk, ,

k=0 =1

where in the last step we used Plancherel’s theorem for L 2 (S d−1 ). A weak formulation of the elliptic PDE is a(u, v) =  f, v ∀v ∈ W 1 (S d−1 ).

(5.2)

The bilinear form is bounded and coercive by the following lemma. Lemma 5.1 For u, v ∈ W 1 (S d−1 ), there are positive constants A0 ≤ 1 and A1 ≥ 1 such that a(u, v) ≤ A1 u W 1 (S d−1 ) v W 1 (S d−1 ) and a(u, u) ≥ A0 u 2W 1 (S d−1 ) . Proof In terms of Fourier series, we have a(u, v) =

Nk,d ∞  

(λk + ω2 ) uk, vk,

k=0 =1

≤ max{1, ω } 2

Nk,d ∞  

(λk + 1) uk, vk,

k=0 =1

⎛ ⎞1/2 ⎛ ⎞1/2 Nk,d Nk,d ∞  ∞    ≤ A1 ⎝ (λk + 1)| uk, |2 ⎠ ⎝ (λk + 1)| vk, |2 ⎠ k=0 =1

= A1 u W 1 (S d−1 ) v W 1 (S d−1 ) ,

k=0 =1

100

5 Fast Iterative Solvers for PDEs on Spheres

where A1 = max{1, ω2 }. For the second inequality, we have a(u, u) =

Nk,d ∞  

(λk + ω2 )| uk, |2

k=0 =1

≥ min{1, ω2 } =

Nk,d ∞  

(λk + 1)| uk, |2

k=0 =1 2 A0 u W 1 (S d−1 ) ,

where A0 = min{1, ω2 }.



We remark that a direct consequence of the lemma is that a(u, u) is equivalent to

u 2W 1 (S d−1 ) , that is A0 u 2W 1 (S d−1 ) ≤ a(u, u) ≤ A1 u 2W 1 (S d−1 ) . There are a number of methods for finding an approximate solution for (5.2). Here, we choose the Ritz-Galerkin method, which involves projecting Eq. (5.2) onto some finite dimensional subspace of W 1 (S d−1 ), which we choose to be a finite dimensional space spanned by a zonal kernel Ψ : S d−1 × S d−1 → R. Note, we will generally consider kernels as being induced by some underlying SBF, i.e., of the form (1.31). N be a set of distinct points on S d−1 and let us define the To this end, let Ξ = {ξ i }i=1 following space VΞ = span{Ψ (ξ i , ·) : ξ i ∈ Ξ }. The Ritz-Galerkin approximation problem is: find uΞ ∈ VΞ such that a(uΞ , v) =  f, v ∀v ∈ VΞ .

(5.3)

The problem will reduce to the problem of solving the following linear system Bc = f,

(5.4)

where the entries of the matrix B are given as Bi, j = a(Ψi , Ψ j ) and the vector f is given as f = [ f j ] Nj=1 in which f j = f, Ψ j S d−1 for i, j = 1, . . . , N . The following lemma, which is a variant of the Funk-Hecke formula (see [Mü66]), facilitates the computation of each entry of the matrix B. of the form (1.31), we have the Lemma 5.2 For any two zonal kernels Ψ and Ψ following relation

S d−1

(z, η)dωd−1 (z) = Ψ (ξ , z)Ψ

1

∞ 

ωd−1

k=0

 k Pk,d (ξ T η). k ψ Nk,d ψ

5.2 The Weak Formulation of the PDE

101

have the following expansions in terms of spherical Proof The two kernels Ψ and Ψ harmonics: Ψ (ξ , z) =

Nk,d ∞  

k Yk, (ξ )Yk, (z), ψ

k=0 =1

(z, η) = Ψ

Nk,d ∞  

 k Yk, (z)Yk, (η). ψ

k=0 =1

Then, by using the orthogonality of spherical harmonics and the addition formula (1.24), we obtain

S d−1

(z, η)dωd−1 (z) = Ψ (ξ , z)Ψ =

Nk,d ∞  

 k Yk, (ξ )Yk, (η) k ψ ψ

k=0 =1 ∞ 

1 ωd−1

 k Pk,d (ξ T η). k ψ Nk,d ψ

=0

 If the supports of Ψi and Ψ j do

 not overlap then Bi, j = 0. This follows from a(Ψi , Ψ j ) = −Δ∗ Ψi + ω2 Ψi , Ψ j , together with the fact that the support of Δ∗ Ψi is the same as that of Ψi . This follows from the second-order differential operator form of the Laplace-Beltrami operator. If the supports of Ψi and Ψ j overlap then, by applying Lemma 5.2 we have

Bi, j = =

S d−1

(−Δ∗ + ω2 )Ψ (ξ i , ξ )Ψ (ξ j , ξ )dωd−1 (ξ )

1

∞ 

ωd−1

k=0

k ]2 Pk,d (ξ iT ξ j ). Nk,d (λk + ω2 )[ψ

(5.5)

5.3 The Additive Schwarz Method Additive Schwarz methods provide fast solutions to equation (5.3) by solving, at the same time, problems of smaller size. Let the space VΞ be decomposed as VΞ = V0 + · · · + V J ,

(5.6)

where V j , j = 0, . . . , J , are subspaces of VΞ , and let P j : VΞ → V j , j = 0, . . . , J , be projections defined by

102

5 Fast Iterative Solvers for PDEs on Spheres

a(P j v, w) = a(v, w) ∀v ∈ VΞ , ∀w ∈ V j .

(5.7)

P := P0 + · · · + P J ,

(5.8)

If we define

then the additive Schwarz method for equation (5.3) consists of solving, by an iterative method, the equation f, (5.9) PuΞ = where the right-hand side is given by f =

J j=0

f j ∈ V j being solutions of f j , with

a( f j , w) =  f, w , for any w ∈ V j .

(5.10)

The well-known equivalence of (5.3) and (5.9) was discussed explicitly in [TS96]. f j we deduce In fact, if uΞ is a solution of (5.3) then from the definition of P j and f j , w) for any w ∈ V j , a(P j uΞ , w) = a(uΞ , w) =  f, w = a( f j . Hence PuΞ = f . On the other hand, if P : VΞ → VΞ is invertible i.e. P j uΞ = and uΞ is the solution of (5.9), then by using successively the symmetry of P and (5.7) and (5.10), we obtain f , v) = a( f , P −1 v) a(uΞ , v) = a(P −1 =

J 

a( f j , P −1 v) =

j=0

=

J  

J 

a( f j , P j P −1 v)

j=0

 f, P j P −1 v =  f, v for any v ∈ VΞ .

j=0

A practical method to solve (5.9) is the conjugate gradient method; the additive Schwarz method (see Sect. 5.6) can be viewed as a preconditioned conjugate gradient method. Bounds for λmin (P) and λmax (P), the minimum and maximum eigenvalues of the additive Schwarz operator P, can be obtained by using the following lemma, see [Wid89]. Lemma 5.3 (i) Assume that there exists a constant c1 > 0 such that, for any u ∈ VΞ satisfying u = Jj=0 u j with u j ∈ V j for j = 0, . . . , J the following inequality a(u, u) ≤ c1

J  j=0

holds. Then

a(u j , u j )

5.3 The Additive Schwarz Method

103

λmax (P) ≤ c1 . (ii) Assume that  there exists a constant c2 > 0 such that any u ∈ VΞ has a decomposition u = Jj=0 u j , u j ∈ V j , satisfying J 

a(u j , u j ) ≤ c2 a(u, u).

j=0

Then

λmin (P) ≥ c2−1 .

In the following sections we will define a subspace decomposition of the form (5.6), and in this way define the additive Schwarz operator for problem (5.3). We will then present the main theoretical result of the chapter, namely an estimate for the condition number of the additive Schwarz operator.

5.4 A Subspace Decomposition Algorithm Let α be a fixed number satisfying 0 < α < π/2, and let Ξ0 := {ρ j : j = 1, . . . , J } be a subset of Ξ such that J  G(ρ j , α). (5.11) S d−1 = j=1

Assume that the support of Ψ (ρ, ·), which is a spherical cap centered at ρ, has radius γ . In particular, in the case where Ψ is induced by one of Wendland’s compactly supported RBFs (see the close of Sect. 2.2) restricted to the sphere, then we choose γ = π/3. For j = 1, . . . , J , the subset Ξ j is defined as Ξ j := {ξ k ∈ Ξ : g(ξ k , ρ j ) ≤ α}.

(5.12)

The sets Ξ j may have different numbers of elements and may overlap each other. Because of (5.11) it is clear that Ξ is decomposed into J overlapping subsets {Ξ j : j = 1, . . . , J } of discrete points such that Ξ=

J  j=1

Ξj.

104

5 Fast Iterative Solvers for PDEs on Spheres

We define V j = VΞ j , j = 0, . . . , J , i.e. V j = span {Ψk = Ψ (ξ k , ·) : ξ k ∈ Ξ j }, so that VΞ = V0 + · · · + V J . The Schwarz operator P is then defined by (5.7) and (5.8). Functions in V j have supports in Γ j , where Γ j := G(ρ j , α + γ ),

j = 1, . . . , J.

