Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems [Reprint 2013 ed.] 9783110960716

Table of contents :
Main definitions and notations
Introduction
Chapter 1. Finite-difference scheme inversion (FDSI)
1.1. Introduction
1.2. Volterra operator equations
1.3. Definitions and examples
1.4. Convergence of FDSI
1.5. Numerical examples
Chapter 2. Linearized multidimensional inverse problem for the wave equation
2.1. Introduction
2.2. Problem formulation
2.3. Linearization
2.4. Analyzing the structure of the solution to one-dimensional direct problem
2.5. Existence theorem for the direct problem
2.6. Uniqueness of solutions to the inverse problem and regularization
2.7. Numerical examples
Chapter 3. Methods of I. M. GePfand, B. M. Levitan and M. G. Krein
3.1. Introduction
3.2. Gel’fand-Levitan-Krein (GLK) equation for one-dimensional inverse problem
3.3. Multidimensional analog of GLK-equations
3.4. Gel’fand-Levitan method for wave equation
3.5. Discrete analog of the Gel'fand-Levitan equation
3.6. Multidimensional discrete analog
3.7. Numerical examples
Chapter 4. Boundary control method (BC method)
4.1. Introduction. Statement of the problem
4.2. BC method in one-dimensional case
4.3. BC method for 2D acoustic inverse problem
4.4. Numerical examples
Chapter 5. Projection method
5.1. Introduction
5.2. Projection method for solving inverse problem for the wave equation
5.3. Projection method for solving inverse acoustic problem
5.4. Numerical examples
Appendix A
Appendix B
Bibliography

Citation preview

Inverse and Ill-Posed Problems Series Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Also available in the Inverse and Ill-Posed Problems Series: Characterisation of Bio-Particles from Light Scattering V.P. Maltsev and K.A. Semyanov Carleman Estimates for Coefficient Inverse Problems and Numerical Applications M.V. KlibanoY and A. A Timonov

Counterexamples in Optimal Control Theory S.Ya. Serovaiskii Inverse Problems of Mathematical Physics ΛΊ.ΛΊ. Lavrentiev, A.V. Avdeev, MM. Lavrentiev, Jr., and V.l. Priimenko Ill-Posed Boundary-Value Problems S.E. Temirbolat Linear Sobolev Type Equations and Degenerate Semigroups of Operators G.A. Sviridyuk and V.E. Fedorov Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis Editors: MM. Lavrent'ev and S.I. Kabanikhin Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations A.G. Megrabov Nonclassical Linear Volterra Equations of the First Kind A.S. Apartsyn Poorly Visible Media in X-ray Tomography D.S. Anikonov, V.G. Nazarov, and I.V. Prokhorov Dynamical Inverse Problems of Distributed Systems V.l. Maksimov Theory of Linear Ill-Posed Problems and its Applications V.K. Ivanov, V.V. Vasin and V.P. Tanana Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation M.A. Atakhodzhaev Investigation Methods for Inverse Problems V.G. Romanov Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya. Be/ov Method of Spectral Mappings in the Inverse Problem Theory V. Yurko Theory of Linear Optimization I.I. Eremin Integral Geometry and Inverse Problems for Kinetic Equations A.Kh. Amirov Computer Modelling in Tomography and Ill-Posed Problems MM. Lavrent'ev, S.M. Zerkal and O.E. Trofimov An Introduction to Identification Problems via Functional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yu.E. Anikonov

Inverse Problems of Wave Processes A.S. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and E.R. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S. Antyufeev Introduction to the Theory of Inverse Problems A.L. Bukhgeim Identification Problems of Wave Phenomena Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems of Vibrational Spectroscopy A.G. Yagola, I.V. Kochikov, GM. Kuramshina and Yu.A. Pentin Elements of the Theory of Inverse Problems A.M. Denisov Volterra Equations and Inverse Problems A.L. Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P. Tanana Inverse and Ill-Posed Sources Problems Yu.E. Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P. Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A. Asanov and E.R. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E. Anikonov Ill-Posed Problems with A Priori Information V.V. Vasin and A.L. Ageev Integral Geometry of Tensor Fields V.A. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

S.I. Kabanikhin, A.D. Satybaev and M.A. Shishlenin

IIIVSPIII UTRECHT · BOSTON

2004

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First published in 2005 ISBN 90-6764-416-1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

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Preface The problems of determining coefficients of hyperbolic equations and systems from additional information on their solutions are of a great practical significances. As a rule, the desired coefficients are important characteristics of the media under consideration. For example, in inverse problems of the theory of elasticity we seek for the Lamé parameters and density; the tensors of dielectric permittivity, magnetic permeability and conductivity are to be found in electromagnetic inverse problems; the velocity of wave propagation in the medium and its density are the sought for parameters in inverse acoustic problems; and so on. We consider dynamic type of inverse problems in which the additional information is given by the trace of the direct problem solution on a (usually time-like) surface of the domain. This kind of inverse problems were originally formulated and investigated by M.M. Lavrent'ev and V.G. Romanov (1966). The technique developed by V.G. Romanov for proving local theorems of unique solvability for dynamic inverse problems and also theorems of uniqueness and conditional stability "on the whole" was applied in the elaboration of numerical methods for solving a wide range of inverse problems of acoustics, seismics, electrodynamics, transport theory and so on (Kabanikhin, 1976, 1988b, 1992, 1995). A majority of the papers and books devoted to the study of dynamic inverse problems deal with one of the following basic methods: the method of Volterra operator equations, Newton-Kantorovich method; Landweber iteration and optimization; the Gel'fand-Levitan-Krein and boundary control methods, the method of finite-difference scheme inversion and linearization. The first group of methods, namely, Volterra operator equations, Newton-Kantorovich, Landweber iteration and optimization produce the iterative algorithms where one should solve the corresponding direct (forward) problem and adjoint (or linear inverse) problem on every step of the iterative process. On the contrary, the Gel'fand-Levitan-Krein method, the method

