Inverse Problems (Mathematical and Analytical Techniques with Applications to Engineering).pdf

INVERSE PROBLEMS

MATHEMATICAL AND ANALYTICAL TECHNIQUESWITH APPLICATIONS TO ENGINEERING

MATHEMATICAL AND ANALYTICAL TECHNIQUESWITH APPLICATIONS TO ENGINEERING

Alan Jeffrey, Consulting Editor

Published:Inverse ProblemsA. G. Ramm

Singular Perturbation TheoryR. S. Johnson

Forthcoming:Methods for Constructing Exact Solutions of Partial Differential Equationswith ApplicationsS. V. Meleshko

The Fast Solution of Boundary Integral EquationsS. Rjasanow and O. Steinbach

Stochastic Differential Equations with ApplicationsR. Situ

INVERSE PROBLEMS

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITHAPPLICATIONS TO ENGINEERING

ALEXANDER G. RAMM

Springer

eBook ISBN: 0-387-23218-4Print ISBN: 0-387-23195-1

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CONTENTS

Foreword

1. Introduction 1

1.1 Why are inverse problems interesting and practically important? 1

1.2 Examples of inverse problems 21.2.11.2.21.2.3

Inverse problems of potential theory 2Inverse spectral problems 2Inverse scattering problems in quantum physics; finding the potentialfrom the impedance function 2

1.2.41.2.51.2.61.2.7

Inverse problems of interest in geophysics 3Inverse problems for the heat and wave equations 3Inverse obstacle scattering 4Finding small subsurface inhomogeneities from the measurementsof the scattered field on the surface 5

1.2.81.2.9

1.2.101.2.111.2.12

Inverse problem of radiomeasurements 5Impedance tomography (inverse conductivity) problem 5Tomography and other integral geometry problems 5Inverse problems with incomplete data 6The Pompeiu problem, Schiffers conjecture, and inverse problemof plasma theory 7Multidimensional inverse potential scattering 8Ground-penetrating radar 8A geometrical inverse problem 9Inverse source problems 10

1.2.131.2.141.2.151.2.16

Preface xvii

xv

viii Contents

1.2.171.2.181.2.191.2.201.2.211.2.221.2.23

Identification problems for integral-differential equations 12Inverse problem for an abstract evolution equation 12Inverse gravimetry problem 12Phase retrieval problem (PRP) 12Non-overdetermined inverse problems 12Image processing, deconvolution 13Inverse problem of electrodynamics, recovery of layered mediumfrom the surface scattering data 13Finding ODE from a trajectory 131.2.24

1.3 Ill-posed problems 14

1.4 Examples of Ill-posed problems 151.4.11.4.2

1.4.31.4.41.4.51.4.61.4.71.4.8

Stable numerical differentiation of noisy data 15Stable summation of the Fourier series and integrals with randomlyperturbed coefficients 15Solving ill-conditioned linear algebraic systems 15Fredholm and Volterra integral equations of the first kind 16Deconvolution problems 16Minimization problems 16The Cauchy problem for Laplaces equation 16The backwards heat equation 17

2. Methods of solving ill-posed problems 19

2.1 Variational regularization 192.1.12.1.22.1.32.1.42.1.52.1.62.1.72.1.8

2.1.9

Pseudoinverse. Singular values decomposition 19Variational (Phillips-Tikhonov) regularization 20Discrepancy principle 22Nonlinear ill-posed problems 23Regularization of nonlinear, possibly unbounded, operator 24Regularization based on spectral theory 25On the notion of ill-posedness for nonlinear equations 26Discrepancy principle for nonlinear ill-posed problems withmonotone operators 26Regularizers for Ill-posed problems must depend on the noiselevel 29

2.2 Quasisolutions, quasinversion, and Backus-Gilbert method 302.2.12.2.22.2.32.2.4

Quasisolutions for continuous operator 30Quasisolution for unbounded operators 31Quasiinversion 32A Backus-Gilbert-type method: Recovery of signals from discreteand noisy data 32

2.3 Iterative methods 40

2.4 Dynamical system method (DSM) 412.4.12.4.22.4.32.4.42.4.52.4.6

2.4.7

The idea of the DSM 41DSM for well-posed problems 42Linear ill-posed problems 45Nonlinear ill-posed problems with monotone operators 49Nonlinear ill-posed problems with non-monotone operators 57Nonlinear ill-posed problems: avoiding inverting of operators in theNewton-type continuous schemes 59Iterative schemes 62

2.4.82.4.9

2.4.10

A spectral assumption 64Nonlinear integral inequality 65Riccati equation 70

ix

2.5 Examples of solutions of ill-posed problems 712.5.12.5.2

Stable numerical differentiation: when is it possible? 71Stable summation of the Fourier series and integrals withperturbed coefficients 85Stable solution of some Volterra equations of the first kind 87Deconvolution problems 87Ill-conditioned linear algebraic systems 88

2.5.32.5.42.5.5

2.6 Projection methods for ill-posed problems 893. One-dimensional inverse scattering and spectral problems 91

3.1 Introduction 923.1.13.1.23.1.3

3.1.43.1.5

What is new in this chapter? 92Auxiliary results 92Statement of the inverse scattering and inverse spectralproblems 97Property C for ODE 98A brief description of the basic results 99

3.2 Property C for ODE 1043.2.13.2.2

Property 104Properties and 105

3.3 Inverse problem with I-function as the data 1083.3.13.3.23.3.33.3.4

Uniqueness theorem 108Characterization of the I-functions 110Inversion procedures 112Properties of 112

3.4 Inverse spectral problem 1223.4.13.4.23.4.33.4.43.4.5

Auxiliary results 122Uniqueness theorem 124Reconstruction procedure 126Invertibility of the reconstruction steps 128Characterization of the class of spectral functions of theSturm-Liouville operators 130Relation to the inverse scattering problem 1303.4.6

3.5 Inverse scattering on half-line 1323.5.13.5.2

Auxiliary material 132Statement of the inverse scattering problem on the half-line.Uniqueness theorem 137Reconstruction procedure 139Invertibility of the steps of the reconstruction procedure 143Characterization of the scattering data 145A new Marchenko-type equation 147Inequalities for the transformation operators and applications 148

3.5.33.5.43.5.53.5.63.5.7

3.6 Inverse scattering problem with fixed-energy phase shifts as the data 1563.6.13.6.2

Introduction 156Existence and uniqueness of the transformation operatorsindependent of angular momentum 157

x Contents

3.6.33.6.4

3.6.5

Uniqueness theorem 165Why is the Newton-Sabatier (NS) procedure fundamentallywrong? 166Formula for the radius of the support of the potential in termsof scattering data 172

3.7 Inverse scattering with incomplete data 1763.7.13.7.23.7.3

Uniqueness results 176Uniqueness results: compactly supported potentials 180Inverse scattering on the full line by a potential vanishing on ahalf-line 181

3.8 Recovery of quarkonium systems 1813.8.13.8.23.8.3

Statement of the inverse problem 181Proof 183Reconstruction method 185

3.9 Kreins method in inverse scattering 1863.9.13.9.23.9.33.9.43.9.5

Introduction and description of the method 186Proofs 192Numerical aspects of the Krein inversion procedure 200Discussion of the ISP when the bound states are present 201Relation between Kreins and GLs methods 201

3.10 Inverse problems for the heat and wave equations 2023.10.13.10.23.10.3

Inverse problem for the heat equation 202What are the correct measurements? 203Inverse problem for the wave equation 204

3.11 Inverse problem for an inhomogeneous Schrdinger equation 204

3.12 An inverse problem of ocean acoustics 2083.12.13.12.23.12.3

The problem 208Introduction 209Proofs: uniqueness theorem and inversion algorithm 212

3.13 Theory of ground-penetrating radars 2163.13.13.13.23.13.33.13.43.13.53.13.6

Introduction 216Derivation of the basic equations 217Basic analytical results 219Numerical results 221The case of a source which is a loop of current 222Basic analytical results 225

4. Inverse obstacle scattering 227

4.1

4.2

4.3

4.4

4.5

4.6

Statement of the problem 227

Inverse obstacle scattering problems 234

Stability estimates for the solution to IOSP 240

High-frequency asymptotics of the scattering amplitude and inversescattering problem 243

Remarks about numerical methods for finding S from the scatteringdata 245

Analysis of a method for identification of obstacles 247

xi

5. Stability of the solutions to 3D Inverse scattering problems with fixed-energydata 255

5.1 Introduction 2555.1.15.1.2

The direct potential scattering problem 256Review of the known results 256

5.2 Inverse potential scattering problem with fixed-energy data 2645.2.15.2.25.2.35.2.45.2.55.2.6

Uniqueness theorem 264Reconstruction formula for exact data 264Stability estimate for inversion of the exact data 267Stability estimate for inversion of noisy data 270Stability estimate for the scattering solutions 273Spherically symmetric potentials 274

5.3 Inverse geophysical scattering with fixed-frequencydata 275

5.4 Proofs of some estimates 2775.4.1 Proof of (5.1.18) 2775.4.2 Proof of (5.1.20) and (5.1.21) 2785.4.3 Proof of (5.2.17) 2835.4.4 Proof of (5.4.49) 2855.4.5 Proof of (5.4.51) 2865.4.6 Proof of (5.2.13) 2875.4.7 Proof of (5.2.23) 2895.4.8 Proof of (5.1.30) 292