We assume that: Assumption 5.1 We can partition the index set {1, . . . , J } into M (for 1 ≤ M ≤ J ) sets Jm , for 1 ≤ m ≤ M such that if i, j ∈ Jm and i = j then Γi ∩ Γ j = ∅. The partitioning problem mentioned in Assumption 5.1 is related to the graph colouring problem [Bol98]. We can define an undirected graph G = (V, E) in which the set of vertices V = {ν1 , . . . , ν J } is identified with the set of caps Γ j , and E is the set of edges, where if Γi ∩ Γ j = ∅ then there is an edge between νi and ν j . A partition satisfying Assumption 5.1 is equivalent to a colouring of the vertices of G so that adjacent vertices have different colours. The minimal number of colours needed is called the chromatic number of G , and is denoted by δ(G ). In general, it is difficult to determine the chromatic number of a graph. However, it is easy to see that δ(G ) ≥ ω(G ), where ω(G ) is the maximal order of a complete subgraph of G , that is, it is the maximal number of vertices all of which are mutually connected. In terms of the caps, every point on the sphere S d−1 lies in at most ω(G ) spherical caps Γ j . An upper bound of δ(G ) is given in [Bol98, Theorem3, Chap. 5]: when G is neither a complete graph nor an odd cycle, then δ(G ) ≤ Δ(G ), with Δ(G ) being the maximal degree of G . In terms of our spherical caps, each cap Γ j intersects at most Δ(G ) other caps. Therefore, for a given set Ξ0 and parameters α, γ , we can compute the lower bound M1 := ω(G ) and the upper bound M2 := Δ(G ) so that 1 ≤ M1 ≤ M ≤ M2 ≤ J.

(5.13)

5.5 An Upper Bound for the Condition Number κ(P) We now state a lemma that will lead to an upper bound of the maximum eigenvalue λmax (P). Lemma 5.4 There exists a positive  constant c independent of the set Ξ such that for any u ∈ VΞ satisfying u = Jj=0 u j with u j ∈ V j for j = 0, . . . , J there holds a(u, u) ≤ cM

J  j=0

a(u j , u j ).

5.5 An Upper Bound for the Condition Number κ(P )

105

Proof Using the inequality |a + b|2 ≤ 2(|a|2 + |b|2 ), we have ⎞

⎛

u 2W 1 (S d−1 )

 2   J   ⎜ 2  ≤ 2 ⎝ u0 W 1 (S d−1 ) +  uj   j=1 

⎟ ⎠.

W 1 (S d−1 )

From the definition of the Sobolev norm (1.66),  2    J   uj    j=1 

W 1 (S d−1 )

 2    J   = π1 (χ1 u j )   j=1 

W 1 (Rd−1 )

 2    J   + π2 (χ2 u j )   j=1 

. (5.14)

W 1 (Rd−1 )

Now, from the fact that u j ∈ V j together with Assumption 5.1 we can partition the index set {1, . . . , J } to M sets of indices Jm so that if i, j ∈ Jm then supp ui ∩ supp u j = ∅. Then, in this proof only, let w j = π1 (χ1 u j ). By using the Cauchy-Schwarz inequality, we have 2    J     w j   j=1 

W 1 (Rd−1 )

 2  M      = w j  m=1 j∈Jm 

W 1 (Rd−1 )

 2  M       ≤M w j   m=1  j∈Jm

.

W 1 (Rd−1 )

(5.15) Since the supports of wi and w j are disjoint for i, j ∈ Jm , i = j, 2        w j   j∈Jm 



=

W 1 (Rd−1 )

w j 2W 1 (Rd−1 ) .

j∈Jm

Thus, 2    J    wj    j=1 

≤M

M  

w j 2W 1 (Rd−1 ) = M

m=1 j∈Jm

W 1 (Rd−1 )

J 

w j 2W 1 (Rd−1 ) .

j=1

Hence, by using similar arguments for π2 (χ2 u j ), we conclude 2      J  uj    j=1 

⎛ ⎞ J J   2 2 ≤ M⎝

π1 (χ1 u j ) W 1 (Rd−1 ) +

π2 (χ2 u j ) W 1 (Rd−1 ) ⎠ j=1

W 1 (S d−1 )

=M

J  j=1

u j 2W 1 (S d−1 ) .

j=1

106

5 Fast Iterative Solvers for PDEs on Spheres

Therefore,

u 2W 1 (S d−1 ) ≤ cM

J 

u j 2W 1 (S d−1 ) .

j=0

Using the fact that a(u, u) ∼ u 2W 1 (S d−1 ) we obtain the result.



From this lemma and Lemma 5.3 it follows that λmax (P) ≤ cM,

(5.16)

where c is a constant independent of M, J and the set Ξ . We now find a lower bound for the minimum eigenvalue of P. In the finite element and boundary element literature, a lower bound is usually obtained by using the interpolation operator and a partition of unity; see e.g. [TW05, TS04]. More precisely, in the FEM and BEM cases a decomposition of a function u ∈ VΞ satisfying the condition of Lemma 5.3 (ii) takes the following form u0 = P0 u, u j = ΠΞ (ϕ j (u − u0 )), j = 1, . . . , J, where P0 is the H 1 -projection onto V0 , ΠΞ is the interpolation operator at the data points, and {ϕ j : j = 1, . . . , J } is a partition of unity satisfying suppϕ j ⊂ Γ j . This approach cannot be used in the present case because with radial basis functions the support of ΠΞ (v) may be the whole sphere even though the support of v is in Γ j . The method of alternating projections [Hal62, vN50] will be used instead. Before introducing this method we need the following definition. Definition 5.1 Let V be a Hilbert space with inner product and norm denoted by ·, · and · , respectively. Assume that U1 and U2 are two closed subspaces of V. The angle α between U1 and U2 is the angle in [0, π/2] whose cosine is given by cos α = sup{v, w : v ∈ U1 ∩ U ⊥ , w ∈ U2 ∩ U ⊥ , v ≤ 1, w ≤ 1}, where U = U1 ∩ U2 , and U ⊥ is its orthogonal complement, namely, U ⊥ := { f ∈ V :  f, v = 0 ∀v ∈ U}. The following result is standard; the proof is included for completeness. Lemma 5.5 If U1 and U2 are two closed subspaces of a Hilbert space V then (U1 + U2 )⊥ = U1⊥ ∩ U2⊥ .

5.5 An Upper Bound for the Condition Number κ(P )

107

Proof It follows from the definition of orthogonal complement that (U1 + U2 )⊥ = { f ∈ V :  f, v = 0 ∀v ∈ U1 + U2 } = { f ∈ V :  f, v1  = 0 =  f, v2  ∀v1 ∈ U1 and v2 ∈ U2 } = U1⊥ ∩ U2⊥ .  The following theorem is crucial in our estimate of the minimum eigenvalue of P. Theorem 5.1 (SSW77, Theorem 2.2) Let V1 , . . . , V J be closed subspaces of a Hilbert space V, and Wi := ∩ Jj=i V j , i = 1, . . . , J . If Qi : V → Vi is the orthogonal projection onto Vi , i = 1, . . . , J , and Q : V → W1 is the orthogonal projection onto W1 , then l f − Q f ≤ cl f − Q f , ∀ f ∈ V, l = 1, 2, . . . ,

Q := Q J · · · Q1 and where Q c2 = 1 −

J −1

sin2 αi ,

i=1

with αi being the angle between Vi and Wi+1 . We shall apply Theorem 5.1 with V being VΞ which is equipped with the inner product a(·, ·) and induced norm · a , and V j being V j⊥ , j = 1, . . . , J . If T is a linear operator on VΞ , we denote by T a the norm of T defined by · a , i.e.,

T a = sup T v a . v∈VΞ

v a ≤1

:= Q J · · · Q1 where Qi is the orthogonal projection from Proposition 5.1 Let Q ⊥ VΞ onto Vi , and let Wi := ∩ Jj=i V j⊥ , i = 1, . . . , J . Then  a ≤ 1−

Q

J −1

1/2 sin2 αi

< 1,

i=1

where αi is the angle between Vi⊥ and Wi+1 . Proof It follows from Lemma 5.5 that W1 = ∩ Jj=1 V j⊥ = (V1 + · · · + V J )⊥ = VΞ⊥ = {0},

108

5 Fast Iterative Solvers for PDEs on Spheres

which implies that the orthogonal projection Q from VΞ onto W1 is identically zero. Theorem 5.1 with  = 1 then yields  a ≤ 1−

Q

J −1

1/2 sin αi

.