Direct Methods of Solving Multidimensional Inverse Problems of boundary control, the finite-difference scheme inversion and sometimes linearization method do not use the multiple direct problem solution and allow one to find the solution in a specific point of the medium. Therefore we will refer to these methods as the "direct" methods. In the present book we will discuss theoretical and numerical background of the direct methods. We will formulate and prove theorems of convergence, conditional stability and other properties of the mentioned above methods. Namely, we will consider the following methods: 1. Finite-difference scheme inversion (Chapter 1) 2. Linearization (Chapter 2) 3. Method of Gel'fand-Levitan-Krein (Chapter 3) 4. Boundary Control method (Chapter 4) 5. Projection method (Chapter 5) This book is intended for students, postgraduate students, engineers, and researchers who are interested in the theory and numerics of inverse problems for hyperbolic equations. The authors are grateful to Academicians of Russian Academy of Science, Professors M. M. Lavrent'ev and A. S. Alekseev, corresponding members of RAS, Professors V. G. Romanov and V. V. Vasin and to Professors Y. E. Anikonov, M. I. Belishev, A. Lorenzi, O. Scherzer, M. Yamamoto, G. Β. Bakanov and Κ. T. Iskakov for the very fruitful discussions and kind permission to use some of their results. The authors would like to thanks Technical Editor Dr. D. Nechaev for his help in preparation of the final version of this book. We would like to mention that the work of Kabanikhin and Shishlenin was supported, in part, by the Grants of the Russian Foundation of Basic Research (02-01-00818 and 02-01-00809), by the Grant of President of Russian Federation for the support of leading scientific schools (NSh-1172.2003.1) and by the Grant of Ministry of Education of Russian Federation "Universities of Russia" (UR.04.01.026).

Sergey I. Kabanikhin Abdigany D. Satybaev Maxim A. Shishlenin

Contents

Main definitions and notations

1

Introduction

3

Chapter 1. Finite-difference scheme inversion (FDSI)

9

1.1. Introduction

9

1.2. Volterra operator equations

11

1.3. Definitions and examples

12

1.4. Convergence of FDSI

25

1.5. Numerical examples

29

Chapter 2. Linearized multidimensional inverse problem for the wave equation

37

2.1. Introduction

37

2.2. Problem formulation

37

2.3. Linearization

39

2.4. Analyzing the structure of the solution to one-dimensional direct problem

41

2.5. Existence theorem for the direct problem

45

2.6. Uniqueness of solutions to the inverse problem and regularization

49

2.7. Numerical examples

52

Direct Methods of Solving Multidimensional Inverse Problems Chapter 3. Methods of I. M. Gel'fand, B. M. Levitan and M. G. Krein 3.1. Introduction 3.2. Gel'fand-Levitan-Krein (GLK) equation for one-dimensional inverse problem 3.3. Multidimensional analog of GLK-equations 3.4. Gel'fand-Levitan method for wave equation 3.5. Discrete analog of the Gel'fand-Levitan equation 3.6. Multidimensional discrete analog 3.7. Numerical examples

67 70 73 76 84 112

Chapter 4. Boundary control method (BC method) 4.1. Introduction. Statement of the problem 4.2. BC method in one-dimensional case 4.3. BC method for 2D acoustic inverse problem 4.4. Numerical examples

119 119 120 122 129

Chapter 5. Projection method 5.1. Introduction 5.2. Projection method for solving inverse problem for the wave equation 5.3. Projection method for solving inverse acoustic problem . . . .

137 137

5.4. Numerical examples

65 65

138 149 155

Appendix A

163

Appendix Β

167

Bibliography

169

Main definitions and notations M - all real numbers Ζ - all integer numbers Ν - all natural numbers R+ = {t G R : t > 0} £ - the length (depth) on which we would like to solve inverse problem A(£) = {(x,t) Δι(^) = {(x,t)

: χ G (0,£),t €

(x,2£-x)}

: t G (0,£), χ G {-£ +

t,£-t)}

A2(£) = {(ar, t) : χ € (0, £), t € (-£ + x, £ - x ) } j — m,n means, that j runs all integer values from m to n, i.e. k,m = { j G Ζ : j = k, k + 1,..., m — 1, m } Kn{V)

= {-V,V)n

= {y = ( y i , . . . ,yn) G M" : \yá\ < V, j = d2 Ax'y-dx*

+

+

dyn*

Sometimes instead of the usual partial derivative we denote

where Dju means the first partial derivative with respect to the j-th variable

2

Direct Methods of Solving Multidimensional Inverse Problems

θ(·) - Heaviside theta-function f 1,

t>0,

[0,

i 0 0, i < 0

n,k - Kronekker symbol / 1, η = k, on,k = < 10, η ψ k

Introduction In this book our attention is focused on inverse problems for hyperbolic equations (IPHE) for the following reasons. First, the great majority of IPHE can be reduced to Volterra operator equations, thus creating a possibility to extend the theoretical results and numerical methods from the Volterra equations theory to IPHE (Barbu, 1975; Amini et al., 1983; Lubich, 1985). In particular, the Picard and Caratheodory successive approximations developed for Volterra equations are applicable to IPHE. Second, a great number of numerical methods, such as finite-difference scheme inversion (FDSI), linearization (LM) and Newton-Kantorovich type methods (NKM), regularizaron method (RM) and optimization (OM), the dynamical version of the Gel'fand-Levitan-Krein methods (GLK) and many others, were elaborated for and tested on IPHE. A fair amount of experimental results has been accumulated, thereby inducing researchers to systematize and generalize the compiled material. Finally, it is well known that in some important cases the direct problems for elliptic and parabolic equations reduce to those for hyperbolic ones. Therefore, the technique worked out for IPHE may also be useful in studying inverse problems for elliptic or parabolic equations. Since an understanding of numerical methods for inverse problems can only be acquired if one has a good understanding of the theory underlying these problems, the relevant concepts of, and results from the inverse problems theory are introduced. Moreover, in recent time has appeared a new promising line of investigation - the discrete inverse problems theory. In the discrete case the inverse problems for hyperbolic, parabolic or elliptic equations reduce to nonlinear algebraic systems. There one of the most interesting and important questions is whether we can use the usual (continuous) techniques for investigating discrete inverse problems. The results concerning the theory and numerical methods for inverse and ill-posed problems can be found in Tikhonov (1946, 1949, 1963, 1964, 1965), Lavrent'ev (1962), Maslov (1968), Lions and Lattes (1969), Romanov (1972,

