Investigation Methods for Inverse Problems [Reprint 2014 ed.] 9783110943849, 9783110364194

Table of contents :
Preface
1 Introduction
1.1 One-dimensional inverse kinematics problem
1.2 Inverse dynamical problem for a string
1.3 Inverse problems for a layered medium
2 Ray statements of inverse problems
2.1 Posing of the inverse problems
2.2 Asymptotic expansion
2.2.1 Asymptotic expansion of the solution
2.2.2 Reduction of the inverse problem
2.2.3 Construction of τ(x, y)
2.2.4 Proof of the expansion in the odd-dimensional case
2.2.5 Proof of the expansion in the even-dimensional case
2.2.6 Proof of the auxiliary lemma
2.3 Uniqueness theorems for the inverse problem
2.3.1 A proof of the stability estimate for the integral geometry problem
2.3.2 Uniqueness theorem for the integral geometry problem related to a vector field
2.3.3 Proof of the uniqueness theorem for inverse kinematics problem
2.3.4 The wave equation with an attenuation
2.3.5 Concluding remarks
2.4 Inverse problems related to a local heterogeneity
3 Local solvability of some inverse problems
3.1 Banach’s spaces of analytic functions
3.2 Determining coefficients of the lower terms
3.2.1 Determining a coefficient of the lower term
3.2.2 Determining an attenuation coefficient
3.3 Determining the speed of the sound
3.4 A regularization method for solving an inverse problem
3.4.1 Theorems related to the system of integro-differential equations
3.4.2 Estimates of a solution to the algebraic equations
3.4.3 Convergence the approximate solution to the exact one
4 Inverse problems with single measurements
4.1 Determining coefficient of the lowest term
4.1.1 Statement of the problem and stability estimates
4.1.2 Proof of the stability theorems
4.1.3 Proof of Lemma 4.1.3
4.1.4 Proof of Lemma 4.1.4
4.2 Determining coefficients under first derivatives
4.3 Determining the speed of sound in the wave equation
4.3.1 Formulation of the problem and a stability estimate of the solution
4.3.2 Proof of Theorem 4.3.1
4.3.3 Proof of Lemma 4.3.5
4.3.4 Proof of Lemma 4.3.6
4.3.5 Proof of the inequality (4.3.24)
4.4 Case of a point source
4.4.1 Formulations of the problem and results
4.4.2 Proofs of the stability theorems
4.4.3 Properties of a solution to problem (4.4.1)
4.4.4 Proof of Lemma 4.4.3
4.4.5 Proof of Lemma 4.4.4
5 Stability estimates related to inverse problems for the transport equation
5.1 The problem of determining the relaxation and a density of inner sources
5.1.1 Statement of basic and auxiliary problems
5.1.2 The basic results
5.1.3 Proof of Theorem 5.1.1
5.1.4 Proof of the auxiliary lemmas
5.2 A stability estimate in the problem of determining the dispersion index and relaxation in 2D
5.2.1 Statement of the problem and the basic results
5.2.2 Proof of Lemma 5.2.1
5.2.3 A priori estimates
5.2.4 Proof of Theorem 5.2.3
5.3 The problem of determining the dispersion index and relaxation in 3D
5.3.1 Statement of the problem and the main results
5.3.2 Proof of Lemma 5.3.1
5.3.3 A priori estimates for function ω(x, v)
5.3.4 Estimates for functions ώ(x, v) and σ̃(x)
5.3.5 A priori estimates and differential properties of function u¯(x,v,v°)
5.3.6 A priori estimates and properties of function v(x, v,v°)
5.3.7 Equations for the derivatives of function ṽ(x, u, v°)
5.3.8 Proof of inequality (5.3.24)
5.3.9 Proof of inequality (5.3.25)
5.3.10 Proof of inequality (5.3.26)
5.3.11 Auxiliary formulae
Bibliography