2

i=1

It remains to show that αi = 0 for all i = 1, . . . , J − 1. Suppose that αi = 0 for some i ∈ {1, . . . , J − 1}. Then noting that (Vi⊥ ∩ Wi+1 )⊥ = Wi⊥ , we obtain from Definition 5.1 sup{a(v, w) : v ∈ Vi⊥ ∩ Wi⊥ , w ∈ Wi+1 ∩ Wi⊥ , v a ≤ 1, w a ≤ 1} = 1. The spaces being finite dimensional, by compactness there exist v ∈ Vi⊥ ∩ Wi⊥ and w ∈ Wi+1 ∩ Wi⊥ satisfying

v a = w a = 1 and a(v, w) = 1. It follows from the Cauchy-Schwarz inequality that v = w. Thus v ∈ Vi⊥ ∩ Wi+1 = Wi . On the other hand v ∈ Wi⊥ , which implies v = 0. This contradicts a(v, v) = 1, proving the proposition.  Lemma 5.6 For any u ∈ VΞ there exist u j ∈ V j , j = 0, . . . , J , satisfying  u = Jj=0 u j and J 

 a(u j , u j ) ≤ 1 +

j=0

J a )2 (1 − Q

 a(u, u),

is defined in Proposition 5.1. where Q is invertible and satisfies Proof It follows from Proposition 5.1 that I − Q −1 a ≤

(I − Q)

1 , a 1 − Q

where I is the identity operator on VΞ . We define, for any u ∈ VΞ , u0 = P0 u, v = u − u0 , −1 v, u1 = P1 (I − Q) −1 v, j = 2, . . . , J, u j = P j Q j−1 · · · Q1 (I − Q)

5.5 An Upper Bound for the Condition Number κ(P )

109

where Pi := I − Qi is the orthogonal projection from VΞ onto Vi , i = 0, . . . , J . It is easy to check that Jj=1 u j (being a telescoping sum) equals v, and therefore J i=0 ui = u. The crude estimate −1 a v a ≤ (I − Q) −1 a u a ,

u j a ≤ (I − Q)

j = 1, . . . , J,

yields a(u j , u j ) ≤

1 a(u, u), a )2 (1 − Q

j = 1, . . . , J,

resulting in J  j=0

a(u j , u j ) = a(u0 , u0 ) +  ≤ 1+

J 

a(u j , u j )

j=1

J a )2 (1 − Q

 a(u, u), 

proving the lemma.

The above lemma and Lemma 5.3 yield the following estimate for the minimum eigenvalue of P −1  J λmin (P) ≥ 1 + . (5.17) a )2 (1 − Q

This estimate is by no means sharp; in fact the right hand side is not an optimal lower a may be very close bound for λmin (P), as can be seen in Table 5.1, because Q

J J 2 and 1 + (P), 1 + C to 1. In that table we present λ−1 j=1 j min a )2 , where for (1− Q the middle term we explicitly compute C j as the norm of the operator defining u j , namely, )−1 a and C j = P j Q j−1 · · · Q1 (I − Q )−1 a , j = 2, . . . , J. C1 = P1 (I − Q

 It is clear from Table 5.1 that (1 + Jj=1 C 2j )−1 is a better approximant to λmin (P). Our experiments show that the projection P j in the definition of u j plays a key role −1 but we cannot account for this fact. In Table 5.1 in reducing the norm of (I − Q) the norms of the operators were computed by using their matrix representations. a as follows. Recalling the definition of the positive definite E.g., we computed Q

matrix B (see (5.4)) and using the Cholesky factorization B = L T L, we obtain for N ci Ψi ∈ VΞ any u = i=1

u a2 = cT B c = L c 22 ,

110

5 Fast Iterative Solvers for PDEs on Spheres

Table 5.1 Upper bounds for λ−1 min (P ) N

qΞ

cos α

cos β

J

λ−1 min (P ) 1+

1344

π/80

2133

π/100

3458

π/140

4108

π/160

7663

π/200

0.90 0.80 0.70 0.60 0.50 0.90 0.80 0.70 0.60 0.50 0.90 0.80 0.70 0.60 0.50 0.90 0.80 0.70 0.60 0.50 0.80 0.70 0.60 0.50

−0.90 −0.93 −0.86 −0.87 −0.83 −0.85 −0.89 −0.89 −0.85 −0.83 −0.88 −0.80 −0.81 −0.85 −0.83 −0.88 −0.81 −0.87 −0.86 −0.80 −0.89 −0.88 −0.81 −0.84

44 23 17 13 10 44 24 17 13 10 46 22 16 13 10 46 24 17 14 10 23 17 11 10

120.41 285.80 17.44 13.03 3.58 34.97 20.16 6.56 8.18 4.48 4.92 5.31 1.82 1.02 1.70 2.17 6.41 4.57 1.26 4.13 1.37 1.68 5.54 3.27

J

2 j=1 C j

1612.23 461.67 135.84 33.11 12.43 391.90 69.22 24.09 26.44 13.88 68.75 41.58 17.77 13.31 11.67 66.21 36.54 22.21 15.31 16.85 27.47 19.32 17.35 12.78

where c = (c1 · · · c N )T . On the other hand, by writing i = QΨ

N 

di,k Ψk

k=1

one can easily see that a2 = cT Q = cT Q = L Qc

2 , T B Qc T L T L Qc

Qu

2

1 + J/ a )2 (1 − Q 98914208.40 4003258.71 625605.77 7481.56 219.18 1684693.43 57266.95 712.68 4745.12 266.90 3662.59 2649.72 101.96 47.03 58.05 4023.85 1043.01 755.70 86.07 1388.24 154.84 130.77 609.86 170.82

5.5 An Upper Bound for the Condition Number κ(P )

111

with column ith being (di,1 · · · di,N )T . is the matrix representation of Q where Q Therefore, a = sup

Q

u∈VΞ

a 2 L −1 c 2

Qu

L Qc

L Q L −1 2 . = sup = sup = L Q

u a

c 2 c∈R N L c 2 c∈R N

Estimates (5.16) and (5.17) yield an upper bound for the condition number of the Schwarz operator. Theorem 5.2 The condition number of the additive Schwarz operator P is bounded by   J , κ(P) ≤ cM 1 + a )2 (1 − Q

is defined as in where c is a constant independent of M, J and the set Ξ , and Q Proposition 5.1.

5.6 An Overlapping Additive Schwarz Algorithm As has been pointed out in [BLB00], essential ingredients for a domain decomposition algorithm are: (i) A method for subdividing the physical space. (ii) An efficient and scale independent method for solving small sub-problems. The solutions to the small problems will be used to precondition the large problem. (iii) A fast method for computing the action of the large matrices that occur at various scales. (iv) An outer iteration. Suppose we number the scattered data following the satellite track as {1, . . . , N }. Let α and β be parameters satisfying 0 < α < π/3 and α ≤ β ≤ π . The algorithm to partition Ξ can be described as follows. (1) The first center is ρ 1 = ξ 1 . (2) Assume that the centers ρ 1 , . . . , ρ  have been chosen. (3) If Ξ = ∪k=1 Ξk (where Ξk is defined by (5.12)) then stop the algorithm and let J = . Otherwise, choose the new center ρ +1 as a point in Ξ satisfying g(ρ +1 , ρ  ) ≥ β and g(ρ +1 , ρ k ) ≥ α, k = 1, . . . ,  − 1. (4) Let  =  + 1 and repeat step (3). Now Ξ0 is defined by Ξ0 := {ρ 1 , . . . , ρ J }. The parameter β is included so that the condition Ξ = ∪k=1 Ξk in step (3) is quickly satisfied. In our experiments, for

112

5 Fast Iterative Solvers for PDEs on Spheres

a given value of α we chose an appropriate value of β by starting with the value β = π , and decreased its value until the above-mentioned condition holds. For j = 0, . . . , J , let I j be a subset of the index set of {1, . . . , N } such that m ∈ I j ↔ xm ∈ Ξ j . The cardinality of the set I j is denoted by s j and the m-th element of the set I j is denoted by I j (m). For a given vector v = (v1 , . . . , v N )T , the restriction of v on Ξ j is the vector u = (u1 , u2 , . . . , us j )T defined as follows um := v I j (m) ,

m = 1, . . . , s j ,

and we write u = R j (v), thus the restriction operation to Ξ j is denoted by R j . Conversely, for a vector u = (u1 , . . . , us j )T , we extend u to v = (v1 , . . . , v N )T as  um vk := 0

if k = I j (m) for 1 ≤ m ≤ s j , otherwise.

and write v = E j (u), where E j denotes the extension operation. A pseudo code. INPUT Input the scattered set Ξ on the sphere, the right-hand side f , and the desired accuracy . SETUP (1) (2) (3) (4) (5)

Partition the scattered set Ξ into J overlapping subsets Ξ j for j = 1, . . . , J and construct the global coarse set Ξ0 . Set the initial residual vector r = f (see (5.4)). Set the initial pseudo-residual vector p = 0. Set the initial solution vector s = 0. Set the iteration counter iter = 0.

ITERATIVE SOLUTION (1) (2) (3) (4) (5) (6) (7) (8) (9)

while r >  f

Set p = 0. for j = 1 to J Construct the local matrix C with entries Cm,n = B I j (m),I j (n) . Set the restriction residual vector z = R j (r). Solve the linear system Cy = z. Update the pseudo-residual vector p = p + E j (y). end for Construct the coarse global matrix D with entries Dm,n = B I0 (m),I0 (n) .