4

Direct Methods of Solving Multidimensional

Inverse Problems

1982), Romanov et al. (1985), Lavrent'ev et al. (1986), Romanov and Kabanikhin (1991, 1994), Kabanikhin and Lorenzi (1999), Kabanikhin and Shishlenin (2002a). The first results on multidimensional inverse problems for hyperbolic equations were the uniqueness theorems proved by M. M. Lavrent'ev and V. G. Romanov (1966) (see also Romanov, 1989). V. G. Romanov also generalized the theorem on uniqueness and conditional stability to the case of Volterra operator equations and described several classes of uniqueness. The next step in studying multidimensional inverse problems after the uniqueness theorem and the estimates of conditional stability was the development of numerical algorithms for solving this kind of problems. In (Kabanikhin, 1976), an approach to constructing numerical algorithms for solving multidimensional inverse problems for hyperbolic equations based on the projection method (PM) was proposed. This approach later appeared to be applicable to the study of inverse problems for the acoustic equation, the system of Maxwell's equations, and the kinetic transport equation (Kabanikhin, 1988b). A large number of methods for solving hyperbolic inverse problems in electrodynamics, seismology, and acoustics have been developed to date. We will only mention several most important methods. The Gel'fand-Levitan method has been widely used in quantum scattering theory and in seismology. Now there is hope that it will also be applied for constructing numerical methods in electrodynamics (due to works by Α. V. Baev, 2001 and V. A. Yurko, 2002). Optimization methods (Kabanikhin and Iskakov, 2001) and the Newton-Kantorovich methods (Bimuratov and Kabanikhin, 1992) have always been most popular. Methods for solving general operator equations, such as the Landweber iteration method (Engl, Hanke, and Neubauer, 1996; Kabanikhin, Kowar, and Scherzer, 1998; Kabanikhin, Kowar, Scherzer and Vasin, 2001) and the boundary control method (Belishev, 1987, 1990), have begun to be used for solving hyperbolic inverse problems in recent years. As follows from the well-known Gel'fand-Levitan-Krein theory and its dynamical version (Blagoveshchenskii, 1971; Romanov, 1984), the main difficulty in problems of this kind is due to the fact that there is no theorem of existence "in large" (i.e., for arbitrary depth) for arbitrary data. The task of verifying the conditions for the existence of solutions to inverse hyperbolic problems can be so complicate and time-consuming that, in a sense, it may be comparable to solving an inverse problem. As far as multidimensional inverse problems are concerned, even application of the Gel'fand-LevitanKrein method remains problematic in most important practical problems.

5

Introduction

The complexity and great importance of hyperbolic inverse problems for applications, explain the difficulty and diversity of numerical methods being developed for these problems. We will consider two types of Inverse Problems.

Forward (direct) problem 1. ^

= Ax,yU

-

q(x,

y)u,

x>0,

(1)

y = ( y i , . . . , yn) € R n ,

¿>0;

du u{x,y,0)

= ip(x,y), χ >

— (x,y,0)

0,

(2)

= φ(χ,ν),

e Rn.

y

Here R is the set of all real numbers and Δ X y is the Laplacian d2u

Δχ.ν11 =

ñ+ dx2^

d2u

d2u

+ ... + dyi2^···^

συη2·

To solve inverse problem 1 is to find functions u(x,y,t) and q(x,y) by known functions ψ{χ, y), ψ(χ, y) and f(y, t), which is given by the condition u(0,y,t)

y € Rn,

= f(y,t),

t > 0.

(3)

We shall consider a problem of determining the density p(x, y) and the velocity of wave propagation c(x, y) in a medium when the source and the receivers are located at the plane boundary of the medium being studied. Forward (direct) problem 2. Suppose that up to the moment t = 0 the medium was at rest d2v A

x,yv -

e" 2 (x, V)ßjp= χ >

Ht

0;

°·

is the pressure,

(4) (5)

VXtV

is the gradient and / dv

dv

dv

Suppose that at the moment t = 0 at the boundary χ = 0 of the half-space χ > 0 the source of the type dv —

turns on.

(+0,y,t)

= H(y,t),

y e R

n

,

t > 0

(6)

6

Direct Methods of Solving Multidimensional

Inverse

Problems

In the inverse problem 2 it is necessary to determine p(x, y) or c(x, y) or p(x, y) and c(x, y) simultaneously provided that for the solution of direct problem (4)-(6) (a classical or a generalized one) the following additional information is available: u\x=+0 = f(y,t). (7) Note, that the equation (4) can be reduced to (1) when c(x, y) = const (Appendix A). Therefore research results for inverse problems for equation (4) can be applied for corresponding inverse problems for the equation (1) and vice versa. We will consider the following type of sources (6): H(y,t)

=

h(y)S(t).

(8)

Here ô(t) is the Dirac delta-function. Suppose that functions c(x,y), p(x,y) and h(y) are positive and smooth enough. Moreover c(x, y) > co > 0. Note, that in this book inverse problems are considered and investigated only in the time-domain approach. In Chapter 1 the finite-difference scheme inversion (FDSI) is considered for inverse hyperbolic problems. We present and discuss the theoretical and numerical results and justify the convergence. In Chapter 2 we consider linearized inverse problem for inverse problem 2 when p(x,y) = const. The linearization can obviously be treated as the first step of the Newton-Kantorovich method. In the case when solution to an inverse problem is known to differ only a little from a given function, one of the most frequently used methods in the inverse problems theory is the linearization method. In Chapter 3 we consider the Gel'fand-Levitan-Krein methods for equations (1) and (4) when c(x,y) = const. The Gel'fand-Levitan-Krein method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parametric family of linear integral Fredholm equations of the first or second kind. We give an outline of the developments in the research in this field. Note that we consider only the time-domain approach. In the multidimensional case, the application of the Gel'fand-LevitanKrein method makes it possible to reduce the multidimensional nonlinear inverse problem to the system of linear integral equations (multidimensional version of the Gel'fand-Levitan-Krein equation; Kabanikhin, 1988a; Kabanikhin and Lorenzi, 1999).

Introduction

7

Chapter 4 deals with the boundary control method (Belishev, 1990, 2002; Pestov, 1999) for inverse problem 2 when c(x,y) — 1. The method (Belishev, 1997) gives an efficient way to reconstruct a Riemannian manifold via its response operator (dynamical Dirichlet-to-Neumann map) or spectral data (a spectrum of the Beltrami-Laplace operator and traces of normal derivatives of the eigenfunctions). An effective linear method for solving inverse problems, namely, the BC method, can be used to determine analytically the density or velocity profile in a stratified acoustic half-space (He, 1996). In Chapter 5 we use the projection method to study the problem of reconstructing the two-dimensional parameter for the inverse problem 1 and inverse problem 2 when c(x,y) = 1. In both cases, the inverse problem is considered in the form of a nonlinear system of Volterra integral equations. We establish the convergence of the projection method and estimate the rate of convergence. The main idea of the projection method consists in projecting the multidimensional inverse problem onto a finite-dimensional subspace generated by some orthogonal system of functions. The finite system of one-dimensional inverse problems thus obtained may be solved numerically, for example, with the use of the finite-difference scheme inversion method. The main problem on this way is to prove the existence of a solution to the finite (generally speaking, nonlinear) system of one-dimensional inverse problems and get an estimate of convergence of the solution to the finite system of one-dimensional inverse problems to the exact solution of the initial multidimensional inverse problem as Ν tends to infinity, the parameter Ν being the length of the segment of the Fourier series in the expansion in the basis functions (Kabanikhin, 1988b). The inverse problems (l)-(3) and (4)-(7) and its various modifications in one- and multidimensional cases were investigated by Romanov, Blagoveshchenskii, Kabanikhin, Symes, Sacks and Santosa and many others. The one-dimensional version of inverse problems has been investigated more thoroughly. In this case for investigating the inverse problem the dynamic version of the Gel'fand-Levitan method (Blagoveshchenskii, Gopinath and Sondhi, Santosa, Symes, Kabanikhin), the Newton-Kantorovich method (Reznitskaya, Antonenko and Reznitskaya, Sacks and Santosa, Xie, Kabanikhin and Bimuratov), the optimization methods (Bamberger, Chavent and Lailly, Santosa and Symes, Tarantola) were used.