Citation preview

INVERSE A N D ILL-POSED PROBLEMS SERIES

Investigation Methods for Inverse Problems

Also available in the Inverse and Ill-Posed Problems Series: Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya Belov 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 AKh.Amirov Computer Modelling in Tomography and Ill-Posed Problems M.M. Lavrent'ev, S.M. Zerkal and O.ETrofimov A n Introduction to Identification Problems via Functional Analysis Д. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yl/.E Anikonov Inverse Problems ofWave Processes AS. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov

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.L Anikonov, BA. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and LR.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 ALAgeev Integral Geometry ofTensor Fields VA. Sharafutdin ov

Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to the Theory of Inverse Problems AL Bukhgeim Identification Problems ofWave 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 Al. Kozhanov Inverse Problems ofVibrational Spectroscopy A.G.Yagola, I.V. Kochikov, G.M. Kuramshina andYuA. Pentin Elements of the Theory of Inverse Problems AM. Denisov

Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Investigation Methods for Inverse Problems

V.G. Romanov

III УSP III UTRECHT · BOSTON · KÖLN · TOKYO

2002

VSP

Tel: + 3 1 3 0 6 9 2 5 7 9 0

P.O. B o x 3 4 6

Fax: + 3 1 3 0 6 9 3 2 0 8 1

3700AHZeist

[email protected]

The Netherlands

© VSP

www.vsppub.com

2002

First p u b l i s h e d in 2 0 0 2

ISBN 90-6764-360-1 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.

Printed

in The Netherlands

by Ridderprint

bv,

Ridderkerk.

To my wife, Nina

Contents Preface

xi

1 Introduction 1.1 1.2 1.3

1

One-dimensional inverse kinematics problem Inverse dynamical problem for a string Inverse problems for a layered medium

2 Ray statements of inverse problems

3 8 16

19

2.1 2.2

Posing of the inverse problems 19 Asymptotic expansion 21 2.2.1 Asymptotic expansion of the solution 21 2.2.2 Reduction of the inverse problem 23 2.2.3 Construction of т(х, у) 24 2.2.4 Proof of the expansion in the odd-dimensional case . . 27 2.2.5 Proof of the expansion in the even-dimensional case . . 33 2.2.6 Proof of the auxiliary lemma 35 2.3 Uniqueness theorems for the inverse problem 38 2.3.1 A proof of the stability estimate for the integral geometry problem 43 2.3.2 Uniqueness theorem for the integral geometry problem related to a vector field 53 2.3.3 Proof of the uniqueness theorem for inverse kinematics problem 59 2.3.4 The wave equation with an attenuation 61 2.3.5 Concluding remarks 62 2.4 Inverse problems related to a local heterogeneity 62

3 Local solvability of some inverse problems 3.1 3.2

Banach's spaces of analytic functions Determining coefficients of the lower terms 3.2.1 Determining a coefficient of the lower term vii

69 70 71 71

viii

CONTENTS 3.2.2 Determining an attenuation coefficient 3.3 Determining the speed of the sound 3.4 A regularization method for solving an inverse problem . . . . 3.4.1 Theorems related to the system of integro-differential equations 3.4.2 Estimates of a solution to the algebraic equations . . . 3.4.3 Convergence the approximate solution to the exact one

79 82 91 95 102 110

4 Inverse problems with single measurements 121 4.1 Determining coefficient of the lowest term 122 4.1.1 Statement of the problem and stability estimates . . . 122 4.1.2 Proof of the stability theorems 124 4.1.3 Proof of Lemma 4.1.3 126 4.1.4 Proof of Lemma 4.1.4 130 4.2 Determining coefficients under first derivatives 136 4.3 Determining the speed of sound in the wave equation 141 4.3.1 Formulation of the problem and a stability estimate of the solution 142 4.3.2 Proof of Theorem 4.3.1 144 4.3.3 Proof of Lemma 4.3.5 150 4.3.4 Proof of Lemma 4.3.6 155 4.3.5 Proof of the inequality (4.3.24) 162 4.4 Case of a point source 162 4.4.1 Formulations of the problem and results 163 4.4.2 Proofs of the stability theorems 164 4.4.3 Properties of a solution to problem (4.4.1) 167 4.4.4 Proof of Lemma 4.4.3 168 4.4.5 Proof of Lemma 4.4.4 170 5 Stability estimates related to inverse problems for the transport equation 175 5.1 The problem of determining the relaxation and a density of inner sources 176 5.1.1 Statement of basic and auxiliary problems 176 5.1.2 The basic results 179 5.1.3 Proof of Theorem 5.1.1 187 5.1.4 Proof of the auxiliary lemmas 194 5.2 A stability estimate in the problem of determining the dispersion index and relaxation in 2D 201 5.2.1 Statement of the problem and the basic results 202 5.2.2 Proof of Lemma 5.2.1 206