5.6 An Overlapping Additive Schwarz Algorithm

(10) (11) (12) (13) (14) (15) (16) (17)

Set zg = R0 (r). Solve the linear system Dyg = zg . Update the pseudo-residual vector p = p + E 0 (yg ). If iter > 0 then set ζ0 = ζ1 . Set ζ1 = p · r. Update the counter, iter = iter +1. If iter = 1 then define p = p else p = p + (ζ1 /ζ0 )p. Update the residual vector  r=r−

(18)

r·p p · Bp

 Bp.

Update the solution vector  s=s+

(19)

113

 r·p p. p · Bp

end while

5.7 Numerical Results In this section, we present numerical experiments on S 2 based on globally scattered position data extracted from a very large data set collected by NASA’s satellite MAGSAT. The scattered data sets Ξ are extracted so that the separation radius qΞ satisfies qΞ ≥ q, where 0 < q < π/3 is a given number. An example of the set Ξ0 for the decomposition algorithm, as mentioned in Sect. 5.4, is illustrated in Fig. 5.2. The code is written in FORTRAN 90 and was run on computers equipped with dual Opteron 2.0GHz CPU and 4GB RAM (Fig. 5.2). We wish to solve (−Δ∗ + I )u(ξ ) = f (ξ ), ξ = (ξ1 , ξ2 , ξ3 ) ∈ S 2 , with I being the identity operator. The function f is defined to be  f (ξ ) = f (ξ3 ) = (−Δ∗ + I )φ3,2 ( 2 − 2ξ3 ), ξ3 ∈ [−1, 1], with φ3,2 (r ) = (1 − r )6+ (35r 2 + 18r + 3) so that the exact solution is  u = φ3,2 ( 2 − 2ξ3 ). The RBF Φ3,2 (ξ ) = φ3,2 ( ξ ) is one of Wendland’s compactly supported functions (see the close of Sect. 2.2).

114

5 Fast Iterative Solvers for PDEs on Spheres

Fig. 5.2 The set Ξ0 with N = 10,443, cos α = 0.98, and cos β = −0.70 1

0.5

0

-0.5

-1 0.5 0.5

0 -0.5

0 -0.5

In the first set of numerical experiments, the spherical basis functions are also derived from Wendland’s radial basis function Φ3,3 (ξ ) = φ3,3 ( ξ ), which is in C 6 (R3 ),  Ψ (ξ , η) = φ3,3 ( 2 − 2ξ · η), φ3,3 (r ) = (1 − r )8+ (32r 3 + 25r 2 + 8r + 1). Recalling (5.5), and noting that with d = 3 there hold Nk,3 = 2k + 1 and λk = k(k + 1), each entry of the matrix B is given by

Bi, j = =

S2

(−Δ∗ + ω2 )Ψ (ξ i , ξ )Ψ (ξ j , ξ )dω2 (ξ )

∞ 1  k ]2 Pk,3 (ξ i · ξ j ). (2k + 1)(λk + ω2 )[ψ 4π k=0

In our experiments, Bi, j is approximated by the partial sum of the first 3000 terms. The k were computed by approximating Legendre coefficients of the univariate function ψ (1.33) by an appropriate Gaussian quadrature over the interval [−1, 1]. We then test the overlapping method as the preconditioner for the conjugate gradient method with different values of cos α, and hence different values of J and M. The stopping criterion for the iteration is

Bc(m) − f 2 ≤ 10−6 ,

f 2

5.7 Numerical Results

115

where B and f are the stiffness matrix and the right-hand side vector as in (5.4). In the following tables, κ(P) is the condition number, the CPU time is measured in seconds, and iter is the number of iterations. The results for the first experiment are shown in Table 5.2. Here M1 and M2 are the lower and upper bounds on M from (5.13). In the second set of experiments, the spherical basis function is derived from a less smooth radial basis function Φ3,2 ∈ C 4 (R3 ),  Ψ (ξ , η) = φ3,2 ( 2 − 2ξ · η), φ3,2 (r ) = (1 − r )6+ (35r 2 + 18r + 3). The results are shown in Table 5.3. The numbers in both Tables 5.2 and 5.3 suggest that when cos α decreases (meaning that α increases), M1 , M2 and κ(P) decrease, but the CPU time decreases then increases. We note that a larger value of α results in a larger size of the overlap and a smaller value of J (the number of subproblems to be solved), which in turn implies larger sizes of the subproblems. As in the case of finite element methods, this results in a smaller condition number κ(P) because the preconditioner is closer to the inverse of the stiffness matrix. However, for an optimal value of α in terms of CPU time, one has to balance between the number of subproblems and their sizes. Our experiments show that any value of α so that cos α ≤ 0.60 is not recommended. Optimal CPU times seem to occur when cos α = 0.98. The numbers in Tables 5.2 and 5.3 also suggest that the smoothness of the kernel does not affect the algorithm. Table 5.4 shows the CPU time and condition number of the matrix B using the conjugate gradient method without preconditioners. In all cases, both the condition number and the CPU time are much worse than those in Tables 5.2 and 5.3. We also compute the discrete maximum errors and 2 errors over a set G = {ξ g } of 19,075 points over S 2 by err∞ = max |uΞ (ξ g ) − u(ξ g )| ξ g ∈G

and

⎛ err2 =

1 ⎝ |G|



⎞1/2 |u(ξ g ) − uΞ (ξ g )|2 ⎠

,

ξ g ∈G

where |G| = 19075 is the cardinality of G. Tables 5.5 and 5.6 summarize the computed results.

qΞ

π 240

π 280

π 320

N

10443

13775

17078

0.99 0.98 0.97 0.95 0.85 0.60 0.99 0.98 0.97 0.95 0.85 0.60 0.99 0.98 0.97 0.95 0.85 0.60

cos α

−0.52 −0.70 −0.64 −0.63 −0.93 −0.86 −0.40 −0.51 −0.57 −0.62 −0.89 −0.77 −0.44 −0.71 −0.52 −0.81 −0.92 −0.79

cos β 140 86 56 43 19 13 141 85 56 44 21 11 148 85 59 44 20 11

M1 367 205 131 91 30 14 374 206 132 93 33 12 387 207 137 93 31 12

M2 420 223 138 94 31 15 429 226 140 96 34 13 442 227 146 96 32 13

J

Table 5.2 Condition numbers κ(P ) of the Schwarz operator using φ3,3 (r ) as RBF λmin 0.009 0.053 0.034 0.122 0.590 1.076 0.031 0.092 0.069 0.210 0.636 0.909 0.037 0.074 0.070 0.161 0.477 0.623

λmax 44.37 29.48 21.78 17.62 10.19 9.02 44.39 29.20 21.57 18.23 10.73 8.38 46.50 30.58 23.29 17.72 10.47 8.42

κ(P ) 4693.57 559.95 645.84 145.04 17.26 8.38 1420.38 316.61 310.99 86.94 16.86 9.22 1272.70 410.70 333.57 109.74 21.97 13.51

CPU 116.2 68.4 78.2 70.0 141.2 865.0 153.9 103.4 108.2 124.9 350.7 1761.7 222.6 185.3 196.2 233.6 702.0 3924.0

Iter. 171 93 90 51 21 16 132 76 66 45 22 17 128 88 72 48 24 20

116 5 Fast Iterative Solvers for PDEs on Spheres

qΞ

π 240

π 280

π 320

N

10443

13775

17078

0.99 0.98 0.97 0.95 0.85 0.60 0.99 0.98 0.97 0.95 0.85 0.60 0.99 0.98 0.97 0.95 0.85 0.60

cos α

−0.52 −0.70 −0.64 −0.63 −0.93 −0.86 −0.40 −0.51 −0.57 −0.62 −0.89 −0.77 −0.44 −0.71 −0.52 −0.81 −0.92 −0.79

cos β 140 86 56 43 19 13 141 85 56 44 21 11 148 85 59 44 20 11

M1 367 205 131 91 30 14 374 206 132 93 33 12 387 207 137 93 31 12

M2 420 223 138 94 31 15 429 226 140 96 34 13 442 227 146 96 32 13

J

Table 5.3 Condition numbers κ(P ) of the Schwarz operator using φ3,2 (r ) as RBF λmin 0.023 0.107 0.059 0.186 0.862 1.121 0.031 0.090 0.199 0.290 0.815 0.961 0.044 0.188 0.134 0.335 0.709 0.805

λmax 49.02 31.83 23.02 18.57 10.50 9.30 49.04 31.35 22.93 19.24 11.03 8.49 51.27 32.72 24.58 18.66 10.81 8.51

κ(P ) 2171.42 296.84 391.40 99.91 12.19 8.29 1592.04 348.66 115.35 66.25 13.55 8.83 1153.38 173.62 182.98 55.69 15.24 10.58

CPU 100.3 55.9 64.9 62.5 127.6 836.4 150.7 96.7 83.6 112.3 322.5 1732.7 167.6 121.0 138.3 171.1 614.3 3549.5

Iter. 137 71 70 44 19 16 112 63 46 38 20 17 101 60 53 36 21 18

5.7 Numerical Results 117

118

5 Fast Iterative Solvers for PDEs on Spheres

Table 5.4 Condition numbers of the conjugate gradient method without preconditioners N qΞ λmin λmax κ(B) CPU Iter. 10443 13775 17078