8

Direct Methods of Solving Multidimensional Inverse Problems

Chapter 1. Finite-difference scheme inversion

1.1.

INTRODUCTION

The finite-difference scheme inversion (FDSI) is outlined as follows. The inverse problem is replaced by its finite-difference version. After solving the system of nonlinear algebraic equations (using rather simple technique) the obtained solution is taken as an approximate solution of the original inverse problem. The method of finite-difference scheme inversion was originally proposed in Geophysics. The first model to be analyzed was the model of a medium represented by a pack of homogeneous layers with horizontal interfaces (the Goupillaud model). The unknowns were the layer characteristics, such as the velocity of wave propagation and the density in every homogeneous layer (Baranov, Kunetz, 1960; Alekseev, Dobrinskii, 1975). Further generalizations lead to the study of the one-dimensional inverse seismic problem of determining the acoustic stiffness of the medium. The general idea of the finite-difference scheme inversion as one of the possible numeric methods of determining the coefficients of hyperbolic equations was first formulated by Alekseev (1967). Various aspects of implementing this idea in application to the dynamic inverse seismic problem were considered by Antonenko (1967), Alekseev and Dobrinsky (1975), Kabanikhin and Satybaev (1987, 1988), and others. In Kabanikhin (1977), it was demonstrated that introducing an a priori constant into the finite-difference scheme inversion makes it possible to prove that the method converges because of

10

Direct Methods of Solving Multidimensional Inverse Problems

the Volterra property of the integral equation relative to the unknown coefficient. In joint papers by Kabanikhin with Abdiev (1986), Akhmetov (1983), Boboev (1982), Satybaev (1987), Martakov (1988), Bakanov and Shishlenin (2001) the finite-difference scheme inversion was applied in studying inverse problems of geoelectrics, the elasticity theory, and P/v -appr oximat ion of the of the kinetic equation of the transport equation. In Kabanikhin (1988b), the convergence of the finite-difference scheme inversion was justified (see subsection 1.4). Various methods for solving dynamic inverse problems based on FDSI were also investigated by Symes, Santosa, Sacks, Ursin, Berteussen and many others. See also Berryman and Greene (1980), Kabanikhin (1980, 1981, 1984c, 1993, 1995), Santosa and Schwetlick (1982), Bube and Burridge (1983), Dobrinskii and Bushenkov (1983), Symes (1983), Abdiev (1984), Romanov et al. (1985), Santosa et al. (1985), Brookes and Bube (1997) and references there. Let us consider the simple example of FDSI for I D inverse acoustic problem:

d2u

d2u

σ'(χ) du

dt2

dx2

σ(χ)

ΐχ|ί 0,

t> 0;

x>0;

= 7S(t),

u\x=+0 = f(t),

dx'

(1.1.1) (1.1.2)

t > 0; t > 0,

(1.1.3) (1.1.4)

where it is assumed that σ(χ) > 0, χ > 0 and σ e C^O, oo). For the inverse problem (1.1.1)—(1.1.4) it is necessary to find the solution u(x,t) and the acoustic stiffness 0 and is sufficiently smooth for t > χ > 0; θ(·) is Heaviside theta-function and

Chapter

1. Finite-difference

scheme

inversion

11

Substituting (1.1.5) into (1.1.1)—(1.1.4), we obtain the equivalent inverse problem with respect to u(x,t) and s(x)

dt2

(Pu

s'(χ)

du

2

s(x)

dx'

dx

du

= 0,

dx x=0 u(x,x + 0) = s(x), u\x=+0

t >

= m ,

t > χ > 0;

0;

(1.1.7) χ > 0;

t>

(1.1.6)

(1.1.8)

(1.1.9)

o.

Let us write the discrete analog of (1.1.6)—(1.1.9) - 2uf + v^1 h2

- 2v\ h?

+

1 Pi+1 - P¿-i

-

h Pi+i

2h

+Pi-1

^-i.

(1.1.10)

(1.1.11)

«f =

(1.1.12)

v\ = Pi •

Here Λ. = ¿/TV and the discrete inverse problem (1.1.10)—(1.1.12) is solved in the domain Ah(£)

= Α(ί)

Π {(χ, t ) : x = hi, t = hk, i, fc € Z} ;

Δ(£) = {(®,t) : χ G ( 0 , £ ) , t e (x, 2£ — x)}.

The system of nonlinear algebraic equations (1.1.10)-(1.1.12) allows the simple inversion (section 1.5). As a result we find the approximate solution to the inverse problem (1.1.1)—(1.1.4). 1.2.

VOLTERRA OPERATOR EQUATIONS

As is known, a broad class of inverse problems for hyperbolic equations and systems reduces to the Volterra operator equations of the first or the second kind with Volterra and boundedly Lipschitz-continuous kernels. In this section the mentioned properties are shown to ensure the well-posedness of inverse problems locally and in the neighborhood of the exact solution as a whole. The procedure developed allows us to estimate the rate of convergence in the finite-difference scheme inversion method.

12

Direct Methods of Solving Multidimensional Inverse Problems

In Subsection 1.3.1 we will show that inverse problems for hyperbolic equations can be reduced to Volterra operator equation of the second kind zeS=[0,£]

(1.2.1)

It was shown (Kabanikhin, 1984a) that under rather general assumptions concerning the family of operators {Kz}zes the set of data {/(z)} for which there exists a solution of the operator equation (1.2.1) is open in C(S;X). Here, the fact that q belongs to the set C(S;X) implies that q is a continuous function of the parameter ζ G S taking its values in the real Banach space X whose norm will be denoted by ||·||. In the one dimensional case X = R. The integral in (1.2.1) is to be meant as a Bochner integral (Gajewski, Gröger and Zacharias, 1974). 1.3.