CONTENTS 5.2.3 Α priori estimates 5.2.4 Proof of Theorem 5.2.3 5.3 The problem of determining the dispersion index and relaxation in 3D 5.3.1 Statement of the problem and the main results 5.3.2 Proof of Lemma 5.3.1 5.3.3 A priori estimates for function w(x, v) 5.3.4 Estimates for functions w(x, v) and σ(χ) 5.3.5 A priori estimates and differential properties of function ΰ(χ, 5.3.6 A priori estimates and properties of function υ(χ, v,v°) 5.3.7 Equations for the derivatives of function ϋ(χ, ν, v°) 5.3.8 Proof of inequality (5.3.24) 5.3.9 Proof of inequality (5.3.25) 5.3.10 Proof of inequality (5.3.26) 5.3.11 Auxiliary formulae Bibliography

ix 208 221 224 224 230 232 234 238 241 . . 243 249 251 254 256 262

Preface This book deals with some inverse problems of mathematical physics. The choice of material was determined by the author's interest in the latest developments in this field. The results presented have been published in different mathematical journals, but have been partly revised before inclusion in this volume. The book does not aim to give a systematic presentation of the theory of inverse problems but introduces some new methods for studying inverse problems and gives obtained results, which are related to the conditional well posedness of the problems. The main focus lies on time-domain inverse problems for hyperbolic equations. The final chapter deals with the kinetic transport equation. The structure of the book is as follows: in Chapter 1 some examples of onedimensional inverse problems are given as an introduction to such problems. In Chapter 2, ray statements of inverse problems for hyperbolic second order differential equations are introduced. In these statements, only some elements of dynamical information on the wave field are used. Expansion of the solution in the neighbourhood of the characteristic cone is considered and several coefficients of this expansion are taken into account, in order to solve an inverse problem. As a result, one obtains problems, which are very close to the problem of the Radon transform inversion or X-ray tomography (see [74, 138]). The X-ray tomography problem consists of recovering a function via its integrals given along all lines. In contrast to the latter problem, here one has integrals along geodesic lines of the Riemannian metric related to the equation. Such problems are called integral geometry problems (see [65]). A similar problem occurred for the first time as a result of linearization in the inverse kinematic problem [112]. Later, some problems of determination of coefficients in linear second order hyperbolic equations were investigated on this basis [113, 166]. Integral geometry problems were studied in a series of publications [11] - [14], [28, 30], [127] - [131], [155, 167, 170, 191, 192]. Here, new proof is given of the stability estimate for the integral geometry problem and some uniqueness theorems related to the ray statement of an inverse problem for a hyperbolic equation are proven.