5.5 Construction of the Dirichlet-to-Neumann map from the scattering dataand vice versa 293

5.6

5.7

5.8

5.9

Property C 298

Necessary and sufficient condition for scatterers to be sphericallysymmetric 300

The Born inversion 307

Uniqueness theorems for inverse spectral problems 312

6. Non-uniqueness and uniqueness results 317

6.1 Examples of nonuniqueness for an inverse problem of geophysics 3176.1.16.1.2

Statement of the problem 317Example of nonuniqueness of the solution to IP 318

6.2

6.3

6.4

A uniqueness theorem for inverse boundary value problems for parabolicequations 319

Property C and an inverse problem for a hyperbolic equation 3216.3.1 Introduction 3216.3.2 Statement of the result. Proofs 321

Continuation of the data 330

7. Inverse problems of potential theory and other inverse source problems 333

7.1

7.2

7.3

Inverse problem of potential theory 333

Antenna synthesis problems 336

Inverse source problem for hyperbolic equations 337

xii Contents

8. Non-overdetermined inverse problems 339

8.1

8.2

8.3

8.4

8.5

Introduction 339

Assumptions 340

The problem and the result 340

Finding from 342

Appendix 347

9. Low-frequency inversion 349

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

Derivation of the basic equation. Uniqueness results 349

Analytical solution of the basic equation 353

Characterization of the low-frequency data 355

Problems of numerical implementation 355

Half-spaces with different properties 356

Inversion of the data given on a sphere 357

Inversion of the data given on a cylinder 358

Two-dimensional inverse problems 359

One-dimensional inversion 362

9.10 Inversion of the backscattering data and a problem of integralgeometry 363

9.11

9.12

9.13

9.14

9.15

9.16

9.17

9.18

Inversion of the well-to-well data 364

Induction logging problems 366

Examples of non-uniqueness of the solution to an inverse problemof geophysics 369

Scattering in absorptive medium 371

A geometrical inverse problem 371

An inverse problem for a biharmonic equation 373

Inverse scattering when the background is variable 375

Remarks concerning the basic equation 377

10. Wave scattering by small bodies of arbitrary shapes 379

10.1 Wave scattering by small bodies 37910.1.110.1.210.1.310.1.4

Introduction 379Scalar wave scattering by a single body 380Electromagnetic wave scattering by a single body 383Many-body wave scattering 385

10.2 Equations for the self-consistent field in media consisting of manysmall particles 38810.2.110.2.210.2.3

Introduction 388Acoustic fields in random media 390Electromagnetic waves in random media 394

xiii

10.3 Finding small subsurface inhomogeneities from scattering data 39510.3.1 Introduction 39610.3.2 Basic equations 39710.3.3 Justification of the proposed method 398

10.4 Inverse problem of radiomeasurements 401

11. The Pompeiu problem 405

11.1 The Pompeiu problem 40511.1.111.1.2

Introduction 405Proofs 407

11.2 Necessary and sufficient condition for a domain, which fails to havePompeiu property, to be a ball 41411.2.111.2.2

Introduction 414Proof 416

Bibliographical Notes 421

References 425

Index 441

FOREWORD

The importance of mathematics in the study of problems arising from the real world,and the increasing success with which it has been used to model situations rangingfrom the purely deterministic to the stochastic, is well established. The purpose of theset of volumes to which the present one belongs is to make available authoritative, upto date, and self-contained accounts of some of the most important and useful of theseanalytical approaches and techniques. Each volume provides a detailed introduction toa specific subject area of current importance that is summarized below, and then goesbeyond this by reviewing recent contributions, and so serving as a valuable referencesource.

The progress in applicable mathematics has been brought about by the extension anddevelopment of many important analytical approaches and techniques, in areas bothold and new, frequently aided by the use of computers without which the solution ofrealistic problems would otherwise have been impossible.

A case in point is the analytical technique of singular perturbation theory whichhas a long history. In recent years it has been used in many different ways, and itsimportance has been enhanced by it having been used in various fields to derivesequences of asymptotic approximations, each with a higher order of accuracy than itspredecessor. These approximations have, in turn, provided a better understanding ofthe subject and stimulated the development of new methods for the numerical solutionof the higher order approximations. A typical example of this type is to be found inthe general study of nonlinear wave propagation phenomena as typified by the studyof water waves.

xvi Foreword

Elsewhere, as with the identification and emergence of the study of inverse problems,new analytical approaches have stimulated the development of numerical techniquesfor the solution of this major class of practical problems. Such work divides naturallyinto two parts, the first being the identification and formulation of inverse problems,the theory of ill-posed problems and the class of one-dimensional inverse problems,and the second being the study and theory of multidimensional inverse problems.

On occasions the development of analytical results and their implementation bycomputer have proceeded in parallel, as with the development of the fast boundaryelement methods necessary for the numerical solution of partial differential equationsin several dimensions. This work has been stimulated by the study of boundary inte-gral equations, which in turn has involved the study of boundary elements, collocationmethods, Galerkin methods, iterative methods and others, and then on to their im-plementation in the case of the Helmholtz equation, the Lam equations, the Stokesequations, and various other equations of physical significance.

A major development in the theory of partial differential equations has been theuse of group theoretic methods when seeking solutions, and in the introduction ofthe comparatively new method of differential constraints. In addition to the usefulcontributions made by such studies to the understanding of the properties of solu-tions, and to the identification and construction of new analytical solutions for wellestablished equations, the approach has also been of value when seeking numericalsolutions. This is mainly because of the way in many special cases, as with similaritysolutions, a group theoretic approach can enable the number of dimensions occurringin a physical problem to be reduced, thereby resulting in a significant simplificationwhen seeking a numerical solution in several dimensions. Special analytical solutionsfound in this way are also of value when testing the accuracy and efficiency of newnumerical schemes.

A different area in which significant analytical advances have been achieved is inthe field of stochastic differential equations. These equations are finding an increasingnumber of applications in physical problems involving random phenomena, and oth-ers that are only now beginning to emerge, as is happening with the current use ofstochastic models in the financial world. The methods used in the study of stochasticdifferential equations differ somewhat from those employed in the applications men-tioned so far, since they depend for their success on the Ito calculus, martingale theoryand the Doob-Meyer decomposition theorem, the details of which are developed asnecessary in the volume on stochastic differential equations.

There are, of course, other topics in addition to those mentioned above that are ofconsiderable practical importance, and which have experienced significant develop-ments in recent years, but accounts of these must wait until later.

Alan JeffreyUniversity of Newcastle

Newcastle upon TyneUnited Kingdom

PREFACE

This book can be used for courses at various levels in ill-posed problems and inverseproblems. The bibliography of the subject is enormous. It is not possible to compile acomplete bibliography and no attempt was made to do this. The bibliography containssome books where the reader will find additional references. The author has usedextensively his earlier published papers, and referenced these, as well as the papers ofother authors that were used or mentioned.

Let us outline some of the novel features in this book.In Chapter 1 the statement of various inverse problems is given.In Chapter 2 the presentation of the theory of ill-posed problems is shorter and

sometimes simpler than that published earlier, and quite a few new results are included.Regularization for ill-posed operator equations with unbounded nonlinear operatorsis studied. A novel version of the discrepancy principle is formulated for nonlinearoperator equations. Convergence rate estimates are given for Backus-Gilbert-typemethods. The DSM (Dynamical systems method) in ill-posed problems is presentedin detail. The presentation is based on the authors papers and the joint papers of theauthor and his students. These results appear for the first time in book form. Papers[R216], [R217], [R218], [R220], [ARS3], [AR1] have been used in this Chapter.

In Chapter 3 the presentation of one-dimensional inverse problems is based mostlyon the authors papers, especially on [R221]. It contains many novel results, which aredescribed at the beginning of the Chapter. The presentation of the classical results,for example, Gelfand-Levitans theory, and Marchenkos theory, contains many novelpoints. The presentation of M. G. Kreins inversion theory with complete proofs is

xviii Preface

given for the first time. The Newton-Sabatier inversion theory, which has been in theliterature for more than 40 years, and was presented in two monographs [CS], [N], isanalyzed and shown to be fundamentally wrong in the sense that its foundations arewrong (cf. [R206]). This Chapter is based on the papers [R221], [R199], [R197],[R196], [R195], [R192], [R185]. One of the first papers on inverse spectral prob-lems was Ambartsumians paper (1929) [Am], where it was proved that one spectrumdetermines the one-dimensional Neumann Schrdingers operator uniquely. This re-sult is an exceptional one: in general one spectrum does not determine the potentialuniquely (see Section 3.7 and [PT]). Only 63 years later a multidimensional analog ofAmbartsumians result was obtained ([RSt1]). The main technical tool in this Chapterand in Chapter 5 is Property C, that is, completeness of the set of products of solu-tions to homogeneous differential equations. For partial differential equations this toolhas been introduced in [R87] and developed in many papers and in the monograph[R139]. For ordinary differential equations completeness of the products of solutionsto homegeneous ordinary equations has been used in different forms in [B], [L1]. Inour book Property C for ODE is presented in the form introduced and developed bythe author in [R196].

In Chapter 4 the presentation of inverse obstacle scattering problems contains manynovel points. The requirements on the smoothness of the boundary are minimal,stability estimates for the inversion procedure corresponding to fixed-frequency dataare given, the high-frequency inversion formulas are discussed and the error of theinversion from noisy data is estimated. Analysis of the currently used numerical methodsis given. This Chapter is based on [R83], [R155], [R162], [R164], [R167], [R167],[R171], [RSa].