DEFINITIONS A N D EXAMPLES

Let us indicate some properties of the family of operators {Kz}zes which occur for a broad class of inverse problems.

in (1-2.1)

Definition 1.3.1. Let Sz = [0,ζ], ζ € S. A (possibly) nonlinear operator Kz : C(SZ·, X) —> C(SZ; X) will be referred to as a Volterra operator if from the relation q(X) = r(X) for all λ € SZl, for all z\ E Sz q, r G C(S; X) it follows that the equality (Kzq)(X) = (Kzr)(X) holds for all Λ G SZl . Definition 1.3.2. A family of (possibly) nonlinear operators Kz : C(SZ;X) —> C(SZ;X), ζ G S, will be referred to as a boundedly Lipschitzcontinuous family of operators if there is a real function μ(υι, 1/2) increasing in both y\ and such that for any ζ E S and , 92 G C(S; X) the estimate \\Kzq! - Kzq2\\c{sz-X)
2)|| 0, k > 0, q G C(S\X). (C, fc)-norm and the ball of radius ε by

We define the

IMIc.fc = s u P z e S {||g||c(sz;x) e- f c z };

g

Bk(i, q, ε) = {r G C(S; X) : \\q- r\\c,k < ε}.

1.3.1. The relations between inverse problems and Volterra equations A relation between inverse problems and Volterra equations was used in the earlier works on the theory of inverse problems. In 1979 at the World International Mathematical Congress in Nice M. M. Lavrent'ev posed the problem of investigating the Volterra general operator equations with the aim of studying a wide class of inverse problems. The main results in this field can be found in books by Lavrent'ev, Romanov and Shishat'skii, Romanov, Bukhgeim, Kabanikhin. The main idea of the method of Volterra operator equation (MVOE) as applied to the inverse dynamic problems for hyperbolic equations is that for a broad class of the corresponding direct problems the representations of solutions by means of Volterra integral operators acting on the data are known. Using these representations, as well as the additional information on the solution of a direct problem we can obtain the Volterra operator equation relative to the unknown coefficients. Let us consider examples of inverse problems which show a relation between inverse problems and Volterra equations. Example 1.3.1 [Inverse problem for wave equation]. We consider the problem of the finding q(x) in Δχ(^) from the relations d2u d2u ~dt* = d t f +

q { x ) u

>

(Μ)€ΞΔΙ(*);

) = )) - ^TO M 4E T O V M ,

with M4 = ||ç||c(R).

(2.4.6)

43

Chapter 2. Linearized multidimensional inverse problem

Proof. We shall represent the desired solution of the integral equation (2.4.2) in the form of the series 00

p(x,t)

(x,t) e Δ^χο,ίο),

= ^2pk(x,t), k=0

(2.4.7)

whose terms will be determined by induction with respect to k. For k = 0, we assume po{x,t) — I(x,t). Suppose that pk{x,t) is already determined. Let Pk+i(x,t) = AXit[qpk], (x,t) Ε Δι(χο,ίο) and set Pk(t)=

sup \Pk(x,t)\, X € [XO —to+t,Xo+to—t]

t€(0,to)·

We now prove the following estimate by induction: W )

< ll^llc(Al(xo,fo))

¿ G (0' ίο)·

( 2 A 8 )

For k = 0, the estimate (2.4.8) is obvious. Let (2.4.8) hold for some k > 0. We shall prove that it also holds for k + 1. The norm ||^|Ιο(Δχ(χ010)) estimated as follows: l l ' I W c t o ) ) < l \\Α*,ΜΟΘ(τ - I£I)]|Ic(Ai(xo,ío)) ^ 1 * 0 ^ 4 . We have MA fto \pk+i(x,t)\ < Ax,t[\qpk\] < -γ J oo, we see that p(x, t) is a solution of the integral equation (2.4.2) which is continuous in Ai(xo,to). • Lemma 2.4.2. Let q E C(R) and let C(R χ R+) be a set of functions p(x,t) continuous everywhere in R χ R+ except maybe the line t = \x\. Then the solution of the integral equation (2.4.5) has partial derivatives of the fìrst order with respect to t and χ which belong to C(R χ R+) and satisfy the inequalities sup

«,T)eAi(x,t),|C|#r

P(m) & r) I < 7ÍM4 [\ +

-

Ί

t2M4e^

where t € R+, m = 1,2,

Lemma 2.4.3. If q G C(R), then then the solution of (2.4.2) has a second derivative with respect to χ which belongs to the class C(R+ x R_) and has the form d2p

^

=

d2 dx2

1 f* + 2 J [Φ +

t

^ ~ q(x)p(x'

- T)P(2)(X + t-T,r)

^

+

+ q(x-t

+ τ)ρ[2){χ - t + r,τ)] dr,

where t € R+ and the derivative satisfìes the

d2P,

sup 2 (ί,τ)€Δι(«,ί),|€|#τ δξ with Ms = Ail!3/2 + Τ2Μ4 + Τ2Μ4(1 +

(ξ,

inequality τ)


ψ(ζ) >0) and has the following structure: u0(z,t)

= S[iP(z)](ye(t-'(x,t)

L2(Kn(V)) = 0

holds for all pairs (x,t) 6 Δ(£). To prove the theorem of existence of the generalized solution of the problem (2.5.4)-(2.5.6), we use obvious modifications of well-known statements. Lemma 2.5.2. Let a sequence {um(x,y,t)}^=1, um € V(£,T>) converge to a function u(x, y, t) in L2{Kn{V)) uniformly with respect to (x, t) € As(£). Then u e V{£,V). Lemma 2.5.3. Let the sequence of functions {um(x,y,t)}^=1, um € V{£,V) converge in itself in L2(/Cn(V)) uniformly with respect to (x,t) € Azil). Then there exists a function u G V(£, V) such that the sequence {um(x,y,t)} converges to u(x,y,t) in L2(/Cn(X>)) uniformly with respect to (x,t)eA3(£). Lemma 2.5.4. Let co(z) and ci(z, y) satisfy the conditions Ao and Ai, respectively. Suppose that there exists a sequence of functions {am(x, y)}, am € C1([—£, £] χ K,n(D)), m = 1 , 2 , . . . , that satisGes the conditions 1) linim-Kjo ||a - am\\wi{[-t,e\xKn{v)) -

1,2,...,

2) for any m = there exists a classical solution wm(x, y, t) of the problem (2.5.4)-(2.5.6) for a = am(x,y).

