| Preface |
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vii | |
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Chapter 1 Inverse Problems |
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1 | (19) |
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1.1 The inverse problem of gravimetry |
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1 | (4) |
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1.2 The inverse conductivity problem |
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5 | (2) |
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7 | (3) |
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1.4 Tomography and the inverse seismic problem |
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10 | (4) |
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1.5 Inverse spectral problems |
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14 | (6) |
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Chapter 2 Ill-Posed Problems and Regularization |
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20 | (19) |
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2.1 Well-and ill-posed problems |
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20 | (3) |
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2.2 Conditional correctness. Regularization |
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23 | (3) |
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2.3 Construction of regularizers |
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26 | (6) |
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2.4 Convergence of regularization algorithms |
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32 | (4) |
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36 | (3) |
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Chapter 3 Uniqueness and Stability in the Cauchy Problem |
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39 | (34) |
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3.1 The backward parabolic equation |
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39 | (9) |
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3.2 General Carleman type estimates and the Cauchy problem |
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48 | (5) |
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3.3 Elliptic and parabolic equations |
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53 | (6) |
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3.4 Hyperbolic and Schrodinger equations |
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59 | (12) |
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71 | (2) |
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Chapter 4 Elliptic Equations: Single Boundary Measurements |
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73 | (32) |
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4.0 Results on elliptic boundary value problems |
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73 | (3) |
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76 | (5) |
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4.2 Reconstruction of lower-order terms |
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81 | (4) |
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4.3 The inverse conductivity problem |
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85 | (8) |
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4.4 Methods of the theory of one complex variable |
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93 | (5) |
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4.5 Linearization of the coefficients problem |
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98 | (2) |
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4.6 Some problems of nondestructive evaluation |
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100 | (3) |
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103 | (2) |
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Chapter 5 Elliptic Equations: Many Boundary Measurements |
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105 | (39) |
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5.0 The Dirichlet-to-Neumann map |
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105 | (3) |
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5.1 Boundary Reconstruction |
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108 | (4) |
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5.2 Reconstruction in XXX |
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112 | (2) |
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5.3 Completeness of products of solutions of PDE |
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114 | (4) |
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118 | (3) |
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5.5 Recovery of several coefficients |
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121 | (6) |
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127 | (4) |
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5.7 Discontinuous conductivities |
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131 | (6) |
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5.8 Maxwell's and elasticity systems |
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137 | (4) |
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141 | (3) |
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Chapter 6 Scattering Problems |
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144 | (19) |
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144 | (3) |
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147 | (4) |
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6.2 Scattering by a medium |
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151 | (4) |
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6.3 Scattering by obstacles |
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155 | (6) |
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161 | (2) |
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Chapter 7 Integral Geometry and Tomography |
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163 | (21) |
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7.1 The Radon transform and its inverse |
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163 | (9) |
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7.2 The energy integrals methods |
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172 | (3) |
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7.3 Boman's counterexample |
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175 | (4) |
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7.4 The transport equation |
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179 | (2) |
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181 | (3) |
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Chapter 8 Hyperbolic Equations |
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184 | (33) |
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184 | (3) |
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8.1 The one-dimensional case |
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187 | (8) |
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8.2 Single boundary measurements |
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195 | (4) |
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8.3 Many measurements: use of beam solutions |
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199 | (6) |
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8.4 Many measurements: methods of boundary control |
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205 | (6) |
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8.5 Recovery of discontinuity of the speed of propagation |
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211 | (4) |
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215 | (2) |
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Chapter 9 Parabolic Equations |
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217 | (29) |
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217 | (3) |
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9.1 Final overdetermination |
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220 | (6) |
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9.2 Lateral overdetermination: single measurements |
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226 | (2) |
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9.3 Lateral overdetermination: many measurements |
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228 | (4) |
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9.4 Discontinuous principal coefficient |
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232 | (5) |
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237 | (5) |
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242 | (2) |
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244 | (2) |
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Chapter 10 Some Numerical Methods |
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246 | (19) |
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247 | (5) |
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10.2 Variational regularization of the Cauchy problem |
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252 | (4) |
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256 | (2) |
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258 | (3) |
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261 | (4) |
| Appendix. Functional Spaces |
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265 | (4) |
| References |
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269 | (14) |
| Index |
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283 | |