In Chapter 5 a presentation of the solution of the 3D inverse potential scatteringproblem with fixed-energy noisy data is given. This Chapter is based on the seriesof the authors papers, especially on the paper [R203]. The basic concept used in theanalysis of the inverse scattering problem in Chapter 5 is the concept of Property C,i.e., completeness of the set of products of solutions to homogeneous partial differentialequations. This concept was introduced by the author ([R87]) and applied to manyinverse problems (see [R139] and references therein). An important part of the theoryconsists of obtaining stability estimates for the potential, reconstructed from fixed-energy noisy data (and from exact data). Error estimates for the Born inversion aregiven under suitable assumptions. It is shown that the Born inversion may fail whilethe Born approximation works well. In other words, the Born approximation maybe applicable for solving the direct scattering problem, while the Born inversion, thatis, inversion based on the Born approximation, may fail. The Born inversion is stillpopular in applications, therefore these error estimates will hopefully be useful forpractitioners.

The authors inversion method for fixed-energy scattering data, is compared withthat based on the usage of the Dirichlet-to-Neumann map. The author shows whythe difficulties in numerical implementation of his method are less formidable than thedifficulties in implementing the inversion method based on the Dirichlet-to-Neumannmap.

xix

A necessary and sufficient condition for a scatterer to be spherically symmetric isgiven ([R128]).

In Chapter 6 an example of non-uniqueness of the solution to a 3D problem ofgeophysics is given. It illustrates the crucial role of the uniqueness theorems in a studyof inverse problems. One may try to solve numerically such a problem, by a parameter-fitting, which is very popular among practitioners. But if the uniqueness result is notestablished, the numerical results may be meaningless. Some uniqueness theorems forinverse boundary value problem and for an inverse problem for hyperbolic equationsare established in this Chapter.

In Chapter 7, inverse problems of potential theory and antenna synthesis are brieflydiscussed. The presentation of the theory on this topic is not complete: there arebooks and many papers on antenna synthesis (e.g., [MJ], [ZK], [AVST], [R21], [R26])including nonlinear problems of antenna synthesis [R23], [R27]).

Chapter 8 contains a discussion of non-overdetermined problems. These are, roughlyspeaking, the inverse problems in which the unknown function depends on the samenumber of variables as the data function. Examples of such problems are given. Most ofthese problems are open: even uniqueness theorems are not available. Such a problem,namely, recovery of an unknown coefficient in a Schrdinger equation in a boundeddomain from the knowledge of the values of the spectral function on theboundary is discussed under the assumption that all the eigenvalues are simple, that is,the corresponding eigenspaces are one-dimensional. The presentation follows [R198].

In Chapter 9 the theory of the inversion of low-frequency data is presented. Thistheory is based on the series of authors papers, starting with [R68], [R77], and usesthe presentation in [R83] and [R139]. Almost all of the results in this Chapter are fromthe above papers and books.

Chapter 10 is a summary of the authors results regarding the theory of wavescattering by small bodies of arbitrary shapes. These results have been obtained ina series of the authors papers and are summarized in [R65], [R50]. The solutionof inverse radiomeasurements problem ([R33], [R65]) is based on these results. Also,these results are used in the solution of the problem of finding small subsurface inho-mogeneities from the scattering data, measured on the surface. The solution to thisproblem can be used in modeling ultrasound mammography, in finding small holes inmetallic objects, and in many other applied problems.

In Chapter 11 the classical Pompeiu problem is presented following the papers[R177], [R186].

The author thanks several publishers of his papers, mentioned above, for the per-mission to use these papers in the book.

There are many questions that the author did not discuss in this book: inverse scat-tering for periodic potentials and other periodic objects, such as gratings, periodicobjects, (see, e.g., [L] for one-dimensional scattering problems for periodic potentials),the Carleman estimates and their applications to inverse problems ([Bu2], [H], [LRS]),the inverse problems for elasticity and Maxwells equations ([RK], [Ya]), the methodsbased on controllability results ([Bel]), problems of tomography and integral geom-etry ([RKa], [R139]), etc. Numerous parameter-fitting schemes for solving various

xx Preface

engineering problems are not discussed. There are many papers published, which useparameter-fitting for solving inverse problems. However, in most cases there are noerror estimates for parameter-fitting schemes for solving inverse problems, and onecannot guarantee any accuracy of the inversion result. In [GRS] the concept of sta-bility index is introduced and applied to a parameter-fitting scheme for solving aone-dimensional inverse scattering problem in quantum physics. This concept allowsone to get some idea about the error estimate in a parameter-fitting scheme.

The applications of inverse scattering to integration of nonlinear evolution equationsare not discussed as there are many books on this topic (see e.g., [M], [FT] and referencestherein).

1. INTRODUCTION

1.1 WHY ARE INVERSE PROBLEMS INTERESTING ANDPRACTICALLY IMPORTANT?

Inverse problems are the problems that consist of finding an unknown property of anobject, or a medium, from the observation of a response of this object, or medium,to a probing signal. Thus, the theory of inverse problems yields a theoretical basis forremote sensing and non-destructive evaluation. For example, if an acoustic plane waveis scattered by an obstacle, and one observes the scattered field far from the obstacle,or in some exterior region, then the inverse problem is to find the shape and materialproperties of the obstacle. Such problems are important in identification of flyingobjects (airplanes missiles, etc.), objects immersed in water (submarines, paces of fish,etc.), and in many other situations.

In geophysics one sends an acoustic wave from the surface of the earth and collects thescattered field on the surface for various positions of the source of the field for a fixedfrequency, or for several frequencies. The inverse problem is to find the subsurfaceinhomogeneities. In technology one measures the eigenfrequencies of a piece of amaterial, and the inverse problem is to find a defect in this material, for example, ahole in a metal. In geophysics the inhomogeneity can be an oil deposit, a cave, a mine.In medicine it may be a tumor, or some abnormality in a human body.

If one is able to find inhomogeneities in a medium by processing the scattered fieldon the surface, then one does not have to drill a hole in a medium. This, in turn, avoidsexpensive and destructive evaluation. The practical advantages of remote sensing arewhat makes the inverse problems important.

2 1. Introduction

1.2 EXAMPLES OF INVERSE PROBLEMS

1.2.1 Inverse problems of potential theory

Suppose a body with a density generates gravitational potential

Is it possible to find given the potential u (x) for far awayfrom D? A point mass m and a uniformly distributed mass m in a ball of radius aproduce the same potential Thus, it is not possible tofind uniquely from the knowledge of u in However, if one knows a priorithat in D, then it is possible to find D from the knowledge of u (x) inprovided that D is, for example, star-shaped, that is, every ray issued from some interiorpoint intersects the boundary of D at only one point.

1.2.2 Inverse spectral problems

Let be the Sturm-Liouville operator defined by the Dirichlet bound-ary conditions as self-adjoint operator in and be its eigen-values. To what extent does the set of these eigenvalues determine q (x )? Roughlyspeaking, one spectrum, that is, the set determines half ofq (x), in the sense that if q (x) is known on then one spectrum determinesuniquely q (x) on A classical result due to Borg [B]and Marchenko [M] says thattwo spectra uniquely determine the operator i.e., the potential q and the boundaryconditions at x = 0 and x = 1 of the type andwhere and are constants, and one assumes that the two spectra correspond to thesame and two distinct The author (see [R196]) asked the following question: ifq (x) is known on the segment [b, 1], 0 < b < 1, then what part of the spectrum oneneeds to know in order to uniquely recover q (x) on [0, b]? It is assumed that q is realvalued: and

Let be the spectral measure of the self-adjoint operator l. This notion is definedin Chapter 3. The inverse spectral problem is: given find q (x), and the boundaryconditions, characterizing

Similar problems can be formulated in the multidimensional cases, when a boundeddomain D plays the role of the segment [0, 1], the role of the spectral data is playedby the eigenvalues and the values on the boundary S of D of the normal derivativesof the normalized eigenfunctions, One may choose other spectral data.

1.2.3 Inverse scattering problems in quantum physics; finding the potentialfrom the impedance function

Consider the Dirichlet operatorin Denote by the Jost solution,

by the Jost function, by the I-function (impedancefunction), and by the scattering data (see Chapter 3).

3The inverse problem of quantum scattering on the half-axis consists of finding q (x), givenIt was studied in [M].

The inverse problem of finding q (x), given is of interest in many appli-cations. The I-function has the physical meaning of the impedance function, it isthe ratio in the problem of electromagnetic wave falling perpendicularly onto theearth, when the dielectric permittivity and conductivity of the earth depend on thevertical coordinate only. One can prove that coincides with the Weyl function

(Chapter 3). It turns out that known determines uniquely q (x), andone can explicitly calculate and given ([R196]).

1.2.4 Inverse problems of interest in geophysics

There are many inverse problems of interest in geophysics. A typical one consists offinding an unknown inhomogeneity in the velocity profile (refraction coefficient) from the scatteredacoustic field measured on the surface of the earth and generated by a point source, situated onthe surface of the earth at varying positions.

Its mathematical formulation (in a simplified form) is:

where is the known background refraction coefficient,where supp v(x) is the support of v,

and v is an inhomogeneity in the refraction coefficient (or in the velocity profile), uis the acoustic pressure, u satisfies the radiation condition at infinity (or the limitingabsorption principle). An inverse problem of geophysics consists of finding the functionv from the knowledge of the scattered field on the surface of the Earth, that is from

known for all at a fixed or for allwhere is a small number (the case of low-frequency surface data) (cf [LRS],[Ro], [R83], [R139]).