48

Direct Methods of Solving Multidimensional Inverse Problems Then there is a function w G V(£,T>) such that lim II«; - io m || = 0.

The function w(x, y, t) is said to be a generalized solution of the problem (2.5.4)-(2.5.6). Using (2.5.8) and Lemma 2.5.2, we can show that the generalized solution defined this way satisfies equation (2.5.4) in the generalized sense. Theorem 2.5.1. Let CQ(Z) and c\(Z, y) satisfy the conditions Ao and Ai, respectively. Then there exists a unique generalized solution of the problem (2.5.4)-(2.5.6) for every I € (0, Th). Proof. We represent a(x, y) in the form a{x,y) = Yiak{x)Yk{y),

(2.5.9)

where Yk{y) = exp

(fc,y)),

η (k,y) = ^ k j y j . J=1

k = (ku ...,kn),

The condition A i implies that the series in (2.5.9) converges to a(x, y) in L{ Xi

^

(2 7 7)

-

' '

where (x,t) € Δ(£); y € (-V,T>); w(x,y,\x\) *>\v=-v = H«=o =

= la(x,y),

x € (-£,£)·,

(2.7.8) (2·7·9) (2.7.10)

= 1) •·= f(y>*) - p(o,ί).

It is required to find α(χ, y) from (2.7.7)-(2.7.10) from the given functions p(x,t), q(x), b(x), g(y,t), and the parameter 7. Suppose that £ is a fixed positive number; Ν and iVi are natural numbers, h = i/Ν and = V/Nx. Put w

i,j

in

WX-

=

w(ih,jhi,kh), -

Pi = p(ih, kh), ,

t„_ _ Wt-

= q(ih),

- Ki -

1

w

,

WJ

Wyy-

tj

aitj = +

a{ih,jh\)\

i~Hj+ ^

w

tj-i

,

and so on. Consider a discrete analog of the inverse problem (2.7.7)-(2.7.10) (Satybaev, 2001): find the mesh function ä i j from the relations Wtt = Wxx + [bi]2Wyy + Qi^ij + äitjpf, i = -Ν, N, k ~|¿| 7 w i = -N,N, j = -N1,N1·, i,j ~k w i = —Ν, Ν, k=\i\,2N-\i\-, i- Νι — wi,Ni := 0, ~k k £ÌZ - p i k= j = -WJfi. = 9j •= fi

(2.7.11) (2.7.12) (2.7.13) (2.7.14)

In what follows, we assume that the mesh functions w^ •, ó¿j are 2^1-periodic in the discrete variable j and even.

54

Direct Methods of Solving Multidimensional Inverse Problems

Let gj be a mesh function defined for j = —Νχ,Νι. (Kabanikhin and Bakanov, 1999) Ni 9j = Σ 9 -i m m=l

Represent gj as

(τη)cosTfljhx'K • 2

where the Fourier coefficients g a r e defined by the formula (rn)

2/n

mjh\K j=—Ni+l

Lemma 2.7.1. The following formula holds: , njhiTT mjhiir Í 0, hi > cos — 2- — cos — 2- — =

(2.7.23) (2.7.24)

The evenness of w\ in i yields an extra relation: W^^W^-Xmh2)-^-1.

(2.7.25)

Thus, the algorithm of solving the finite difference two-dimensional direct problem (2.7.21), (2.7.22) consists of the following steps:

Chapter 2. Linearized multidimensional

inverse problem

59

Fig. 2.3: Regularized solution to the linearized inverse problem A with 5% noise in data

60

Direct Methods of Solving Multidimensional Inverse Problems

1.6

1.4

12

Fig. 2.4: Exact solution to the two-dimensional inverse problem Β (1) (2) (3) (4) (5) (6) (7)

knowing oJ m) for m = Ü725, calculate w\ by (2.7.24) for i = ÖJV; determine WQ from (2.7.25); determine w\ from (2.7.23); calculate by (2.7.25); using (2.7.23), find involving (2.7.23), find wf; calculate w$ by (2.7.25); etc.

For each m = 0,25, we define the trace of the solution f(m)k Then, / is reconstructed by the formula

=

.

25

At the second step of the model computations, we solved the linearized two-dimensional inverse problem (2.7.7)-(2.7.10) using the Fourier method. The inverse problem was reduced to solving the one-dimensional inverse problems for the Fourier coefficients of w(x, y, t) 1

25

m= 1 We reconstructed 25 Fourier coefficients from the known data

f(y,t).

Chapter 2. Linearized multidimensional inverse problem

61

Fig. 2.6: Regularized solution to the linearized inverse problem Β with 5% noise in data

Direct Methods of Solving Multidimensional Inverse Problems

Fig. 2.8: Regularized solution to the linearized inverse problem C

Chapter 2. Linearized multidimensional

inverse problem

63

Fig. 2.9: Regularized solution to the linearized inverse problem C with 5% noise in data We obtained the solution for the linearized two-dimensional inverse problem (Fig. 2.3, 2.6, 2.9) for the perturbed data taken in the form r(y, t) = f(y, t) + 0.05(sin (39y) + sin (53t)).

(2.7.26)

Direct Methods of Solving Multidimensional Inverse Problems

Chapter 3. Methods of I. M. Gel'fand, Β. M. Levitan and M. G. Krein

3.1.

INTRODUCTION

The Gel'fand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parametric family of linear integral Fredholm equations of the first or second kind. We give an outline of the developments in the research in this field. I. M. Gel'fand and Β. M. Levitan (1951) presented a method of reconstructing the Sturm-Liouville operator from the spectrum function and gave sufficient conditions for a given monotonie function to be a spectrum function of the operator. It is also important to mention the papers by Krein (1951, 1954), in which the physical statement of the string tension problem is considered and theorems on solving the inverse boundary value problem axe formulated. In Blagoveschenskii (1966), a new proof of M. G. Krein's results on the theory of inverse problems for the string equation was obtained. The new proof has the advantage of being local (nonstationary). In Blagoschenskii's approach, the local dependence of the unknown coefficient on the additional data is explicitly taken into account. The origin of the method due to Gel'fand and Levitan (1951) dates back to early 50-th (see also Krein, 1951, 1954; Levitan, 1984). This method is one of those extensively used in applications. It enjoys the possibility of