xii

Preface

In Chapter 3 the solvability problem for multidimensional inverse problems is discussed. Existence theorems for multidimensional inverse problems are almost absent, due to over-determination of inverse problems and large difficulties in characterization of the data. A special class of functions is considered, which are analytic with respect to a part of variables and are smooth with respect to remaining ones. For this class, some local solvability theorems are proven and a numerical algorithm for the solution of an inverse problem is proposed. As mentioned above, multidimensional inverse problems are often over-determined. Uniqueness and stability theorems are proven, mainly for the case when the Dirichlet-to-Neumann map is known (i.e. when for a solution of a boundary value problem with an arbitrary Dirichlet data, the Neumann data on the boundary is given). It corresponds to using many observations. In Chapter 4 some multidimensional inverse problems for hyperbolic equations with single observations are considered. In this case, only single information on the solution of a direct problem is used for the determination of one unknown coefficient. Conditional stability estimates for considered problems are derived. Finally, in Chapter 5 inverse problems related to the transport equation are considered. These problems consist of determining the density of special inner sources and the relaxation, or the dispersion index and the relaxation. The classical tomography problem can be considered as a partial case of an inverse problem for the reduced transport equation when one neglects the scattering. Different formulations of inverse problems for the transport equation were proposed and uniqueness questions were investigated in a series of publications [6] - [10], [34] - [36], [49, 50, 160]. In this volume, conditional stability estimates for inverse problems are stated. For a long time I have been working together with my colleagues Ju. E. Anikonov, A.L. Bukhgeim, S.I. Kabanikhin, R.G. Mukhometov, V.G. Yakhno and many others. The creative relationships with them and our fruitful discussions have always stimulated me. I am grateful to them all. I would like to express my hearty thanks, especially, to the leader of the Siberian scientific school on inverse and ill-posed problems, Academician M.M. Lavrent'ev, who has friendly supported me from my first steps in this field of science. This work was partly supported by the Russian Foundation for Basic Research under grant No. 99-01-00602. Novosibirsk, December 2001.

Chapter 1 Introduction Inverse problems for differential equations are considered in this book. So we should explain what is mean the term an inverse problem. At first, note that many authors use this term in different senses and, moreover, an explicit definition can be only given for each concrete inverse problem. Nevertheless, there exist some typical features of such problems. The term implies that there are some direct (forward) problems. As a role, the direct problems are usual well-posed problems of the mathematical physics, i.e., the problems for that solutions are unique, exist and stable (in a some sense) to small variations of data. It is supposed that differential equations are given for the such problems as well as some suitable initial and boundary data. However, many applications stimulate studying problems in which differential equations are only known partially, namely, some functions entering the differential equations (or sometimes a right-hand side or boundary conditions) are still unknown. For example, if the differential equation is linear then a part of coefficients or even all coefficients may be unknown. These coefficients are functions of independent variables, in general, and they characterize properties of a medium (say, its density, elastic moduli, electric or heat conductivity, etc.). The problems in which these unknown functions should be found from a given information on solutions of direct problems for differential equations are named inverse problems. An inverse problem is called one-dimensional if unknown functions depend on one variable and multidimensional if they are functions of more than an one variable. Differential equations are usually consequences of physical laws and therefore a structure of a differential equation is determined by the observed physical phenomenon while coefficients of this equation depend on concrete medium and they are often unknown if the medium is inhomogeneous. For instance, if domain Ω is inaccessible for direct measurements then the coeffi1

2

1.

Introduction

cients are unknown in Ω. At the same time, the phenomenon can be observed outside Ω and sometimes the observations can be repeated for different data initiating this phenomenon. These observations give a motivation to posing the inverse problem: find unknown coefficients inside Ω using the observations of the phenomenon (solutions of the differential equation) outside Ω. Note that the similar problems occur in different applications: in physics, astronomy, geophysics, biology, medicine, mechanics etc. For example, the main goal of geophysics is studying of the Earth structure. However, direct measurements in deep layers of the Earth are impossible because the deepest pore is about 13 kilometers only. So studying properties of deep regions inside of the Earth can be based on indirect methods only. Using observations of surface elastic or electro-magnetic oscillations, heat flux through the surface, gravity potential of Earth, one can try to find corresponding properties of the Earth interior. Then mathematical formulations lead to some typical inverse problems. One of the simplest formulation is the inverse kinematics problem. In this problem one needs to determine a speed of a signal inside of a bounded domain via the given travel times between each couple points of the domain boundary. An one-dimensional variant of this problem is considered below in the next section. The multidimensional problem is also studied in the chapter 2. It was noted for a long time ago that the inverse kinematics problem has no an unique solution if the medium has inner wave guides. Therefore it is interesting use the more complete dynamical information on the physical phenomenon in order to avoid the non-uniqueness. Simplest statements of one-dimension dynamical inverse problems are considered in this chapter for a string and a layered medium. To the present time a bibliography of different one-dimensional inverse problems consists of several hundreds papers (see papers [1, 2, 18, 19, 21, 31, 32, 33, 40, 44, 54, 59, 60, 66, 92, 95, 97, 105, 124, 205, 218] and books [5, 26, 41, 68, 69, 93, 111, 149, 185, 215, 221, 222] and references therein). The multidimensional inverse problems were intensively studied in the end of the last century. The earliest such problem is related to the geophysics problem of recovering a shape of a body via its given potential outside of the body. We refer to following works [48, 78, 84, 109, 143], [156] - [159], [193, 194] where different statements and corresponding results can be found. The other famous problem came from the quantum physics. It is the problem of finding a potential in the Schrödinger equation from scattering data. The corresponding one-dimension problem was studied in 1950th (see [42, 66], [102] -[104], [119] - [122]). The main results related to uniqueness, existence and stability of the solution were obtained in these papers. However