Another problem is to find the conductivity of the medium from the measurementsof the electromagnetic waves, scattered by a source that moves in a borehole along thevertical line. ([R83], [R139])

1.2.5 Inverse problems for the heat and wave equations

A typical inverse problem for the heat equation

is to find q (x) from the flux measurements: The extra data (measureddata), allow one to find q ( x ) . Another inverse problem is to find theunknown conductivity from boundary measurements. For example, let

Can one findgiven a(t) and

4 1. Introduction

Consider the inverse conductivity problem: let inon where N is the outer unit normal to S, the extra data is

the flux g at the boundary. Suppose that the set is known. Can onedetermine uniquely? Here D is a bounded domain with a sufficiently smoothboundary S, and is the Sobolev space. In applications in medicine, f is theelectrostatic potential, which can be applied to a human chest, and g is the flux ofthe electrostatic field, which can be measured. If one can determine from thesemeasurements then some diagnostic information is obtained ([R157], [R139], [R131],[R103], [Gro], [LRS], [Ro], [Is1]).

There are many inverse problem for the wave equation. One of them is to find thevelocity c (x) in the equation at t = 0, u = 0on S, given the extra data on S for a fixed and all t > 0, or for varying on S,and where T > 0 is some number. ([LRS], [Ro], [RRa], [RSj]).

1.2.6 Inverse obstacle scattering

Let be bounded domain with a Lipschitz boundary be theexterior domain, be the unit sphere in The scattering problem consists offinding the scattering solution, i.e., the solution to the problem

The coefficient A is called the scattering amplitude.Existence and uniqueness of the solution to (1.2.2)(1.2.3) are proved in [RSa]

without any assumption on the smoothness of the boundary. If the Neumann boundarycondition

is used in place of (1.2.3), then the existence and uniqueness of the solution to (1.2.2)(1.2.3N) are proved in [RSa] under the assumption of compactness of the embedding

where R > 0 is such thatand is the Sobolev space. See also [GoR]. The inverse obstacle

scattering problem consists of finding S and the boundary condition on S, givenin the following cases:

(1) either at a fixed for all and all or,(2) at a fixed for all and running through open subsets of or,(3) for fixed and and all

Uniqueness of the solution of the first inverse problem is proved by M. Schiffer(1964), (see [R83]) of the second by A. G. Ramm (1986) (see [R83]), and the third

5problem is still open. See also [R154], [R155], [R162], [R159], [R164], [R167],[R171], [CK].

One may consider a penetrable layered obstacle, and ask if the scattering amplitudeat a fixed allows one to determine the boundaries of all the layers uniquely,and the constant velocity profiles in each of the layers. See [RPY] for an answer to thisquestion.

1.2.7 Finding small subsurface inhomogeneities from the measurementsof the scattered field on the surface

Suppose there are few small, in comparison with the wave-length, holes in the metallicbody. A source of acoustic waves is on the surface of the body, and the scattered fieldis measured on the surface of the body for various positions of the acoustic source, ata fixed frequency.

The inverse problem is to find the number of the small holes, their locations, andtheir volume. A similar problem is important in medicine, where the small bodies arethe cancer cells to be found in the healthy tissue of a humans body. In the ultrasoundmammography modeling, one deals with the tissue of a womans breast ([R193],[GR1]).

1.2.8 Inverse problem of radiomeasurements

Suppose a complicated electromagnetic field distribution (E, H) exists in the apertureof a mirror antenna. For many practical reasons one wants to know this distribution.Let be the field scattered by a small probe placed at a point x in the aperture ofthe antenna. Given the shape and electromagnetic constants and of the probe,the inverse problem of radiomeasurements consists of finding (E(x), H(x)) from theknowledge of (See [R65] for a solution to this problem).

1.2.9 Impedance tomography (inverse conductivity) problemThis problem was briefly mentioned in Section 1.2.5.

1.2.10 Tomography and other integral geometry problems

Define whereis the unit sphere in The function is called the Radon

transform of f. The function f can be assumed piecewise-continuous and absolutelyintegrable over every plane so that the Radon transform would be well definedin the classical sense. But in fact, one can define the Radon transform for much largersets of functions and on distributions [R170], [RKa], [Hel].

Given one can uniquely recover f (x) provided, for example, thator where is the weighted space with the norm

Practically interesting questions are:

(a) How are singularities of f and related?(b) Given the noisy measurements of f at a grid, how does one find the discontinuities

of f ?

6 1. Introduction

A grid is a set of points whereThe noisy measurement are where

are identically distributed, independent random variables with zero mean value anda finite variance See [R176], [RKa] for a detailed investigation of the aboveproblem. An open problem is: what are the minimal assumptions on the growth of f (x)at infinity that guarantee the injectivity of the Radon transform? There is an exampleof a smooth function such that for all p, and[RKa].

In many applications one integrates f not over the planes but over some otherfamily of manifolds. The problem of integral geometry is to recover f from theknowledge of its integral over a family of manifolds.

For example, if the family of manifolds is a family of spheres of various radii r > 0and centers s running over some surface S, then arethe spherical means of f, and the problem is to recover f from the knowledgeof and

Conditions on S that guarantee the injectivity of the operator M are given in [R211],where some inversion formulas are also derived.

1.2.11 Inverse problems with incomplete data

Suppose that not all the scattering data in Section 1.2.3 are given, for example,is given, but and J are unknown.

In general, one cannot recover a from these incomplete data. However,if one knows a priori that q (x) has compact support, or thenthe data alone determine q (x) uniquely. Such type of inverse problemswe call inverse problems with incomplete data. The incompleteness of the datais remedied by the additional a priori assumption about q (x), so, in fact, the data arecomplete in the sense that q (x) is determined uniquely by these data.

Another example of an inverse problem with incomplete data, is recovery ofq (x) = 0 for x < 0, from the

knowledge of the reflection coefficient in the full-axis (full-line) scatteringproblem:

The coefficients and are reflection and transmission coefficients, respecti-vely.

A general cannot be uniquely recovered from the knowledge ofalone: one needs to know additionally the bound states and norming constants torecover q uniquely. However if one knows a priori that q (x) = 0 for forexample, for x < 0, then q (x) is uniquely determined by alone.

71.2.12 The Pompeiu problem, Schiffers conjecture, and inverse problemof plasma theory

Let Assume

where SO(n) is the group of rotations, and is a bounded domain. The problem(going back to Pompeiu (1929)) is to prove that (1.2.4) implies that D is a ball.Originally Pompeiu claimed that (1.2.4) implies that f = 0, but this claim is wrong.References related to this problem are given in [R186], [R177], [Z]. One can provethat (1.2.4) holds iff (= if and only if) for all and some

where and Iff and

some then the overdetermined problem

has a solution.The Schiffers conjecture is: if D is a bounded connected domain homeomorphic

to a ball, and

then D is a ball.The Pompeiu problem in the form (1.2.4) is equivalent to the following conjecture:

if

and D is homeomorphic to a ball, then D is a ball.An inverse problem of plasma theory consists of the following.Let

where u is a non trivial solution to (1.2.8), (i.e., if f (0) = 0 then and letthe extra data (measured data) be the value Assume thatf (u) is an entire function of u. The inverse problem, of interest in plasma theory,is: given can one recover f (u) uniquely. Even for

j = 0, 1, the problem is open (cf [Vog]).

8 1. Introduction

1.2.13 Multidimensional inverse potential scattering

Let

where is given, and The coefficient is called the scatteringamplitude, and the solution to (1.2.9)(1.2.10) is called the scattering solution. Thedirect scattering problem is: given q, and find and, inparticular, This problem has been studied in great detail (see, e.g., [CFKS],[[R121], Appendix]) under various assumptions on q ( x ) . We assume that

for and often we assume additionally that

The inverse scattering problem (ISP) consists of finding q ( x ) , givenConsider several cases:

(1) A is given for all and all(2) A is given for all and a fixed(3) A is given for a fixed all and all(4) is given for all and all (back scattering data).

In case (1) uniqueness of the solution to ISP has been established long ago, and followseasily from the asymptotics of A as An inversion formula based on high-energy asymptotics of A is known (Born inversion), (cf [Sai]). In case (2) the uniquenessof the solution to ISP is proved by Ramm [R109], [R100], (see also [R105], [R112],[R114], [R115], [R120], [R125], [R130], [R133], [R140], [R142], [R143], [RSt2],[R203]), an inversion formula for the exact data is derived in [R109], [R143], aninversion formula for the noisy data is derived by in [R143], and stability estimates forthe inversion formulas for the exact and noisy data are derived by in [R143], [R203].In case (3) and (4) uniqueness of the solution to ISP is an open problem, but in thecase (4) uniqueness holds if one assumes a priori that q is sufficiently small. A genericuniqueness result is given in [St1]. See also [StU].

1.2.14 Ground-penetrating radar

Let the source of electromagnetic waves be located above the ground, and the scatteredfield be observed on the ground. From these data one wants to get information aboutthe properties of the ground. Mathematical modeling of this problem is based on theMaxwell equations

where is the vertical coordinate,is the region of the ground, is the source, f (t)

9describes the shape of the pulse of the current j along a wire going along theaxis at the height above the ground. Assume for

(in the air), for f (t) = 0 for t < 0and t > T, and are dielectric and magnetic constants,for Differentiate the second equation (1.2.1) with respect to t, and get

LetThen

where for ground-penetrating radar. The ground-penetrating radar inverse problem is: given

find andOne may use the source which is a current along a

loop of wire, is the unit vector in cylindrical coordinates. In this case, one looks forE of the form: and from (1.2.11) one gets:

Let where is the Bessel function. SetThen u solves (1.2.12), and the inverse problem is the same as above

([R185]).

1.2.15 A geometrical inverse problem

Let

Here D is domain homeomorphic to an annulus, is its inner boundary, is itsouter boundary. The geometrical inverse problem is: given and find

10 1. Introduction

One can interpret the data as the Cauchy data on for an electrostatic potentialu, and then is the surface on which the potential is vanishing if or thecharge distribution is vanishing if ([R139]).