66

Direct Methods of Solving Multidimensional

Inverse

Problems

formulating necessary and sufficient conditions for existence in the large of the inverse problem solution. It is important in view of nonlinearity of inverse problems. The essence of this method lies in reduction of the nonlinear inverse problem (investigated originally in spectral domain by Gel'fand and Levitan, 1951) to a one-parameter set of linear integral Fredholm equations of the first or second kind. Ideas of Gel'fand-Levitan method were actively used in the theory of inverse dynamic seismic problems starting with the papers by Alekseev (1967), Kunetz (1961), Pariiskii (1968). Blagoveshchenskii (1969, 1971), Gopinath and Sondhi (1971) developed the dynamic (time-domain) version of Gel'fand-Levitan method for an acoustic inverse problem. Aleekseev and Dobrinskii (1975) used the discrete analogy of the Gel'fand-Levitan method while investigating the numerical algorithms for a one-dimensional dynamic inverse problem of seismics (see also Goupillaud, 1961; Kunetz, 1963; Symes, 1979; Santosa, 1982). The comparison of the Gel'fand-Levitan method with others intended for solving onedimensional inverse problems can be found in Burridge (1980) and Ursin and Berteussen (1986). Pariiskii (1977, 1978) analyzed the numerical algorithms for solving the Gel'fand-Levitan equations. In Kabanikhin (1979) an algorithm is proposed for numerical solution of Gel'fand-Levitan equation which uses the sufficient condition for the inverse problem solvability. In Kabanikhin (1988a, 1990, 1992) the multi-dimensional analogy of the Gel'fand-Levitan equation was obtained. In Romanov and Kabanikhin (1989) the dynamical version of Gel'fand-Levitan method is applied to the one-dimensional inverse geoelectric problem for quasistationary approximation of Maxwell's equations. We mention also Berryman and Greene (1980), Case and Kack (1983), Bube (1984), Gladwell and Willms (1989), Natterer (1994). The ideas of the Gel'fand-Levitan method have been extensively used in inverse dynamic seismic problems starting from the works by Alekseev (1967), Kunetz (1961), and Pariiskii (1968, 1977). A.S. Blagoveschenskii developed a dynamic (in time) version of the Gel'fand-Levitan method for the inverse acoustic problem (Blagoveschenskii, 1966, 1971). A.S.Alekseev and V . l . Dobrinskii (1975) used a discrete version of the Gel'fand-Levitan method for studying numerical algorithms of solving the one-dimensional inverse dynamic seismic problem. Detailed survey of the numerical methods for solving equations of the Gel'fand-Levitan type is given in Pariiskii (1977). In Kabanikhin (1988a), a new algorithm for solving the Gel'fand-Levitan equation is proposed, which involves the application of the sufficient con-

Chapter 3. Methods of Gel'fand, Levitan and Krein

67

dition for the solvability of an inverse problem. In the monograph by Romanov and Kabanikhin (1991), the dynamic version of the Gel'fand-Levitan method is applied to the one-dimensional inverse geoelectric problem for the quasi-stationary approximation of Maxwell's equations. The multidimensional version of the Gel'fand-Levitan equation is obtained in Belishev (1987), Kabanikhin (1988b, 1989), Belishev and Blagoveschenskii (1992). The multidimensional analogs of Gel'fand-Levitan-Krein equations of section 3.3 were established by Kabanikhin (1988). The one-dimensional inverse acoustic problem was reduced to the GLK-equation by Blagoveschenskii (1971). Rakesh (1993) proved the uniqueness theorem which can be applied to the inverse problem of sections 3.2 and 3.3. Romanov (1972) established the uniqueness and the conditional stability and Kabanikhin (1976) proved the local solvability in the set of coefficients p(x, y) which can be represented in the form M k=1

When p(x, y) is analytic with respect to y the local solvability and global uniqueness can be proved as well (Romanov, 1989) in scale of Banach spaces. The analyticity with respect to y made it possible to justify the projection method (Kabanikhin, 1987), i.e. to reduce multidimensional inverse problems of type (4)-(7) to the finite system of multidimensional inverse problems (see the last chapter). We would like to mention a lot of well-known results concerning inverse problem (4)-(7) by Symes (1979), Sacks and Santosa (1985, 1987).

3.2.

G E L ' F A N D - L E V I T A N - K R E I N (GLK) E Q U A T I O N FOR ONE-DIMENSIONAL INVERSE PROBLEM

Let us describe Gel'fand-Levitan-Krein (GLK) approach to inverse acoustic problem: d2u dt2

d2u dx2

σ'(χ) du σ(χ) dx'

χ > 0,

t > 0;

(3.2.1) (3.2.2) (3.2.3)

u(0,t)

= f(t).

(3.2.4)

68

Direct Methods of Solving Multidimensional Inverse Problems

following to Blagoveschenskii (1971). Here 6(t) is Dirac delta-function. Let function ü(x, t) be odd continuation of function u(x,t) for negative t: ü(x,t) = u(x,t),

t> 0;

ü(x,t) = —u(x,—t),

t> 0;

/(i) = - / ( - < ) ,

t < 0.

The similar way f(t) = f(t),

ί < 0.

One can obtain that ü(x, t) solves the following problem: d2ü dt2

d2u dx2

σ'(χ) dû σ{χ) dx du

-ι tu\ «u=o = m , Function u(x,t) |£| < χ.

^

x=0

has the form: u(x,t)

χ > 0,

t€

= 0.

ξ 0 for t < x. So ü(x,t)

= 0 for

Let W(x, t) be a solution to the following problem d2W dt2

d2W dx2

w\x=0

= ô(t),

σ'(χ) dW σ(χ) dx ' dW — =0. dx i=0

χ > 0,

tel;

(3.2.5) (3.2.6)

Function W(x, t) is W(x,t) t] =

= 0,

ì

\t\>x;

+ x ) + s { t

"

+W{x t}

'

'

where W(x, t) is smooth enough for |i| < x. Note, that if we continue the functions σ(χ) and u(x, t) by even way on negative x, then function u(x, t) is solution of the following problem: d2u dt2

d2u dx2

ι u\t=o = 0,

σ'{χ) du σ(χ) dx' 9u Έ

=

t > 0;

-2ηδ(χ).

ί=0

We have the equality on ü(x, t) and ü(x,t)=

iE!,

W(x,t):

f W(x,s)f(tJR

It is obvious that Φ(χ, t) = 0 for χ < |ί|.

s)ds.

(3.2.7)

69

Chapter 3. Methods of Gel'fand, Levitan and Krein We apply to both parts of (3.2.7) the operation d_ Γ dt Jo

( ' ) ΛΡ W)^·

Then we derive F(x,t)

=

^Jw(x,a)f(t-8)da

= 2f(+0)V(x,

t) + Γ f{t J—X

Here F(x,t)

= l

X d

^ , t )

- s)V(x, s) ds.