1.1 One-dimensional inverse kinematics problem

3

different modifications of this problem are discussing till the present time. The first result for multidimensional inverse problems was obtained by Ju. M. Berezansky [27]. Later this problem and closed to it were studied by many researches [46, 47, 51, 57, 58, 61, 62, 75, 76, 132, 133, 135, 140, 142, 144, 145, 146, 164]. New statements of inverse problems were given and investigated in the works [3, 4, 56], [22] - [25], [29, 39], [85] - [91], [107, 108, 113], [115] - [118], [123, 136, 137, 139, 147, 148], [161] - [165], [170, 185, 187, 188], [198] - [201], [202, 203, 210, 211, 216, 213, 214, 219, 220]. Considered multidimensional inverse problems clearly demonstrate that solutions of these problems are unstable as role. Therefore they are ill-posed problems. A. N. Tikhonov was the first who clearly formulated requirements (an additional a priory information) to an ill-posed problem in order it can be solved stable [204]. Later he introduced the concept of the regularizing operator for an ill-posed problem [206]. The similar ideas were developed by Μ. M. Lavrent'ev [109] and V. K. Ivanov [78] - [80]. We refer to the following papers [16, 17, 20, 63, 81, 82], [90] - [94], [106, 109, 113, 126, 150, 207, 208, 209, 212] where many methods were suggested for solving ill-posed problems. In this book some methods for investigation of inverse problems are presented. These problems related to hyperbolic second order differential equations and the transport kinetic equation. Their studying was carried out in the 1990th, mainly. Of course, the presented in the book results are only very small part of the extensive theory of inverse problems. However, recently the series of books related to methods of solving inverse problems has appeared [5,10,12,13,17, 26, 37, 38, 43, 48, 51, 53, 55, 69, 71, 84, 96, 98,114,149,159]. Therefore a reader can find the most convenient method for researching a new problem in these books.

1.1

One-dimensional inverse kinematics problem

Let R+ {(χ, у) € R 2 | у > 0} and a speed с of a signal in this half-space be a positive function of у only, i.e., с = c(y). Suppose that c(y) € C2[0, oo) and its derivative d(y) is positive on [0, oo). Then the signal from the origin (0,0) reaches a point (£, 0), ξ > 0, along a smooth curve which belongs (except of its ends) to K2 and is symmetric with respect to the line χ = ξ/2. In geophysics this curve is named the ray. With mathematical point of view, it is a geodesic line for the Riemannian metric in which element of a length dτ is determined by the formula dr y/dx2 + dy2/c(y). Denote by ί(ξ) the

4

1. Introduction

corresponding travel time along the ray. For ξ small enough £(£) is a singlevalued monotonic increasing function with a negative second derivative t" (ξ) (a proof see below). It may be false if ξ is not small. Assume here that ί(ξ) is a single-valued function on interval (0, ξ 0 )· Consider the following problem: given ί(ξ) for ξ G (0, £0) find c(y) in where it is possible. The problem is named the one-dimensional inverse kinematics problems. It was solved by G. Herglotz [77] in the beginning last century. He obtains an explicit formula for the solution to the problem. Below we explain the main ideas which solve the problem. Note that the problem is really related with a differential equation. It is well known in the mathematical physics that the travel time τ(χ, x°) between to arbitrary points and x° € Mn satisfies to the eikonal equation: \4xr(x,x°)\2