π 240 π 280 π 320

0.2585E-03 0.6767E+04 0.2618E+08 2204.98 0.3362E-04 0.8082E+03 0.2404E+08 2802.73 0.4090E-04 0.1022E+04 0.2499E+08 4545.70

Table 5.5 Errors using φ3,3 (r ) as RBF N cos α cos β 10443

13775

17078

0.99 0.98 0.97 0.95 0.85 0.75 0.99 0.98 0.97 0.95 0.85 0.75 0.99 0.98 0.97 0.95 0.85 0.75

−0.52 −0.70 −0.64 −0.63 −0.93 −0.90 −0.40 −0.51 −0.57 −0.62 −0.89 −0.91 −0.44 −0.71 −0.52 −0.81 −0.92 −0.91

3512 2845 2981

M1

M2

J

err∞

err2

140 86 56 43 19 14 141 85 56 44 21 15 148 85 59 44 20 15

367 205 131 91 30 18 374 206 132 93 33 18 387 207 137 93 31 18

420 223 138 94 31 19 429 226 140 96 34 19 442 227 146 96 32 19

8.4101e-04 8.4257e-04 8.4283e-04 8.4294e-04 8.4320e-04 8.4336e-04 7.9116e-04 7.9160e-04 7.9139e-04 7.9172e-04 7.9103e-04 7.9144e-04 7.0749e-04 7.0767e-04 7.0722e-04 7.0739e-04 7.0728e-04 7.0796e-04

4.3393e-07 4.3417e-07 4.3421e-07 4.3421e-07 4.3420e-07 4.3420e-07 3.9836e-07 3.9836e-07 3.9833e-07 3.9838e-07 3.9834e-07 3.9836e-07 3.5143e-07 3.5141e-07 3.5151e-07 3.5149e-07 3.5144e-07 3.5147e-07

5.7 Numerical Results

119

Table 5.6 Errors using φ3,2 (r ) as RBF N cos α cos β M1 10443

13775

17078

0.99 0.98 0.97 0.95 0.85 0.75 0.99 0.98 0.97 0.95 0.85 0.75 0.99 0.98 0.97 0.95 0.85 0.75

−0.52 −0.70 −0.64 −0.63 −0.93 −0.90 −0.40 −0.51 −0.57 −0.62 −0.89 −0.91 −0.44 −0.71 −0.52 −0.81 −0.92 −0.91

140 86 56 43 19 14 141 85 56 44 21 15 148 85 59 44 20 15

M2

J

err∞

err2

367 205 131 91 30 18 374 206 132 93 33 18 387 207 137 93 31 18

420 223 138 94 31 19 429 226 140 96 34 19 442 227 146 96 32 19

1.0816e-03 1.0802e-03 1.0798e-03 1.0796e-03 1.0795e-03 1.0794e-03 1.0601e-03 1.0606e-03 1.0603e-03 1.0606e-03 1.0605e-03 1.0610e-03 9.6261e-04 9.6174e-04 9.6151e-04 9.6209e-04 9.6185e-04 9.6209e-04

4.5627e-07 4.5629e-07 4.5628e-07 4.5624e-07 4.5621e-07 4.5623e-07 4.3020e-07 4.3021e-07 4.3013e-07 4.3020e-07 4.3016e-07 4.3019e-07 3.7418e-07 3.7406e-07 3.7400e-07 3.7400e-07 3.7400e-07 3.7406e-07

Chapter 6

Parabolic PDEs on Spheres

6.1 Introduction In this chapter we consider the following parabolic partial differential equation defined on the unit sphere S d−1 ⊂ Rd : 

∂ ∗ ∂t u(ξ, t) − Δ u(ξ, t)

= F(ξ, t) u(ξ, 0) = f (ξ),

f ∈ W 2s (S d−1 ),

(6.1)

where Δ∗ is the Laplace-Beltrami operator on S d−1 and W 2s (S d−1 ) is the Sobolev space defined on S d−1 (see Chap. 1). It is known that Eq. (6.1) describes the heat diffusion process on the surface of the sphere with external heat source F(ξ, t). In many applications in geophysics and global weather forecasting, it is common that the function f is not known analytically everywhere but only at a finite set of scattered points. We propose a collocation method in which SBFs are used to construct the approximate solution. The approximate solution of the PDE will be of the form u Ξ (ξ, t) =

N 

ci (t)Ψ (ξi , ξ),

(6.2)

i=1

subject to the initial condition u Ξ (ξ, 0) = s f (ξ), where Ψ is a zonal kernel induced by a suitable SBF, (see (1.31)) and s f is the SBF interpolant of the function f . In cases where the zonal kernel Ψ satisfies certain regularity conditions, we are able to obtain error estimates in certain Sobolev norms.

© The Author(s) 2015 S. Hubbert et al., Spherical Radial Basis Functions, Theory and Applications, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-17939-1_6

121

122

6 Parabolic PDEs on Spheres

First, we will consider the semi-discrete problem, in which the exact solution u(ξ, t) is approximated by u Ξ of the form (6.2). By using a collocation method, in which we force the approximate solution u Ξ and the exact solution u to agree on a set of collocation points, the problem is reduced to solving a system of ordinary differential equations. Second, we discretize (6.1) also in time variable to produce a fully discrete scheme for the approximate solution using the backward Euler method and the Crank-Nicolson method. A detailed convergence analysis will be given, using the mathematical tools developed in previous chapters. Finally, the chapter will be concluded with some numerical experiments. The numerical analysis here follows a framework set out in [Tho97], which was used to analyze the approximation of solutions of the heat equation on a bounded domain Ω ⊂ Rd for the finite element method. However, the framework of [Tho97] is modified significantly with the structure of the reproducing kernel Hilbert space Hψ for a collocation method on S d−1 .

6.2 The Homogeneous Semi-discrete Problem By the method of separation of variables, see [Rau91], Sect. 5.7, the exact solution for the homogeneous problem: 

∂ ∂t u(ξ, t)

u(ξ, 0)

= Δ∗ u(ξ, t) = f (ξ), f ∈ L 2 (S d−1 ),

is given as the infinite series u(ξ, t) =

∞ 

e

−λk t

Nk,d 

 f k, Yk, (ξ).

=1

k=0

Let the approximate solution be of the following form: u Ξ (ξ, t) =

N 

ci (t)Ψ (ξi , ξ),

i=1

where Ψ is a spherical basis function. The homogeneous semi-discrete problem is formulated as the following: we require the Eq. (6.1) to be exact on the set Ξ , i.e. 

∂ ∂t u Ξ (ξ j , t) u Ξ (ξ, 0)

= Δ∗ u Ξ (ξ j , t) ∀ξ j ∈ Ξ, = s f (ξ),

(6.3)

6.2 The Homogeneous Semi-discrete Problem

123

where s f is the interpolant of f in VΞ . Equation (6.3) can be rewritten as the following: N N  d  ci (t)Ψ (ξi , ξ j ) = ci (t)Δ∗ Ψ (ξi , ξ j ), ∀ξ j ∈ Ξ, (6.4) dt i=1

i=1

subject to the following initial condition: N 

ci (0)Ψ (ξi , ξ j ) = f (ξ j ), ∀ξ j ∈ Ξ.

i=1

If we set A := [Ψ (ξi , ξ j )]i, j=1,...,N and B := [Δ∗ Ψ (ξi , ξ j )]i, j=1,...,N then Eq. (6.4) can be written as the following system of ordinary differential equations in time: d c(t) = A−1 Bc(t), (6.5) dt where c(t) = [c1 (t), . . . , c N (t)]T . It is known that (see, for example, [NSS00]), in order to solve the system (6.5), we have to compute the distinct eigenvalues r1 , . . . , rk of the matrix A−1 B with multiplicities n 1 , . . . , n k . For each eigenvalue ri , we find n i linearly independent generalized eigenvectors. Each independent solution of (6.5) is of the form   t 2 −1 −1 rt −1 2 v + t (A B − r I )v + (A B − r I ) v + . . . , exp(A Bt)v = e 2 where r is an eigenvalue and v is a corresponding generalized eigenvector. If r has multiplicity n i , then the above series reduces to the first n i terms. The linearly independent solutions form column vectors of a matrix E(t), and then the fundamental matrix exp(A−1 Bt) is given as exp(A−1 Bt) = E(t)E −1 (0). The solution of the homogeneous semi-discrete problem is u Ξ (ξ, t) = [Ψ (ξ 1 , ξ) . . . Ψ (ξ N , ξ)] exp(A−1 Bt)c(0) where c(0) = A−1 f |Ξ . We shall express the solution u Ξ (ξ, t) in terms of some evolution operator. Lemma 6.1 For small t > 0 we have u Ξ (ξ j , t) = exp(B A−1 t)[ f (ξ j )] Nj=1

(6.6)

124

6 Parabolic PDEs on Spheres

Here and thereafter, the notation [a j ] Nj=1 stands for [a1 , . . . , a N ]T which is a vector in R N . Proof Using Eq. (6.6), we have [u Ξ (ξ j , t)] Nj=1 = A exp(A−1 Bt)A−1 [ f (ξ j )] Nj=1   tn = A I + t A−1 B + . . . (A−1 B)n + . . . A−1 [ f (ξ j )] Nj=1 n! tn = [ f (ξ j )] Nj=1 + t B A−1 [ f (ξ j )] Nj=1 + . . . + (B A−1 )n [ f (ξ j )] Nj=1 + . . . n! = exp(B A−1 t)[ f (ξ j )] Nj=1 .