1.2.16 Inverse source problems

(1) Inverse source problems in acousticsLet in f = 0 for satisfies the radiation condition

uniformly in Define the radiation pat-tern A by the formula: as Then

The inverse source problems are:(i) Given and find f (x).

(ii) Given and a fixed find f (x).Clearly, by (1.2.15) problem (i) has at most one solution, but an a priorigiven function may be not of the form (1.2.15): the right-hand sideof (1.2.15) is an entire function of exponential type of the vectorProblem (ii), in general, may have many solutions, since may vanishfor all at some

(2) Inverse source problem in electrodynamicsConsider Maxwells equations (1.2.11) in and assume j = 0 for Theradiation condition for (E, H) is:

where E and H in (1.2.11) are assumed monochromatic with time de-pendence, and where are the constantvalues of and near infinity.

The inverse source problem is: given find j. Again, one should specifyfor what and the function is known. One can derive the relation betweenA and j. Namely, assuming and constants, and j smooth and compactly supported,one starts with the equations

then gets

11

then

so

Thus where [a, b] is the vector product. So

andIt is now clear, that even if A is known for all and all vector J is

not uniquely determined, but only its component orthogonal to is determined.Therefore, the solution to the inverse source problem in electrodynamics is not uniqueand may not exist, in general. The antenna synthesis problems are inverse source-typeproblems of electrodynamics. For example, if j is the current along a linear antenna(which is a wire along is the length of the antenna,

then

so is determined uniquely by the data A. Findingwhich produces the desired diagram is the problem of linear antenna

synthesis. There is a large body of literature on this subject.Let

forThe inverse source problem is: given find f (x, t),

The questions mentioned in this subsection were discussed in many papers andbooks ([AVST], [MJ], [ZK], [R11], [R21], [R26], [R27], [R28], [R73], [Is2]).

Let and Then

12 1. Introduction

1.2.17 Identification problems for integral-differential equations

Consider a Cauchy problem.

whereis a closed, linear, densely defined in a Hilbert space H, operator,

D(A) is its domain of definition, Assumethat h (t) is unknown, and the extra data are given for

The inverse problem is: given find h(t).See [LR].

1.2.18 Inverse problem for an abstract evolution equation

Consider a Cauchy problem with the extra data(measured data) Here A(t) is a one-parametric family of closed,densely defined, linear, operators on a Banach space X, which generates an evolutionfamily is a given function on [0, T] with values in X,and is an unknown scalar function. The inverse problem is: given f(t), andw, find (see [RKo]).

In applications may be a control function which should be chosen so that themeasured data are reproduced.

1.2.19 Inverse gravimetry problem

Let be the of the gravitational field generated by some masses,located in the region Assume that the values u(x, 0) := f ( x ) are known,

in the region and u(x, h) := g(x). Thenis the equation for g. The inverse gravimetry problem is: given f, find g.

See [VA], [RSm1].

1.2.20 Phase retrieval problem (PRP)Let The PRP consists of finding arg givenClearly, the solution to this problem is not unique: produces thesame Under suitable assumptions on f ( x ) and one can get uniquenessresults for PRP.

See [KST], [R139].

1.2.21 Non-overdetermined inverse problems

Formally we call an inverse problem non-overdetermined if the unknown function,which is to be found, depends on the same number of variables as the data. For example,problems in cases (1) and (2) in Section 1.2.13 are overdetermined, while in cases (3)and (4) they are not overdetermined. In multidimensional inverse scattering problemsuniqueness of the solution is an open problem for most of the non-overdetermined

13

problems. For example, in Section 1.2.13 case (3) is a non-overdetermined problem,and uniqueness of its solution is an open problem.

Recently Ramm ([R198]) proved that the spectral data and alldetermine q (x) uniquely provided that all the eigenvalues are simple. Here

in is thekernel of the resolution of the identity of the selfadjoint Neumann operator

inThe above inverse problem is not overdetermined, because depends on

three variables in and q (x) is also a function of three variables in It is anopen problem to find out if this result remains valid without the assumption about thesimplicity of all the eigenvalues.

1.2.22 Image processing, deconvolution

In many applications one is interested in the following inverse problem: given the propertiesof a linear device and the output signal, find the input signal. By the properties of a lineardevice one means its point-spread function (scattering function) or transfer function.For example, Given and f (x), one wants to find

In practice the output signal f (x) is noisy, i.e., is given in place of f(x),where the norm depends on the problem at hand.

See [BG], [RSm6], [RG].

1.2.23 Inverse problem of electrodynamics, recovery of layered mediumfrom the surface scattering data

There are many inverse problems arising in electrodynamics. If (1.2.11) are the gov-erning equations,

where e is a constant vector, and are known constants:and respectively, outside a bounded domain and

the data are measured on the surface of the Earth forvarious orientations of e then the inverse problem is to determine and fromthe above data.

See [RSo], [RK].

1.2.24 Finding ODE from a trajectoryLet and are constants, u = u ( t ) , Can one find

and uniquely? In general, the answer is no. A trivial example is u ( t ) = 0. Whattrajectory allows one to find and uniquely? Suppose

14 1. Introduction

Then the system of equations

for finding and is uniquely solvable, so that (1.2.20) guarantees that u(t),determines and uniquely. More generally, consider a system

where is a constant matrix, u (0) = v. If there arepoints such that the system is alinearly independent system of vectors, then u (t) determines the matrixuniquely. The reader can easily prove this. One can find more details in [Den].

1.3 ILL-POSED PROBLEMS

Why are ill-posed problems important in applications? How are they related to inverseproblems? Let

where X and Y are Banach spaces, or metric spaces, and A is a nonlinear operator, ingeneral. Problem (1.3.1) is called well-posed if A is a homeomorphism of X onto Y .In other words, the solution to (1.3.1) exists for any is unique, and depends onf continuously, so that is a continuous map. If some of these conditions do nothold, then the problem is called ill-posed.

Ill-posed problems are important in many application, in which one may reduce aphysical problem to equation (1.3.1) where A is not boundedly invertible. For example,consider the equation

If then A is not boundedly invertible: A is injective, its rangebelongs to the Sobolev space is an unbounded operator in

If noisy data are given, then may be not in therange of A. A practically interesting problem is: can one find an operator such that

as In other words, can one estimate stably, givenandAny Fredholm first-kind integral equation with linear compact operator

is of the form (1.3.1). Such an operator in an infinite-dimensional space cannot haveclosed range and cannot be boundedly invertible. Since many inverse problems canbe reduced to ill-posed equations (1.3.1), these inverse problems are ill-posed. That ishow Ill-posed problems are related to inverse problems. Methods for stable solution ofIll-posed problems are developed in Chapter 2. The literature on Ill-posed problemsis enormous ([IVT], [EHN], [Gro], [TLY], [VV], [VA], [R58]).

15

1.4 EXAMPLES OF ILL-POSED PROBLEMS

1.4.1 Stable numerical differentiation of noisy data

This example has been mentioned in Section 1.3. Methods for stable numerical dif-ferentiation of noisy data are given in Chapter 2.

In navigation a ship receives a navigation signal which is a univalent function f (x)(that is, a smooth function which has precisely one point of maximum), and thecourse of the ship is determined by this point. The function f is observed in anadditive noise. Given noisy data one wants to find A possibleapproach to this problem, is to search for a point at which One can see froma simple example that small perturbations of f can lead to large perturbations oflet Then No matter how small is,one can choose so large that will take arbitrary large values at somepoints x.

1.4.2 Stable summation of the Fourier series and integralswith randomly perturbed coefficients

Let be an orthonormal basis of a Hilbert space H and L be a linear system(a linear operator) such that where are some numbers. Due to theinner noise in L, one observes the noisy output where is thenoise. Thus, if the input signal is the output isOne has noisy Fourier coefficients and one wants to recover the function

If one has a Fourier integral, one can formulate a similar problem. To see that thisproblem is ill-posed, in general, let us take denote and assumethat is playing the role of noise, Then one has the problem: given thenoisy Fourier coefficients find This problemis ill-posed because the series may diverge. For example, ifand then the series diverges. Similarly, if isthe Fourier transform of a function and is the noisy data,

then the problem is to calculate f (x) with minimal error,given noisy data The notion of minimal error should be specified.

1.4.3 Solving ill-conditioned linear algebraic systems

Let be a linear operator such that its condition numberis large. Then the linear algebraic system Au = f can be considered practi-

cally as an ill-posed problem, because small perturbation of the data f may lead toa large perturbation in the solution u. One has:

so that the relative error in the data may result in rela-tive error in the solution. In the above derivation we use the inequality

Methods for stable solution of ill-conditioned algebraic systems are givenin Chapter 2.

16 1. Introduction

1.4.4 Fredholm and Volterra integral equations of the first kind

If or and is a continuouskernel in D := [a, b] [a, b], then the operators A and V are compact in

these operator are not boundedly invertible in H. Therefore problems Au =f and Vu = f are ill-posed.

1.4.5 Deconvolution problems

These are problems, arising in applications: an input signal u generates an output signalf by the formula

Often one has:

The deconvolution problem consists of finding u, given f and For (1.4.2)the identification problem is of practical interest: given u (t) and f (t), find A (t). Thefunction A(t) characterizes the linear system which generates the output f (t) givesthe input u(t).

Mathematically the deconvolution problems are the problems from Section 1.4.4.