= c~2(x)

and to the condition r(x,x°) = 0(\x — x°|) as χ —> x°. The direct problem for this equation is a calculation of function τ(χ,χ°) for different χ and a;0. It is also interesting a finding of lines which are orthogonal to the surfaces t(x, x°)=constant because along this lines the signal from x° is propagated towards x. The inverse problem consists in finding c(x) inside a domain Ω given r(x,x°) for points χ and x° belonging to the boundary of Ω. For the considered one-dimensional case Ω = R+ and one can fix the point x° assuming that it coincides with the origin. Denote n(y) := 1 /c(y) the refractive index of the medium. Then the equation of any ray is determined by the formula n(y) sin 0 = p ,

(1.1.1)

where θ 6 [0, π] is the angle between a tangent line to the ray at the point (x, y) and axis у and ρ is a positive constant along the ray. Hence, each ray going from the origin under the angle θο to axis у may be characterized by the parameter ρ — n(0) sin 0O- On the other hand, the parameter ρ is determined by ξ, i.e., ρ = ρ(ξ)· Suppose that the equation n(y) = ρ has the solution у = η > 0, η = η (ρ)· Then the ray tangents the line у = η and belongs the strip {(x, y) \ 0 < у < η}, so it has a vertex at у = η and is symmetric with respect to this vertex. Therefore the coordinates of the vertex are (ξ/2, η). Because dx/dy — tan θ and 1 + cot2 9 = 1/ sin2 θ, one finds from here that ^ = ± dУ

,

P

Vn2(y)~P2

,

(1.1.2)

where the sign (+) corresponds to the part of the ray from the origin to point (ξ/2, η) and (—) to the remain part. Integrating (1.1.2), one finds the

1.1

One-dimensional

inverse

kinematics

5

problem

relation between ξ and ρ in the form = 2J J о

Pdy

, v

n

2

•

( y ) - p

(1.1.3)

2

Recall that here τι(η) — p. Along the ray an element of travel time dt is determined by the formula =

(1.1.4)

Therefore function t(£) is given as follows

W = 2 / J

Ш

? V

n

2

У

·

( y ) - p

(1.1.5)

2

The equations (1.1.3), (1.1.5) form the completed system of relations for the inverse problems. To give an analysis of this system, it is useful to make a change of variables in the integrals by introducing the new integration variable q — n(y). Because n(y) is a monotonic decreasing function, there exists an inverse monotonic function у = f ( q ) such that n { f ( q ) ) = q. Note that the derivative f'(q) is negative. Then the relations (1.1.3), (1.1.5) can be written in the form ,

f p f b )

„

Μ

„

go

/

V

/

'

M

d q

.

.

go

where qo = n(0). The first of this relations presents ξ as a function of p, i.e., ξ — ξ (ρ), while the second one determines the other function t = t(p) Κζ(ρ))· The couple ξ(ρ), ί ( ξ ( ρ ) ) gives the parametric representation the function t = ί(ξ). Integrating by parts one can reduce the first relation to the following form

ζ ( ρ ) =

- 2 q f ( q )

a r c s i n ^ )

'

+

qo ?0

2 j

( q f ( q ) ) '

arcsin ( - ) dq

·' go go go

=

- i r p f ' ( p )

+

2qof'(qQ)

arcsin

-2

f

( q f ' ( q ) ) '

arcsin j - 1 do. 9/ (1.1.7)

6

1.