 Since u Ξ ∈ VΞ , this implies u Ξ (ξ, t) = E Ξ (t) f (ξ) for some evolution operator E Ξ (t). We can show that E Ξ (t) is a stable operator in VΞ in  · ψ norm by the following lemma: Lemma 6.2 For every ϕ ∈ VΞ , E Ξ (t)ϕψ ≤ ϕψ . Proof Let θ(ξ, t) be defined as θ(ξ, t) =

N 

ci (t)Ψ (ξi , ξ).

i=1

We wish to solve the following PDE by a collocation method ∂ θ(ξ, t) = Δ∗ θ(ξ, t), ∂t subject to the initial condition θ(ξ, 0) = ϕ(ξ). In our collocation method, it is required that the PDE is exact on the set of given points Ξ , i.e. ∂ θ(ξ j , t) = Δ∗ θ(ξ j , t), ∀ξ j ∈ Ξ, ∂t subject to the initial condition

6.2 The Homogeneous Semi-discrete Problem

125

θ(ξ j , 0) = ϕ(ξ j ), ∀ξ j ∈ Ξ. Since Ψ is the reproducing kernel in the Hilbert space Hψ , 

∂ θ(·, t), Ψ (·, ξ j ) ∂t

 ψ



= Δ∗ θ(·, t), Ψ (·, ξ j ) ψ , ∀ξ j ∈ Ξ.

(6.7)

Since VΞ is spanned by Ψ (ξ, ξ j ), for j = 1, . . . , N , Eq. (6.7) implies that for every function v ∈ VΞ ,  

∂θ , v = Δ∗ θ, v ψ . ∂t ψ Since θ ∈ VΞ , we can take v = θ to obtain  

1 ∂ ∂θ 2 θψ = , θ = Δ∗ θ, θ ψ . 2 ∂t ∂t ψ From the definition of ·, · ψ , Nk,d ∞  

∗ −λk | θk, |2 ≤ 0, ∀θ ∈ VΞ . Δ θ, θ ψ = k ψ k=0 =1

(6.8)

Thus, we obtain the result θ(·, t)ψ ≤ θ(·, 0)ψ or in other words E Ξ (t)ϕψ ≤ ϕψ . 

6.3 The Inhomogeneous Semi-discrete Problem The approximation of the inhomogeneous equation will be tackled via an elliptic projection from the space of the exact solution u to the finite dimensional space VΞ , which is somehow similar to the Ritz projection in the finite element method. To begin, let us define the following operator: P : W 2s+2 (S d−1 ) → VΞ u → u P , where



∗ = Δ∗ u(ξ j ) ∀ξ j ∈ Ξ, Δ u P (ξ j ) u dω = udω . d−1 d−1 P d−1 d−1 S S

(6.9)

126

6 Parabolic PDEs on Spheres

It can be checked that u P is well-defined. Lemma 6.3 Let u ∈ W 2s+2 (S d−1 ), and u P ∈ VΞ be constructed from a linear k ∼ combination of shifts of a zonal kernel Ψ induced by an SBF ψ (1.31) with ψ −s (1 + λk ) . Then there is a constant C independent of h Ξ so that u P − uψ ≤ Ch sΞ uW 2s+2 (S d−1 ) . Proof Since Δ∗ u P is the interpolation of Δ∗ u, by (3.60) in the proof of Theorem 3.5 we have Δ∗ u P − Δ∗ uψ ≤ Ch sΞ Δ∗ uW 2s (S d−1 ) ≤ Ch sΞ uW 2s+2 (S d−1 ) .

(6.10)

Let ϕ = u P − u, then from the definition (6.9) (u P − u)dωd−1 = 0. ϕ 0 = S d−1

Hence, ϕ2ψ =

Nk,d ∞  

⎛ k ≤ ⎝| | ϕk, |2 /ψ ϕ0 | 2 +

k=0 =1

Nk,d ∞  

⎞ k ⎠ λ2k | ϕk, |2 /ψ

k=1 =1

≤ Δ

∗

ϕ2ψ .

Combining with (6.10), we have u P − uψ ≤ Δ∗ u P − Δ∗ uψ ≤ Ch sΞ uW 2s+2 (S d−1 ) .  The collocation semi-discrete equation (6.3) now takes the following form ∂ u Ξ (ξ j , t) − Δ∗ u Ξ (ξ j , t) = F(ξ j , t), ∀ξ j ∈ Ξ, ∂t

(6.11)

subject to the initial condition u Ξ (ξ, 0) = s f (ξ).

Theorem 6.1 Let f, u t ∈ W 2s+2 (S d−1 ) and u, u Ξ be the solution for (6.1) and (6.11) respectively. The approximate solution u Ξ is constructed as a linear combination of shifts of a spherical basis function Ψ (x, y) = ψ(x · y) which satisfies k ∼ (1 + λk )−s . Then there is a positive constant C, independent of h Ξ , so that ψ the following error estimate holds:

6.3 The Inhomogeneous Semi-discrete Problem

u(·, T ) − u Ξ (·, T )ψ ≤

Ch sΞ

127

  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 )



T

+ 0

 u t (·, σ)W 2s+2 (S d−1 ) dσ .

Proof Let θ = θ(ξ, t) := u Ξ − u P , and let γ := u P − u. Note that θ ∈ VΞ . When being restricted on the set Ξ , using the relationship Δ∗ u P |Ξ = Δ∗ u|Ξ we have the following equations: 

        ∂ ∂ ∂ ∗ ∗ ∗   θ−Δ θ  = uΞ − Δ uΞ  − u P − Δ u P  ∂t ∂t ∂t Ξ Ξ Ξ    ∂u P − Δ∗ u  = F|Ξ − ∂t    Ξ   ∂u P  ∂u ∂u ∗  −Δ u  + − = F|Ξ − ∂t ∂t ∂t Ξ Ξ   ∂ (u − u P ) , = ∂t Ξ

or in terms of a PDE in the finite dimensional space VΞ , ∂θ (ξ, t) − Δ∗ θ(ξ, t) = −Γ (ξ, t), ∂t

(6.12)

∂γ . By Duhamel’s principle, see [Rau91], where Γ (ξ, t) is the SBF interpolant of ∂t Sect. 3.11, we have θ(ξ, T ) = E X (T )θ(ξ, 0) −

T

E X (T − σ)Γ (ξ, σ)dσ

0

Since E X (T )vψ ≤ vψ for all v ∈ VΞ (by Lemma 6.2), we have θ(·, T )ψ ≤ θ(·, 0)ψ + 0

T

Γ (·, σ)ψ dσ.

Here, θ(·, 0)ψ = s f − P f ψ ≤ s f − f ψ + P f − f ψ   ≤ Ch sΞ  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) .

128

6 Parabolic PDEs on Spheres

We can use Lemma 6.3 to obtain   ∂  s  γt (·, t)ψ =  (u − u P )  ≤ Ch Ξ u t (·, t)W 2s+2 (S d−1 ) . ∂t ψ Using Lemma 2.1, we obtain Γ (·, t)ψ ≤ γt ψ ≤ Ch sΞ u t W 2s+2 (S d−1 ) . We know from Lemma 6.3 that γ(·, T )ψ = u(T ) − u P (T )ψ ≤ Ch sΞ u(T )W 2s+2 (S d−1 )   T   s  f + ≤ Ch Ξ  u (·, σ)dσ t   0

 s ≤ Ch Ξ  f W 2s+2 (S d−1 ) +

T 0

 W 2s+2 (S d−1 ))



u t (·, σ)W 2s+2 (S d−1 ) dσ .

Therefore, after adjusting the constant C, we obtain u(·, T ) − u Ξ (·, T )ψ ≤ θ(·, T )ψ + γ(·, T )ψ  ≤ Ch sΞ  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) +

T 0

 u t (·, σ)W 2s+2 (S d−1 ) dσ .

 In the following two sections, we will consider two time discretization schemes for the inhomogeneous heat equation defined in (6.1), namely the backward Euler method and the Crank-Nicolson method. We will provide an error analysis for the fully discrete solutions.

6.4 Time Discretization Using the Backward Euler Method Let us discretize the time derivative using the backward Euler method as u(ξ, t) − u(ξ, t − τ ) + o(1) − Δ∗ u(ξ, t) = F(ξ, t). τ The collocation equation for u Ξ is u Ξ (ξ j , t) − u Ξ (ξ j , t − τ ) − τ Δ∗ u Ξ (ξ j , t) = τ F(ξ j , t), ∀ξ j ∈ Ξ.