1.4.6 Minimization problems

Consider the minimization problem Suppose that isthe infimum of If f is perturbed, that is,

then the infimum of may be not attained, or it may be attainedat an element which is far away from Thus the map may be not contin-uous. In this case the minimization problem is ill-posed. Such problems were studied[Vas].

1.4.7 The Cauchy problem for Laplaces equation

Claim 1. The Cauchy problem for Laplaces equation is an ill-posed problem.

Consider the problem: in the half-planeIt is clear that

solves the above problem, and this solution is unique (by the uniqueness of the solutionto the Cauchy problem for elliptic equations). This example belongs to J. Hadamard,and it shows that the Cauchy data may be arbitrarily small (take while thesolution tends to infinity, as at any point Thus, theclaim is verified (cf. [LRS], [R139]).

17

1.4.8 The backwards heat equation

Consider the backwards heat equation problem:

Given v(x), one wants to find u (x, 0) := w (x).By separation of variables one finds

Therefore, pro-vided that this series converges, in that is, provided that

This cannot happen unless decays sufficiently fast. Therefore the backwards heatequation problem is ill-posed: it is not solvable for a given v (x) unless (1.4.3) holds,and small perturbations of the data v in may lead to arbitrary largeperturbations of the function w(x), but also may lead to a function v for which thesolution u(x, t) does not exist for t < T (cf. [LRS], [IVT]).

2. METHODS OF SOLVING ILL-POSED PROBLEMS

2.1 VARIATIONAL REGULARIZATION

2.1.1 Pseudoinverse. Singular values decomposition

Consider linear Equation (1.3.1). Let be a linear closed operator, D(A)and R(A) be its domain and range, and be Hilbert spaces, N(A) :={u : Au = 0}, is the adjoint operator, the bar stands for theclosure, and is the orthogonal sum. If A is injective, i.e., N(A) = {0}, and surjective,i.e. and then the inverse operator is defined on

I is the identity operator. A closed, linear, defined on all ofoperator, is bounded, so A is an isomorphism of onto if it is injective, surjective,and

If P is the orthoprojector onto N(A) in and Q is the or-thoprojector onto then one defines a pseudoinverse (generalized inverse)

Thus,and for The operator is bounded

iff If i.e., for some then the problemhas a solution every element is also a so-

lution, and there is a unique solution with minimal norm, namely the solutionsuch that If then the infimum of isnot attained. If A is bounded and then the element solvesthe equation and is the minimal norm solution to this equation,

20 2. Methods of solving ill-posed problems

i.e., Indeed, if and thenConversely, if and with then

and Thus, if One canprove the formula: where is a regulariza-tion parameter (see Section 2.1.2) and

Let us define the singular value decomposition. Let be a linear com-pact operator, is a compact selfadjoint operator,

are called s -values of A. If and thenThus,

If is arbitrary, then(PicardsThus, an element

test). If then where are eigenvalues of Then

If dim then A can be written aswhere A is an m n matrix, U and V are unitary matrices (n n and m m,respectively), whose columns are vectors and respectively, and S is an m nmatrix with the diagonal elements r is the rank of the matrix A,and other elements of S are zeros. The matrix can be calculated by the formula

where is an n m matrix with diagonal elementsand other elements of are zeros.

2.1.2 Variational (Phillips-Tikhonov) regularizationAssume is linear, is not nec-essarily in R(A). The problem Au = f is assumed ill-posed (cf. Sec. 1.3). Considerthe problem:

where is a parameter.

Theorem 2.1.1. Assume Au = f, and Then:

(i) The minimizer of (2.1.1) does exist and is unique(ii) If and satisfies the condition as then

where

21

Proof. Functional (2.1.1) is quadratic. A necessary and sufficient condition for its min-imizer is the Eulers equation:

which has a unique solution Claim (i) is proved. Onehas if

Thus Below stands for various positive constants.Choose so that as and let Thenso as and This implies and we claimthat This claim we prove later. Thus Let usprove that One has

Here the estimate and therelation as were used. To prove the first estimate,one uses the formula: and thepolar representation of A* yields, where V is an isometry,One has where the spectralrepresentation for T was used.Let us prove the second relation:

N(A), andwhere P is the orthoprojector onto

is the resolution of the identity of the self-adjoint operator B (see[KA]).

If thenFinally, let us prove the claim used above.

Lemma 2.1.2. If B is a monotone hemicontinuous operator in a Hilbert space H, D(B) = H,then

In our proof above, is a linear operator, so B is monotone, that is,Recall that a nonlinear operator A is called

hemicontinuous, if is a continuous function of for any

Proof of Lemma 2.1.2. Clearly, If then that is,monotone operator is w -closed. Indeed, if B is monotone, then

for any Passing to the limit and usinghemicontinuity of B, one gets: This impliesso Lemma 2.1.2 is proved.

The claim is proved for nonlinear monotone oper-ators in Theorem 2.1.6 below. Theorem 2.1.1 is proved.


2.1.3 Discrepancy principle

Theorem 2.1.1 gives an a priori choice of which guarantees convergenceAn a posteriori choice of is given by Theorem 2.1.3 below.

Theorem 2.1.3. (Discrepancy principle). Assume If is the root of theequation

then

Proof. First, let us prove that equation (2.1.3) has a unique solution. Write this equationas

where is the resolution of the identity of the selfadjoint operatorand the commutation formula was used. One checksthis formula easily. If then If then

where is the orthoprojector on Indeed,because and since

and Thus, if then equation (2.1.3) has a solution. Thissolution is unique because is a monotone increasing function of for each fixed

Now let us prove One hasSince it follows that Therefore (*)

If then one can select a weakly convergent se-quence as In the proof of Theorem 2.1.1 it was proved thatwhere is the unique minimal-norm solution of the equation By thelower semicontinuity of the norm in H, one has Togetherwith (*), one gets This and the weak convergence imply

Our proof is based on the following useful result.

Theorem 2.1.4. If and then

Proof. If then Also one hasThus, and

as

2.1.4 Nonlinear ill-posed problems

Lemma 2.1.5. Assume that A in (1.3.1) is a closed, nonlinear, injective map. If K is acompactum, then the inverse operator is continuous on A(K).

Proof. Since A is injective, is well-defined on A(K ). LetThen where is a subsequence denoted again Since A is

closed, and imply and, by the injectivity ofA, one has Lemma 2.1.5 is proved.

Claim: Let us assume that is monotone, continuous, D(A) = H, A(u) = f,and Then

Proof. Indeed, Multiply this equation by and usethe monotonicity of A, to get Thus, Let Selecta sequence, denoted again by such that as ThenSince A is monotone, it is w-closed (see the proof of Theorem 2.1.1), soand is the minimal norm solution to equation (1.3.1).

Theorem 2.1.6. If is monotone and hemicontinuous, if D(A) = H, ifwhere is the minimal-norm solution to A(u) = f, and if

then the minimal-norm solution to (1.3.1) is unique and

Proof. We have Thus,Let Then and

Thus, This and the weak convergence imply strongconvergence as in Theorem 2.1.3.

The minimal norm solution to (1.3.1) is unique if A is monotone and continuous,because in this case the set of solutions N is convex and closed. Its closedness is obvious, ifA is continuous. Its convexity follows from the monotonicity of A and the followinglemma:

Lemma 2.1.7 (Minty). If A is monotone and continuous, then (a)is equivalent to (b)

Proof. If (a) holds, then and (b) holds by the monotonicity of A. If (b) holds,then take where w is arbitrary, and get

Take and get Thus, A(u) = f, and (a) holds.Lemma 2.1.7 is proved.

To prove that N is convex, one assumes that and derives thatIndeed, if then, by Lemma 2.1.7,

Thus,Thus,

23


To prove uniqueness of the minimal norm element of a convex and closed set N in aHilbert space, one assumes that there are two such elements, and Then

and so that any elementof the segment joining and has minimal norm m. Since Hilbert space is strictlyconvex, this implies Indeed, take t = 1/2. Then

So Thus, Theorem 2.1.6is proved.

Consider the equation

Theorem 2.1.8. Assume that A is monotone and continuous, and equation (1.3.1) has asolution. If and as then the unique solution to (2.1.4)converges strongly to u, the unique solution to (1.3.1) of minimal norm.

Proof. Because A is monotone and the equation has a so-lution, and this solution is unique. Let solve (2.1.4), and Onehas By Theorem 2.1.6, Letus prove We haveMultiply this by and use the monotonicity of A to get

This implies if Theorem 2.1.8is proved.

2.1.5 Regularization of nonlinear, possibly unbounded, operator

Assume that :

(1)(2)

(3)(4)

is a closed, injective, possibly nonlinear, map in Banach space X.is a functional such that the set is precompact in X for any

constantEquation (1.3.1) has a solution

The last assumption can be replaced in some cases when A is an unboundedoperator, by the assumption.

Define the functional where is a parameter,Consider the minimization problem:

Let where is the smallest integer satisfying the in-equality Denote One has

and By assumption (2), as one can select a

convergent subsequence, denoted again such thatThus, A(u) = f by the closedness of A, and by the injectivity of A. Since thelimit of any subsequence is the same, namely it follows thatWe have proved:

Theorem 2.1.9. If (1.3.1) has a solution, then, under the assumptions (1)(4) (orany sequence such that converges strongly to the solution of (1.3.1)as

Remark 2.1.10. In the proof of Theorem 2.1.9 we do not need existence of the minimizerof the function (2.1.5).

2.1.6 Regularization based on spectral theory

Assume that A in (1.3.1) is a linear bounded operator, andi.e., is the unique minimal-norm solution.

