Second Edition
>>Victor Isakov
Contents
Preface to the Second Edition vii
Preface to the First Edition ix
Chapter 1 Inverse Problems 1
1.1 The inverse problem of gravimetry 1
1.2 The inverse conductivity problem 5
1.3 Inverse scattering 7
1.4 Tomography and the inverse seismic problem 10
1.5 Inverse spectral problems 14
Chapter 2 Ill-Posed Problems and Regularization 20
2.1 Well- and ill-posed problems 20
2.2 Conditional correctness: Regularization 23
2.3 Construction of regularizers 26
2.4 Convergence of regularization algorithms 33
2.5 Iterative algorithms 37
Chapter 3 Uniqueness and Stability in the Cauchy Problem 41
3.1 The backward parabolic equation 42
3.2 General Carleman estimates and the Cauchy problem 51
3.3 Elliptic and parabolic equations 57
3.4 Hyperbolic and Schr ¨ odinger equations 65
3.5 Systems of partial differential equations 80
3.6 Open problems 86
Chapter 4 Elliptic Equations: Single Boundary Measurements 89
4.0 Results on elliptic boundary value problems 89
4.1 Inverse gravimetry 92
4.2 Reconstruction of lower-order terms 97
4.3 The inverse conductivity problem 102
4.4 Methods of the theory of one complex variable 111
4.5 Linearization of the coefficients problem 116
4.6 Some problems of detection of defects 119
4.7 Open problems 125
Chapter 5 Elliptic Equations: Many Boundary Measurements 127
5.0 The Dirichlet-to-Neumann map 127
5.1 Boundary reconstruction 130
5.2 Reconstruction in 134
5.3 Completeness of products of solutions of PDE 138
5.4 Recovery of several coefficients 143
5.5 The plane case 149
5.6 Nonlinear equations 154
5.7 Discontinuous conductivities 160
5.8 Maxwell’s and elasticity systems 166
5.9 Open problems 170
Chapter 6 Scattering Problems 173
6.0 Direct Scattering 173
6.1 From A to near field 176
6.2 Scattering by a medium 180
6.3 Scattering by obstacles 184
6.4 Open problems 190
Chapter 7 Integral Geometry and Tomography 192
7.1 The Radon transform and its inverse 192
7.2 The energy integral methods 201
7.3 Boman’s counterexample 205
7.4 The transport equation 208
7.5 Open problems 215
Chapter 8 Hyperbolic Problems 218
8.0 Introduction 218
8.1 The one-dimensional case 221
8.2 Single boundary measurements 229
8.3 Many measurements: use of beam solutions 236
8.4 Many measurements: methods of boundary control 243
8.5 Recovery of discontinuity of the speed of propagation 249
8.6 Open problems 253
Chapter 9 Inverse parabolic problems 255
9.0 Introduction 255
9.1 Final overdetermination 259
9.2 Lateral overdetermination: single measurements 264
9.3 The inverse problem of option pricing 270
9.4 Lateral overdetermination: many measurements 275
9.5 Discontinuous principal coefficient and recovery of a domain 279
9.6 Nonlinear equations 288
9.7 Interior sources 293
9.8 Open problems 295
Chapter 10 Some Numerical Methods 297
10.1 Linearization 298
10.2 Variational regularization of the Cauchy problem 303
10.3 Relaxation methods 308
10.4 Layer-stripping 310
10.5 Range test algorithms 313
10.6 Discrete methods 318
Appendix. Functional Spaces 321
References 324
Index 343
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