(6.13)

6.4 Time Discretization Using the Backward Euler Method

129

Let us define tm := mτ , Um (ξ) := u Ξ (ξ, tm ) and introduce the notation ∂ t Um :=

Um − Um−1 . τ

The collocation equation (6.13) can be rewritten as ∂ t Um (ξ j ) − Δ∗ Um (ξ j ) = F(ξ j , tm ), ∀ξ j ∈ Ξ,

(6.14)

subject to the initial condition U0 = s f . If we write Um = Sect. 6.3, we have

N

i=1 cm,i Ψ (ξ i , ξ)

then in terms of matrices A and B, defined in

(A − τ B)cm = Acm−1 + τ [F(ξ j , mτ )] Nj=1 ,

(6.15)

with the initial condition Ac0 = [ f (ξ j )] Nj=1 . We now estimate the difference between Um and the exact solution u at the time tm . Theorem 6.2 Let us assume that u tt ∈ W s (S d−1 ) and u t , f ∈ W 2s+2 (S d−1 ) and let Um be the solution of (6.14). The approximate solution Um is constructed from k ∼ (1 + λk )−s . Then there are positive shifts of a spherical basis function Ψ with ψ constants C1 and C2 so that we have the following error estimate: Um − u(·, tm )ψ ≤ C1 h sΞ Γ ( f, u t ) + C2 τ

tm

u tt (·, σ)s dσ.

0

where Γ ( f, u t ) :=  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) + tm m  u t (·, σ)W 2s+2 (S d−1 ) dσ + τ u t (·, t j )W 2s (S d−1 ) . 0

j=1

Proof With a slight abuse of notation, we write u(tm ) for u(·, tm ) where the implication is understood. We have Um − u(tm ) = Um − Pu(tm ) + Pu(tm ) − u(tm ) =: θm + γm .

130

6 Parabolic PDEs on Spheres

We already know Pu(·, tm ) − u(·, tm )ψ = γm ψ ≤ Ch sΞ u(·, tm )W 2s+2 (S d−1 )   tm ≤ Ch sΞ  f W 2s+2 (S d−1 ) + u t (·, σ)W 2s+2 (S d−1 ) dσ . 0

Similar to (6.12), we have ∂ t θm (ξ j ) − Δ∗ θm (ξ j ) = −ωm (ξ j ), ∀ξ j ∈ Ξ,

(6.16)

where ωm = ∂ t Pu(tm ) − IΞ u t (tm ), where IΞ u t (tm ) is the SBF interpolant for u t (tm ). We can rewrite Eq. (6.16) as (1 − τ Δ∗ )θm (ξ j ) = θm−1 (ξ j ) − τ ωm (ξ j ), ∀ξ j ∈ Ξ.

(6.17)

In terms of the inner product ·, · ψ in the reproducing kernel Hilbert space Hψ ,



θm − τ Δ∗ θm , Ψ (ξ j , ·) ψ = θm−1 − τ ωm , Ψ (ξ j , ·) ψ , ∀ξ j ∈ Ξ.

(6.18)

Since VΞ is spanned by Ψ (ξ j , ·)’s, j = 1, . . . , N , this means for every v ∈ VΞ ,

θ m − τ Δ∗ θ m , v

ψ

= θm−1 − τ ωm , v ψ .

By taking v = θm , we have

θm − τ Δ∗ θm , θm ψ = θm−1 − τ ωm , θm ψ

θm 2ψ − τ Δ∗ θm , θm ψ = θm−1 , θm ψ − τ ωm , θm ψ .

Since Δ∗ θm , θm ψ ≤ 0 (cf. inequality (6.8)), we infer that θm 2ψ ≤ θm−1 , θm ψ + τ | ωm , θm ψ | ≤ θm−1 ψ θm ψ + τ ωm ψ θm ψ . Simplifying θm ψ on both sides, we obtain θm ψ ≤ θm−1 ψ + τ ωm ψ .

(6.19)

6.4 Time Discretization Using the Backward Euler Method

131

By repeated application, θm ψ ≤ θ0 ψ + τ

m 

ω j ψ .

j=1

Here, as before, θ0 ψ = s f − P f ψ ≤ Ch sΞ ( f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) ). Now for every 1 ≤ j ≤ m, ω j = ∂ t Pu(t j ) − ∂ t u(t j ) + (∂ t u(t j ) − IΞ (u t (t j ))) =: ω j,1 + ω j,2 , where IΞ (u t (t j )) is the SBF interpolant of u t (t j ). We note that ω j,1 = (P − I )τ −1



tj

u t (σ)dσ = τ −1

t j−1



tj

(P − I )u t (σ)dσ,

t j−1

whence τ

m 

ω j,1 ψ ≤

j=1

m 

Ch sΞ

j=1

=

Ch sΞ



tm 0

tj

t j−1

u t (σ)W 2s+2 (S d−1 ) dσ

u t (σ)W 2s+2 (S d−1 ) dσ.

Further, u(t j ) − u(t j−1 ) − u t (t j ) + u t (t j ) − IΞ u t (t j ) τ tj 1 =− (s − t j−1 )u tt (s)ds + u t (t j ) − IΞ u t (t j ), τ t j−1

ω j,2 =

so that τ

m  j=1

  m  t j m      ω j,2 ψ ≤ (s − t j−1 )u tt (·, σ)dσ  + τ u t (·, t j ) − IΞ u t (·, t j )ψ   t j−1  j=1

≤τ

tm 0

u tt (·, σ)ψ dσ + Cτ h sΞ

ψ m 

j=1

j=1

u t (·, t j )W 2s (S d−1 ) .

132

6 Parabolic PDEs on Spheres

Therefore, by setting C1 := C and noting  · ψ ∼  · W s (S d−1 ) , we obtain a constant C2 so that τ

m 

ω j ψ ≤ τ

j=1

m 

ω j,1 ψ + τ

j=1

⎛

≤ C1 h sΞ ⎝

m 

ω j,2 ψ

j=1



+C2 τ 0

0 tm

tm

u t (·, σ)W 2s+2 (S d−1 ) dσ + τ

m 

⎞ u t (·, t j )W 2s (S d−1 ) ⎠

j=1

u tt (·, σ)W s (S d−1 ) dσ.

Thus u(·, tm ) − Um ψ ≤ γm ψ + θm ψ ≤ γm ψ + θ0 ψ + τ

m 

ω j ψ

j=1



≤ C1 h sΞ Γ ( f, u t ) + C2 τ

tm

0

u tt (·, σ)W s (S d−1 ) dσ,

where Γ ( f, u t ) :=  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) + tm m  u t (·, σ)W 2s+2 (S d−1 ) dσ + τ u t (·, t j )W 2s (S d−1 ) . 0

j=1



6.5 Time Discretization Using the Crank-Nicolson Method We now turn to the Crank-Nicolson method in which the semi-discrete equation is discretized in a symmetric fashion around the point tm−1/2 := (m − 1/2)τ , which will produce a second order in time accurate method. More precisely, Um in VΞ can be defined recursively by ∂ t Um (ξ j ) − Δ∗ (Um (ξ j ) + Um−1 (ξ j ))/2 = F(ξ j , tm−1/2 ), ∀ξ j ∈ Ξ, given that U0 = s f .

(6.20)

6.5 Time Discretization Using the Crank-Nicolson Method

133

In matrix form 1 1 (A − τ B)cm = (A + τ B)cm−1 + τ [F(ξ j , tm−1/2 )] Nj=1 , 2 2 given that Ac0 = [ f (ξ j )] Nj=1 .

Theorem 6.3 Let Um and u be the solutions of (6.20) and (6.1), respectively. We assume that f, u t ∈ W 2s+2 (S d−1 ) and u ttt , Δ∗ u tt ∈ W s (S d−1 ). The approximate k ∼ solution Um is constructed from shifts of a spherical basis function Ψ with ψ (1 + λk )−s . Then there are positive constants C1 and C2 , independent of h Ξ , so that  Um − u(·, tm )ψ ≤C1 h sΞ Γ ( f, u t ) + C2 τ 2

tm 0

u ttt (·, σ)W s (S d−1 )  + Δ∗ u tt (·, σ)W s (S d−1 ) dσ ,

where Γ ( f, u t ) :=  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) + tm m  u t (·, σ)W 2s+2 (S d−1 ) dσ + τ u t (t j−1/2 )W 2s (S d−1 ) . 0

j=1

Proof Let Um − u(tm ) = Um − Pu(tm ) + Pu(tm ) − u(tm ) =: θm + γm . With the above notation we have ∂ t θm (ξ j ) − Δ∗ (θm (ξ j ) + θm−1 (ξ j ))/2 = −ηm (ξ j ), ∀ξ j ∈ Ξ, where now   u(tm ) + u(tm−1 ) ηm = ∂ t Pu(tm ) − ∂t IΞ u(tm−1/2 ) + IΞ Δ∗ u(tm−1/2 ) − 2 = (P − I )∂ t u(tm ) + (∂ t u(tm ) − IΞ u t (tm−1/2 )) +   u(t N ) + u(tm−1 ) IΞ Δ∗ u(tm−1/2 ) − 2 =: ηm,1 + ηm,2 + ηm,3 .