Lemma 2.1.11. Solvable equation (1.3.1) with bounded linear operator A is equivalent tothe equation

Proof. If u solves (1.3.1), apply A* to (1.3.1) and get (2.1.6), so u solves (2.1.6).If u solves (2.1.6) and (1.3.1) is solvable, i.e., then

and This implies so Au = f. Thus usolves (1.3.1).

Equation (2.1.6) is a solvable equation with monotone, continuous operator, so The-orem 2.1.6 is applicable and yields the following theorem:

Theorem 2.1.12. If as and isthe minimal-norm solution to (1.3.1), then where is the uniquesolution of the equation

Lemma 2.1.13. Consider the elements where is theresolution of the identity of B = A* A, does not dependon s and and is apiecewise-continuous function. Let so that as Then

25


Proof. If then and Thus,where

and One hasThus

as If the rate of decay of and the rate of growthof can be estimated, then a quasioptimal choice of can be made byminimizing with respect to for a fixed

Remark 2.1.14. We have used the spectral theorem for a selfadjoint operator B, namelythe formula where is the resolution of the identity of B,

Remark 2.1.15. Similarly, one can use the theory of spectral operators in place of the spectraltheory of selfadjoint operators, in particular Riesz bases formed by the root vectors.

2.1.7 On the notion of ill-posedness for nonlinear equations

If A is a linear operator, then problem (1.3.1) is ill-posed if either oror R(A) is not closed, i.e. is unbounded. If A is nonlinear and Frechet

differentiable, then there are several possibilities. If is boundedly invertible atsome u, then A(u) is a local homeomorphism at this point, but it may be not a globalhomeomorphism. If is not boundedly invertible, this does not imply, in general,that A is not a homeomorphism. For example, a homeomorphism A(u) may havea compact derivative, so its linearization yields an ill-posed problem. On the otherhand, A(u) may be compact, so (1.3.1) is an ill-posed problem, but may be afinite-rank operator, so that the range of is closed. In spite of the above, wewill often call a nonlinear equation problem (1.3.1) ill-posed if is not boundedlyinvertible, and well-posed if is boundedly invertible, deviating therefore fromthe usual terminology.

2.1.8 Discrepancy principle for nonlinear ill-posed problemswith monotone operators

Assume that A in (1.3.1) is monotone, i.e.,D(A) = H, A is continuous, is unbounded or does not exist, so (1.3.1) is anill-posed problem, Consider the discrepancy principle forfinding assuming that A is nonlinear monotone:

where C = const > 1, is any element such thatwhere and plays the role

of the regularization parameter We need three lemmas.

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Lemma 2.1.16. If A is monotone and continuous, and the set isnonempty, then it is convex and closed.

Lemma 2.1.17. If A is monotone and continuous, then it is w-closed, that is, andimply A(u) = f, where and stand for the weak and strong convergence in

H, respectively.

Lemma 2.1.17 in a stronger form (hemicontinuity of A replaces continuity, and itis assumed in this case that the monotone operator A is defined on all of H ) followsfrom the proof of Lemma 2.1.2.

Lemma 2.1.18. If and then

Proof of Lemma 2.1.16. If and then A(u) = f, so is closed.If A is monotone, A(u) = f and A(v) = f, then and

and vice versa. Thus, for any the element

Lemma 2.1.18 is Theorem 2.1.4.

Theorem 2.1.19. Assume:

(i)(ii)

(iii)(j)

A is a monotone, continuous operator, defined on all of H,equation A(u) = f is solvable, is its minimal-norm solution, and

where C > 1 is a constant. Then:the equation

is solvable for for any fixed Here is any element satisfying inequalitywhere

andand

(jj) if solves (2.1.8), and then

Remark 2.1.20. The equation is uniquely solvable for any and anyIf is its solution, and where C = const > 1, then

equation (2.1.8) with replaced by is solvable for If is its solution,then If A is injective, and if thenwhere solves the equation

If A is not injective, then it is not true, in general, that where is theminimal-norm solution to the equation A(u) = f even if one assumes that A is a linearoperator.


Proof of Theorem 2.1.19. If A is monotone, continuous and is defined on all of H, thenthe set is convex and closed, so it has a unique minimal-normelement To prove the existence of a solution to (2.1.8), we prove that the function

is greater than for sufficiently large and smaller thanfor sufficiently small If this is proved, then the continuity of with respect

to on implies that the equation has a solution.Let us give the proof. As we use the inequality:

and, as we use another inequality:

As one gets where c > 0 is a constant depending onThus, by the continuity of A, one obtains

As one gets Thus,Therefore equation has a solution

Let us now prove that if then From the estimate

and from the equation (2.1.8), it follows that Thus, one may assume thatand from (2.1.8) it follows that as By w-closedness of

monotone continuous operators (hemicontinuity in place of continuity would suffice),one gets A(U) = f, and from it follows that Because A ismonotone, the minimal norm solution to the equation A(u) = f in H is unique.Consequently, Thus, and By Theorem 2.1.4, it followsthat

Note that because due to the assumptionwhere C > 1. Theorem 2.1.19 is proved.

Proof of Remark 2.1.20. Let solve the equation letsolve the equation and Then

Multiply this equation by and use the monotonicity ofA to get The triangle inequality yields:Note that and as Thus, Therefore

Fix and let Then and where isthe minimal-norm solution to the equation A(u) = f. If A is injective, then thisequation has only one solution Since one gets the inequality

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Consequently, equation (2.1.8), withw replacing has a solution We claim that in fact,

as Indeed, from (2.1.8), with w replacing one getsand we prove below that where This implies

We now claim that the limit does exist, that u solves the equationA(u) = f, and It is sufficient to check that where c = const doesnot depend on as Indeed, if then a subsequence, denoted again,converges weakly to an element and (2.1.8) impliesSince A is monotone, it is wclosed, so By the injectivity of A, any sub-sequence converges weakly to the same element so Consequently,

as claimed. The inequality follows from the as-sumption

To prove the inequality note thatwhere Since C > 1, this implies where

Thus, whereThe last statement of Remark 2.1.20 is illustrated by the following example:

Example 2.1.21. Let Aw = (w, p )p,where (q, p) = 0, One has where is theminimal-norm solution to the equation Au = p. Equation has theunique solution Equation (2.1.8) isThis equation yields where and we assume1 (see the second inequality in the assumption (iii) of Theorem 2.1.19). LetThen, and Au = p. Therefore is not p,i.e., u is not the minimal-norm solution to the equation Au = p.

Remark 2.1.20 is proved.

Remark 2.1.22. It is easy to prove that if conditions (i) and (ii) of Theorem 2.1.19 holdand and if wherethen where and is the minimal-norm solutionto the equation A(u) = f. In particular, if 0 < a < l, then

Indeed, where is the unique solution tothe equation It is well known that pro-vided that and, clearly, one multiplies the identity

by and uses the monotonicity of A andthe inequality

The result similar to the one in the above remark can be found in [ARy].

2.1.9 Regularizers for ill-posed problems must depend on the noise level

In this Section we prove the following simple claim:

Claim 2. There is no regularizer independent of the noise level to a linear ill-posed problem.If such a regularizer exists, then the problem is well-posed.


Let A be a linear operator in a Banach space X. Assume that A is injective andis unbounded, that equation

is solvable, and g is such that

where is the norm in X and is the noise level. Nothing is assumed aboutthe statistical nature of noise. In particular, we do not assume that the noise has zeromean value or finite variance.

Question: Can one find a linear operator R with the property:

for any where Ran( A) is the range of A, and any satisfying (2.1.10)?Answer: no.

Proof. If such an R is found, then, taking and using the fact thatis arbitrary, one concludes that on the range of A. Secondly, writing

where

and w is arbitrary otherwise, one concludes from (2.1.11) and from the fact thatthat

for any w satisfying (2.1.12). Since R is linear, this implies that R is bounded, whichcontradicts the equation on Ran(A) and the unboundedness of whichis the necessary condition for the ill-posedness of (2.1.9).

A similar result one can find in [LY].

2.2 QUASISOLUTIONS, QUASINVERSION, AND BACKUS-GILBERT METHOD2.2.1 Quasisolutions for continuous operatorAssume that equation (1.3.1) is solvable, its solution where K is a compactumin a Banach space X, and A is continuous. Consider the problem

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where is the infimum of the function of and Aminimizer for (2.2.1) is called a quasisolution to (1.3.1) with Let be aminimizing sequence for (2.2.1). Since K is a compactum, one may assume that

as

Thus so is a minimizer for the problem (2.2.1). Theabove argument shows that if f replaces in (2.2.1), and if equation (1.3.1) is solv-able and its solution belongs to K, then any minimizer for (2.2.1) with is asolution to (1.3.1). Let us prove that where is a minimizerfor (2.2.1), and u is a solution to (1.3.1), where existence of a solution (1.3.1) isassumed.

Indeed, so one may assume that as By continuity of A, onehas Thus, The last conclusionfollows from the solvability of (1.3.1), which yields f = A(u) and from the inequality

We have proved:

Theorem 2.2.1. If equation (1.3.1) is solvable, K is a compactum containing all the solutionsto (1.3.1), and then (2.2.1) has a minimizer andfor every minimizer and some solution u to (1.3.1).

Remark. Suppose that X is strictly convex, i.e., if thenu = v. For example Hilbert spaces H are strictly convex, the spaces arestrictly convex, but and C(D) are not. Suppose that K is a convex compactum,i.e., convex closed compact set. The metric projection of an element ontoK is the element such that If X is strictlyconvex, then is unique, and if K is a convex compactum, then dependscontinuously on f. If A is injective and closed, not necessarily linear, and K is acompactum, then is continuous on the set AK. Indeed, ifand then a subsequence, denoted again converges to u because K iscompact, and if A is closed, then A(u) = f, which proves the claim. Therefore, if X isstrictly convex and K is a convex compactum, and if A is an injective bounded linearoperator, then the quasisolution depends continuously on f in thenorm of X.

