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 III-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 Schrödinger 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 Ω ...................................... 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. В Oman'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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