134

6 Parabolic PDEs on Spheres

Applying arguments similar to (6.18) and (6.19) we arrive at 

θm − θm−1 −

 τ ∗ Δ (θm + θm−1 ), χ = −τ ηm , χ ψ , ∀χ ∈ VΞ . 2 ψ

By taking χ = θm + θm−1 and noting that Δ∗ (θm + θm−1 ), θm + θm−1 ψ ≤ 0 (cf. inequality (6.10)), we have θm 2ψ − θm−1 2ψ ≤ −τ ηm , (θm + θm−1 ) ψ ≤ τ ηm ψ (θm ψ + θm−1 ψ ). Simplifying the common factor (θm ψ + θm−1 ψ ) on both sides of the inequality, we obtain θm ψ ≤ θm−1 ψ + τ ηm ψ . After repeated application this yields θm ψ ≤ θ0 ψ + τ

m  (η j,1 ψ + η j,2 ψ + η j,3 ψ ). j=1

The term θ0 ψ can be estimated as before. For the latter sum, we have η j,1 ψ = (P − I )∂ t u(·, t j )ψ tj −1 s ≤ Cτ h Ξ u t (·, σ)W 2s+2 (S d−1 ) dσ. t j−1

Further, η j,2 ψ = ∂ t u(·, t j ) − IΞ u t (·, t j−1/2 )ψ ≤ ∂ t u(·, t j ) − u t (·, t j−1/2 )ψ + u t (·, t j−1/2 ) − IΞ u t (·, t j−1/2 )ψ   tj  t 1   j−1/2  2 2 = (s − t j−1 ) u ttt (·, σ)dσ + (s − t j ) u ttt (·, σ)dσ    2τ  t j−1 t j−1/2

ψ

+u t (·, t j−1/2 ) − IΞ u t (·, t j−1/2 )ψ tj ≤τ u ttt (·, σ)ψ dσ + Ch sΞ u t (t j−1/2 )W 2s (S d−1 ) . t j−1

Let u(t j ) + u(t j−1 ) ϕ := u(t j−1/2 ) − 2 1 tj 1 t j−1/2 (t j−1 − σ)u tt (σ)dσ + (σ − t j )u tt (σ)dσ. = 2 t j−1 2 t j−1/2

6.5 Time Discretization Using the Crank-Nicolson Method

135

Therefore, we have 1 η j,3 ψ = IΞ Δ∗ (u(·, t j−1/2 ) − (u(·, t j ) + u(·, t j−1 )))ψ 2 = IΞ Δ∗ ϕψ ≤ Δ∗ ϕψ (see Lemma 2.1) tj ≤ C2 τ Δ∗ u tt (·, σ)W s (S d−1 ) dσ, since  · ψ ∼  · W s (S d−1 ) . t j−1

Altogether, with C1 := C, we have τ

m  (η j,1 ψ + η j,2 ψ + η j,3 ψ ) j=1

⎛ ≤

⎝

C1 h sΞ



tm 0



⎞ u t (·, t j−1/2 )W 2s (S d−1 ) ⎠

j=1

tm

+C2 τ 2

u t (·, σ)W 2s+2 (S d−1 ) dσ + τ

m 

0

(u ttt (·, σ)W s (S d−1 ) + Δ∗ u tt (·, σ)W s (S d−1 ) )dσ.

Thus  θm ψ + γm ψ ≤ C1 h sΞ Γ ( f, u t ) + C2 τ 2

tm 0

 u ttt (·, σ)W s (S d−1 ) + Δ∗ u tt (·, σ)W s (S d−1 ) dσ ,

where Γ ( f, u t ) :=  f W 2s (S d−1 ) +  f W 2s+2 (S d−1 ) + +τ

m 

tm

0

u t (·, σ)W 2s+2 (S d−1 ) dσ

u t (·, t j−1/2 )W 2s (S d−1 ) .

j=1



6.6 Numerical Experiments on S2 Let us consider the function    G(z) = 1 − 2 ln 1 + (1 − z)/2 . We can expand G(z) as a series of Legendre polynomials (cf. [Mü66]): G(z) =

∞  =1

1 P (z). ( + 1)

136

6 Parabolic PDEs on Spheres

The following PDE describes the heat diffusion process from the north pole onto the surface of the unit sphere: 

∂ ∂t u(ξ, t)

u(ξ, 0)

= Δ∗ u(ξ, t), ξ ∈ S 2 , = G(ξ · ρ), where ρ = (0, 0, 1)T .

(6.21)

Since the initial condition u(ξ, 0) is a zonal function which depends only on the geodesic distance from any given point on the sphere to the north pole, the solution u(ξ, t) also depends only on the geodesic distance to the north pole. The problem (6.21) is reduced to ∂ ∂u ∂u = (1 − z 2 ) , ∂t ∂z ∂z subject to the following initial condition: u(z, 0) = G(z),

z ∈ [−1, 1].

We know that the Legendre polynomials are eigenfunctions of the operator ∂ ∂ (1 − z 2 ) . ∂z ∂z Thus, by the method of separation of variables, the exact solution of (6.21) is given as ∞ −(+1)t  e u(z, t) = P (z). ( + 1) =1

We can approximate u(z, t) by the truncated series: u L (z, t) =

L  e−(+1)t =1

( + 1)

P (z).

The error is estimated by using the fact that P (z)∞ = 1 (see [Mü66]) u − u L ∞

 ∞    e−(+1)t    P (z) =   ( + 1) =L+1 ∞ ∞ dx −L(L+1)t ≤e L x(x + 1)   1 ≤ e−L(L+1)τ ln 1 + L

For time-step τ = 0.00125, in order to obtain the accuracy of order 10−16 it is required that L ≥ 160. All the code used in the experiments was written in MATLAB.

6.6 Numerical Experiments on S 2

137

Fig. 6.1 A set of 500 equal area points due to [SK97]

The spherical basis functions used to construct the approximate solution are derived from a class of locally supported radial basis function proposed by Wendland [Wen95]. We use the following function Ψ (ξ, η) = ψ(ξ · η) = Φ(ξ − η), ξ, η ∈ S 2 . Note that, due to Theorem 4.1, Ψ (ξ, η) inherits the property of positive definiteness k ∼ (1 + λk )−s for some s > 0. For our numerical study, we use the from Φ, and ψ function Φ(ξ) = φ(ξ) with φ(r ) = (1 − r )4+ (4r + 1). The set of points which are used in constructing the SBFs is generated according to an algorithm in [SK97]. These points are distributed uniformly around the sphere, each point is a center of a rectangular cell on the unit sphere of area 4π/N . A picture of 500 points generated by the algorithm is shown in Fig. 6.1. The iterative equation (6.15) becomes (I − τ A−1 B)cm = cm−1 , with the initial equation

c0 = A−1 f |Ξ .

Since A is positive definite and B has non-positive eigenvalues, it can be shown that all the eigenvalues of the matrix (I − τ A−1 B) are in the interval (0, 1] (see Lemma 6.4 stated and proved as follows). Hence the numerical algorithm is stable.

138

6 Parabolic PDEs on Spheres

Table 6.1 Backward Euler method with different sets of points and time-steps m hΞ qΞ E ∞ (τ = 0.01) E ∞ (τ = 0.005) E ∞ (τ = 0.0025) 200 400 600 800 1000 1200

0.1942 0.1288 0.1122 0.0950 0.0849 0.0789

0.1130 0.0731 0.0675 0.0577 0.0516 0.0476

0.0224 0.0137 0.0088 0.0060 0.0044 0.0034

0.0225 0.0138 0.0089 0.0061 0.0045 0.0036

0.0225 0.0139 0.0090 0.0062 0.0046 0.0036

Lemma 6.4 (cf. [Wil65], Chap. 1, Sect. 31) Let A be a symmetric positive definite matrix and B be a symmetric positive semi-definite (negative semi-definite). Then all of the eigenvalues of AB are non-negative (non-positive). Proof Since A is symmetric positive definite, there is an invertible matrix P such that A = P T P. Let C = P B P T , then C and AB have the same set of eigenvalues since (P T )−1 AB P T = P B P T = C. The matrix C is symmetric since C T = P B T P T = P B P T = C since B is symmetric. Now since B is positive semi-definite, (P T x)T B(P T x) ≥ 0 for all x ∈ R N . Hence C is symmetric semi-positive definite. Hence all of the eigenvalues of C are non-negative, as are the eigenvalues of the matrix AB.  Lemma 6.5 Let A be a symmetric positive definite matrix and B be a symmetric negative semi-definite matrix. Then for any  > 0, all of the eigenvalues of (I − A−1 B)−1 are in the interval (0, 1]. Proof Let μ be an eigenvalue of I − A−1 B, then 1/μ is an eigenvalue of (I − A−1 B)−1 . It is observed that μ = 1 − δ where δ is an eigenvalue of A−1 B. By Lemma 6.4, δ ≤ 0, and therefore, μ ∈ [1, ∞). Thus, 1/μ ∈ (0, 1].  Table 6.1 shows the numerical errors between the iterated solution Um obtained by backward Euler method and u 160 . Here, m = 1.5/τ and E ∞ (τ ) := max |Um − u L |. x∈S 2

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