¿e

and derivative of f(t) is taken in points of continuity, i.e. f'(t) = df/dt ίφ 0.

for

One can show that dF

dF

— (x,t) = 0,

— (x,t)=0,

χ > \t\.

Therefore F(x,t) = const. The value of this constant is ,· m/ \ ,· f dü^ . άξ 1 fx lim F(x,t) = lim / — (Ç,t)—2= lim / t^+o v ' > t-*+oJ0 2 ί->+ο J 7 = lim Γ 2 - ^ - 5 ( 0 άξ =

following (x > 0): du^ . ( -r-(£,t)dt 'σ ( 0

r (+0)

'

Whence Prom the last equality we obtain the integral equation: - 2 / ( + 0 ) Φ ( χ , t) - J

X

f'(t - s)Φ(χ, s) ds =

\t\ < x.

(3.2.8)

Taking into account that W(x,t) is solution of (3.2.5), (3.2.6), we can obtain that Φ(χ,ί) solves the following problem δ2Φ dt2

02Φ σ'(χ) 0Φ + T T « -dx >: 2 dx σ(χ) Λ

Φχ=ο = 0,

9Φ οχ χ=ο

m

σ(0)

η

a; > 0,

m

te R;

, . (3.2.9) (3.2.10)

70

Direct Methods of Solving Multidimensional Inverse Problems

Solution of (3.2.9), (3.2.10) can be represented in the form Φ(χ,ί) =

1

2τ/σ(+0) σ(χ)

9(t + χ) + θ(χ - t)

+Φ(χ,ί),

where θ(·) is Heaviside theta function; Φ(χ, t) is smooth enough for \t\ < x. Therefore Φ(χ, χ - 0) = — , = . 2-\/σ(+0) σ(χ) Then the solution to the inverse acoustic problem (3.2.1)-(3.2.4) is given by the formula: Φ(0,0) σ [ Χ ) ~ 2Φ2(χ,χ)·

3.3.

MULTIDIMENSIONAL A N A L O G OF GEL'FAND-LEVITAN-KREIN EQUATIONS

Let us consider inverse problem of finding density p(x, y) in inverse acoustic problem. The main goal is to reduce multidimensional nonlinear inverse problem to the system of linear integral equations (multidimensional analog of Gel'fand-Levitan-Krein equation (Kabanikhin, 1988a; Kabanikhin and Lorenzi, 1999). We consider the system of forward problems A,uW =

01

at2 -

A

X j

y

+

VXiy\np(x,y)

χ > 0,

y € [-7Γ,7τ],

u( * ) = 0 ,

V:

t > 0,

(3.3.1)

fcG (3.3.2)

< n < 0 = °; duW

(3.3.3)

(+0, y , i ) = eikyô(t); dx u (fc)l|y=7r = uW I\y-—π·

(3.3.4)

We suppose that the trace of forward problem solution (3.3.1)-(3.3.4) exists and can be measured: (3.3.5)

u ( f c ) (0,y,t) = / < % , * ) .

The necessary condition of existence of solution to the inverse problem (3.3.1)-(3.3.5): fW(y,+0)

= -eìk»,

y G [—π, π],

k € Ζ.

Chapter 3. Methods of Gel'fand, Levitan and Krein

71

Let us define (Kabanikhin, 1988a; Kabanikhin and Lorenzi, 1999) the auxiliary system of the forward problems: ApwW

= 0,

w™(0,y,t)

χ > 0,

y € [-π,π],

= e[myô(t),

íeR,

m G Ζ;

— ( 0 , y, t) = 0.

(3.3.6) (3.3.7)

The solution to (3.3.6), (3.3.7) has the form (χ, y, t) = sW (χ, y) [ 0; dv(k) (x, y, 0) = eìky5(x), ιξΚ, y € M"; ~dT dvW (0, y, t) = 0, y € M", t > 0. fW(y,t), dx

= 0,

v(fc)(0 ,y,t) =

(3.4.2) (3.4.3)

Let α(χ, y) is solution Causchy problem for da dx

+

a(0,y) = 0,

da dy

= c z(x,y),

χ € R,

da, g ( 0 , y ) = c- 1 (0,y),

y€

y € Rn.

Let us introduce new variables z = a(x,y),

y= y

and new functions u^{z,y,t)=v^(x,y,t),

b(z,y) = c(x,y).

(3.4.4)

74

Direct Methods

of Solving Multidimensional

Inverse

Problems

It is obvious that for χ e R,

c(x, y) > co > 0,

y e Rn

change of variable (3.4.4) establishes one to one correspondence (z, y) ( a , y) for some χ E (0, h). Then υΡ^ solves the following problem « „ W . r2 ^ - 2Ä - ^ - ^ - p D ^ - O , dt dz ζ E R, u « U = 0,

y E Rn,

(3.4,)

t > 0; (3.4.6)

—gj—(x,y,0) = 6 ( 0 , y ) e i f c ^ ) , ζ € R,

Here

>ôa

φ , y) = Let

y € P(*> 2/) =

6(0, y) ξ 1,

2

& 2 Δχ, ζ α·

yeRn.

(3.4.7)

Additional information about solution to the direct problem (3.4.5), (3.4.6) can be written in the form uW(0,y,t)

y E Rn,

= fW(y,t),

t > 0,

Using the odd continuation of functions obtain Mu(fc) = 0,

and f ^ with respect to t we

y E Rn,

zeR+,

duW dz

z=0

(3.4.8)

k ETL.

t > 0, = 0,

¿fe £ Ζ; y E Rn,

k

eZ.

Then u ^ is connected with auxiliary problem Mffi ( m ) = 0 ,

ζ € R+,

w (m) I

y € Rn, ¿=0

t > 0, = 0,

m Ε Ζ; y G R",

m e Z.

by the following way uW(z,y,t)= for

ζ G

R+, y

G

f ^2f^(t-s)w^\z,y,s)ds, m

R n , t E R, k Ε Ζ.

(3.4.9)

Chapter 3. Methods of Gel'fand, Levitan and Krein

75

In the neighborhood of plane t = ζ the direct problem solution has the form υ,™ (z, y, t) = s^

(t, y)S(z - t ) + Q^

(t, y)9{z - t ) + w^

(z, y, t), (3.4.10)

where w ^ is continuous, s ^ and Q(mi solve the following family of problems m G Ζ: Λ„(τη)

aim) +

y G Rn,

= 0,

¿Gl+;

(3.4.11)

píTiy «

(m)

¿

|í=o = ^ - '

dt

(3.4.12)

_ ô ^ ) dt2

_

q

ÔQM _ dy

dy2

n

yGM,