Introduction

„ 0 (for all ξ > 0 if (logc(y))" < 0 anywhere). If ξ —> 0 then ί'(ξ) qo = n(0). Thus we have the following necessary conditions for function ί(ξ) under the assumed above assumptions on c(y): 1) i(0) = 0, 2 ) f ( 0 > 0, 3) i " ( 0 < 0. The relation (1.1.9) is very important for solving the inverse problem. Indeed, in inverse problem function ί'(ξ) is given, therefore taking its derivative, one finds parameter ρ and this parameter gives value of the refraction index n(y) at у = η. Hence, it is enough to determine the correspondence

1.1 One-dimensional inverse kinematics problem

7

between ρ and η, i.e., η = f(j>), in order to find π(η). To solve the latter problem one can use relation (1.1.6) «р) = 2 / ^ М | ,

„ Po = ϊ'(ζο)· The equation (1.1.10) is the Abel equation with respect to f'(p)· An inversion of equation (1.1.10) is given by the formula 90

m

=

7Γ ./ V S2 - p2 Ρ

pe\p0,qo}·

(l.i.ii)

It means that ρ = π(η) = / _1 (??) is determined for η G [0, f(po)]· Thus, given ί(ξ) for ξ 6 [0, ξο] one can uniquely find function c(y) inside the layer ye[0,/(f(eb)]. If d{y) > 0 for all у however the second logarithmic derivative does not satisfy to the condition (logc(y))" < 0, then function ξ(ρ) may not be a strongly monotonic function and therefore ί(ξ) may be a multi-valued function. Different situations may occur then. For example, if function ξ'(ρ) has only finite number of zeroes on an interval [pi, then function ί(ξ) consists of finite number of single-valued smooth branches with cusp points corresponding to zeroes of ξ'(ρ). In this case, using the function t = i(£) one can calculate function ξ(ρ), where ρ — ί'(ξ). Then the previous method of determining c(y) can be used. A more complicate situation occurs if function ξ(ρ) is a constant, i.e., ξ(ρ) — ξ' for some interval [рьрг]· Then rays corresponding different ρ from this interval are collected at the point (£', 0). Hence, points (0,0) and ( ξ 0 ) are focuses for these rays. In this case function i(£) is essentially multi-valued at point ξ = ξ'. However, only limiting values of ί'(ξ) as ξ —> ξ' ± 0 are necessary in order to find p\ and P2- If they exist, function ξ(ρ) is determined for the interval [рьрг] as the constant ξ'. Thus, in this case one can also use the given above formula (1.1.11 for determining c{y)· If c'(y) < 0 for all y, the signal from the origin reaches point (ξ, 0) along axis x, hence, ί(ξ) = |£|/c(0) and only c(0) can be found in the inverse problem. The more interesting case is when function c'(y) is positive for [0, yo) and changes sign at у — y0 > 0. It was noted that in this case one can uniquely find c(y) for [0, yo], however it is impossible uniquely find c(y) for у > yo using the data of the inverse problem. A characterization of a set of all possible solutions was given by Gerver and Markushevich [67].

8

1.2

1. Introduction

Inverse dynamical problem for a string

Consider the Cauchy problem:

(

д2

\

д2

w|t=0 = δ(χ),

ut\t=o = o,

(1.2.1)

where M+ := {(x, t) Ε M2| t > 0} and ut means the derivative with respect to t. If the function q(x) is given, the problem of finding function u(x, t) satisfying relations (1.2.1) is the usual direct problem of mathematical physics. Note that the more general equation id2 . d2 , Л t, . д 2 а { х ) ь ( х ) с { х ) и {дГ д^Тх) ^0 can be reduced to the equation Lqu — 0 under a suitable change of dependent and independent variables. Consider a simplest inverse problem for operator Lq. Assume that q(x) is an even function, i.e., q(—x) — q(x) and the trace of the solution to problem (1-2.1) is known at χ = 0, i.e,, u|*=0 = /(*),

t Ε (0,T],

(1.2.2)

where Τ > 0. The inverse problem is: given f(t) find q(x). We suppose here that q(x) Ε C(—oo, oo). Under this assumption we shall demonstrate that q(x) can be uniquely and stable found for χ Ε [0,Τ/2], moreover, the solution of the inverse problem exists for small T, if function f ( t ) satisfies some necessary conditions. These necessary conditions follow from properties of the solution to problem (1.2.1). Lemma 1.2.1 Let q(x) Ε С[-Г/2,Г/2], Τ > 0. Then the solution to problem (1.2.1) exists in the characteristic triangle Αχ '•— {(ж, ί)| 0