2.2.2 Quasisolution for unbounded operatorsAssume that A is closed, possibly nonlinear, injective, unbounded operator, ispossibly, unbounded, equation (1.3.1) is solvable, assumptions (1)(4) of Section 2.1.5hold, and K is a compactum containing all the solutions to (1.3.1).

Theorem 2.2.2. Under the above assumptions, if whereK, then where u is a solution to (1.3.1).

Proof. One can assume as because K is a compactum. Note that because Thus,


By closedness of A, one gets A(w) = f, so w is a solution to (1.3.1), and one candenote w by u.

2.2.3 QuasiinversionLet A be a linear bounded operator in (1.3.1), and (1.3.1) is solvable, consider the equa-tion is a parameter, Q is an operatorchosen so that and where u is a solutionto (1.3.1). If A is unbounded, a similar idea can be applied to equation (1.3.1): considerthe equation where Q is chosen so that

The problem is: how does one choose Q with these properties?If A is a linear bounded operator, then Q = I can be used by Theorem 2.1.1. If A isunbounded, some assumptions on its spectrum are needed. See [LL] for details.

2.2.4 A Backus-Gilbert-type method: Recovery of signals from discreteand noisy data

We discuss in this section the following Problem 2. Let D and the bar denote varianceand mean value respectively,

Problem 1. Given estimate

The idea is as follows: the estimate is sought in the form

The problem is to find such that:(1) if then(2) if then where is a certain number. We find

optimal, in a certain sense. Namely, if then are foundfrom the requirements:

Note that

If then are found from {(A) and if this problem is solvableand, if not, one increases so that this problem becomes solvable.

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2.2.4.1. A typical problem we are concerned with is the problem of estimating thespectrum of a compactly supported function from the knowledge of the spectrum ata finite number of frequencies. More precisely, let

Suppose that the numbers:

are given. At the moment we assume that are given exactly, i.e., there is no noise.The case when the data are noisy will be considered below.

Problem 2. Given find an estimate of such that

To be specific let us assume that and that the estimate is of the form

where the functions will be chosen soon. From (2.2.5) and (2.2.2) it follows that

and

where and the star denotes complex conjugate.Property (2.2.4), convergence, holds if is a delta-sequence, i.e.,


Let

and let

be a sequence of functions such that

where

One can interpret (2.2.11) as the requirement that the estimate is exact for f (x) =const. Given (2.2.11), the smaller Q(x), the better is the quality of the delta-sequence

Thus we are led to the optimization problem: Find such a sequencethat

Note that the general problem of the type

where is a linearly independent set of functions, and is a boundeddomain in can be treated in exactly the same way as before.

If the problem (2.2.13) has the unique solution then(2.2.6) is the optimal estimate which, as we prove, has the convergence property(2.2.4).

2.2.4.2. If the data are noisy, that is are given in place of whererandom vector with the covariance matrix

35

where the bar denotes the mean value, then the variance can be computed

where ( , ) is the inner product in Let us fix and require that

The optimization problem for finding the vector can beformulated as follows:

Here Clearly, problem (2.2.18) is not solvable for all Wewill discuss this important point below. If (2.2.18) is solvable, the solution is unique,and the optimal estimate is given by

This estimate has varianceOur arguments so far are close to the usual ones. The new point is our convergence

requirement (2.2.4). We prove the convergence property of our estimate and give therate of convergence. The case when the data are the finite number of moments istreated, and the optimization requirements are introduced.

The problem we discuss is of interest in geophysics and many other applications.

2.2.4.3. Here a solution of the estimation problems is given.We start with problem (2.2.13). Let us write Q(x) as a quadratic form

where


and

is a self-adjoint positive definite matrix.Let us write (2.2.13) as

Using the Lagrange multiplier one obtains the standard necessary and sufficientcondition for the minimizer

Therefore

is uniquely defined by (2.2.25), since is positive definite, and the denominator in(2.2.25) does not vanish. The minimum of Q(x) is

We assume that

If (2.2.27) holds, then (2.2.4) holds. Indeed, (2.2.13) and (2.2.27) imply that

In this argument we assume that the function f (x) satisfies the inequality

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This inequality is satisfied if, for example, the derivative of f (x) exists except at a finitenumber of points and is uniformly bounded.

Let us illustrate the assumption (2.2.27). LetThen

wherewhere is the cofactor corresponding to the element of the

matrixOne can show that (2.2.24) holds at any point at which f (x) is differentiable and

Indeed, if we do not take the but use then theerror of the estimate will be not less that On the other hand, for this choiceof h, the kernel (2.2.8) is the Dirichlet kernel. From the theory of the Fourier seriesone knows that (2.2.9) holds in and at any point at whichf (x) is differentiable.

In practice it is advisable to choose the system in such a way that tendsrapidly to zero. Note that depends only on the system and therefore wecan control this quantity to some extent by choosing the system

Let us note that one can estimate f (x) at a given point optimally using the sameprocedure. In this case the convergence condition (2.2.4) will hold for If is fixed,we can choose the system so that

In this case

and we can choose so that, in addition to (2.2.29), the condition

is satisfied. For example, take then (2.2.29) reduces to


and one can choose which behave nearly like in a small neighborhood ofThen can be made very large, and as in (2.2.31).

2.2.4.4. In this section we solve problem (2.2.18). As we have already mentioned,this problem may not be solvable for every because there may be no h whichsatisfies both restrictions of (2.2.18). Since the set isconvex and Q(h) is a strictly convex function of h, it is clear that the solution to(2.2.18) is unique when it exists. For the solution to exist it is necessary and sufficientthat the set M of h, which satisfy the restrictions (2.2.18), be not empty.

Let us give an analytic solution to problem (2.2.18). If for the optimal h the inequalityholds, then the solution to (2.2.18) is the same as the solution to (2.2.13)

and is given by formula (2.2.25). Therefore, first one checks if the function (2.2.25)satisfies the inequality

If it does, then it is the solution to (2.2.18). If it does not, then the solution satisfiesthe equality

By the Lagrange method the necessary condition for the optimal h, for the solutionto problem (2.2.18), is

where and are the Lagrange multipliers. It follows from (2.2.36) that

Taking the complex conjugate in the first equation (2.2.36) we see that

From (2.2.37) and (2.2.38) one gets

Substituting (2.2.40) into

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yields an equation for

The roots of Eq. (2.2.42), give by formula (2.2.38), and h by formula (2.2.37).Finally choose the for which (Bh, h) = min. This solves problem (2.2.18).

2.2.4.5. One can simplify the solution to problem (2.2.18) in the following way. Ifthe problem (2.2.18) takes the form

Let us choose the coordinate system so that

and normalize so that

In this case (2.2.43) can be written as

where

Thus, problem (2.2.43) in with two constraints is reduced to problem (2.2.46) inwith one constraint. Problem (2.2.46) can be solved by the Lagrange multipliers

method. One has

where is the Lagrange multiplier. Thus,


Substitute (2.2.49) into the constraint equation (2.2.46) to obtain an equation forIf this equation is solved then (2.2.49) gives the corresponding H. If there are severalsolutions then the is the one that minimizes the quadratic form (2.2.46).

2.3 ITERATIVE METHODS

There is a vast literature on iterative methods [VV], [BG], [R65]. First, we prove thefollowing result.

Theorem 2.3.1. Every solvable equation (1.3.1) with bounded linear operator A can besolved by a convergent iterative method.

Proof. It was proved in Section 2.1.6 that if A is a bounded linear operator and equa-tion (1.3.1) is solvable, then it is equivalent to the equation (2.1.6). By y we denotethe minimal norm solution to (1.3.1), i.e., the solution orthogonal to N(A), the nullspace of A. Note that N(B) = N(A). Without loss of generality assume (if

then one can divide by equation (2.1.6)). Consider the iterations

where is arbitrary. Denote and write (2.3.1) as

Thus, Since one gets:

where is the resolution of the identity of B. Thus,where P is the orthoprojector onto N(B) = N(A). If one takes and

then by induction, and In particular,if then (by induction, since and

where A+ is the pseudoinverse of A, defined inSection 2.1.1.

Exercise 2.3.2. Prove that if then in (2.3.1).

Assume now that is given in place of f, Let us show that ifone stops iterations (2.3.1), with in place of f and at then

as if is properly chosen, and is defined by (2.3.1). LetThen and

Thus, because ifand we had assumed which, together with implies

Therefore, if is chosen so that thenIndeed, We have proved in Theorem 2.3.1

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and as Let us summa-rize the result.

Theorem 2.3.3. If thenwhere is the minimal norm solution to (1.3.1), and

is obtained by iterations (2.3.1), with in place of f and

A general approach to construction of convergent iterative methods for nonlinearproblems is developed in Section 2.4.

2.4 DYNAMICAL SYSTEM METHOD (DSM)2.4.1 The idea of the DSM

Consider the equation:

We assume in Section 2.4 that

where is the minimal-norm solution to (2.4.1),H is a real Hilbert space. Many of our results

hold in reflexive Banach spaces X and F : X X*, but we do not go into detail. Theelement in (2.4.2) will be specified later. In Section 2.4 we will call (2.4.1) a well-posed problem if

and ill-posed if is not boundedly invertible. We assume existence of a solutionto (2.4.1) unless otherwise stated, but uniqueness of the solution is not assumed.

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