Preface to the Third Edition |
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vii | |
Preface to the Second Edition |
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ix | |
Preface to the First Edition |
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xi | |
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Chapter 1 Inverse Problems |
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1 | (22) |
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1.1 The Inverse Problem of Gravimetry |
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1 | (5) |
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1.2 The Inverse Conductivity Problem |
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6 | (2) |
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8 | (4) |
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1.4 Tomography and the Inverse Seismic Problem |
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12 | (4) |
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1.5 Inverse Spectral Problems |
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16 | (7) |
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Chapter 2 Ill-Posed Problems and Regularization |
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23 | (24) |
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2.1 Well- and Ill-Posed Problems |
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23 | (4) |
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2.2 Conditional Correctness: Regularization |
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27 | (3) |
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2.3 Construction of Regularizes |
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30 | (7) |
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2.4 Convergence of Regularization Algorithms |
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37 | (5) |
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42 | (5) |
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Chapter 3 Uniqueness and Stability in the Cauchy Problem |
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47 | (58) |
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3.1 The Backward Parabolic Equation |
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48 | (10) |
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3.2 General Carleman Estimates and the Cauchy Problem |
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58 | (7) |
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3.3 Elliptic and Parabolic Equations |
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65 | (8) |
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3.4 Hyperbolic and Schrodinger Equations |
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73 | (22) |
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3.5 Systems of Partial Differential Equations |
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95 | (6) |
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101 | (4) |
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Chapter 4 Elliptic Equations: Single Boundary Measurements |
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105 | (44) |
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4.0 Results on Elliptic Boundary Value Problems |
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105 | (3) |
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108 | (6) |
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4.2 Reconstruction of Lower-Order Terms |
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114 | (5) |
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4.3 The Inverse Conductivity Problem |
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119 | (11) |
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4.4 Methods of the Theory of One Complex Variable |
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130 | (5) |
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4.5 Linearization of the Coefficient Problem |
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135 | (4) |
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4.6 Some Problems of Detection of Defects |
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139 | (7) |
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146 | (3) |
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Chapter 5 Elliptic Equations: Many Boundary Measurements |
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149 | (62) |
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5.0 The Dirichlet-to-Neumann Map and the Cauchy Set |
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149 | (5) |
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5.1 Boundary Reconstruction |
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154 | (5) |
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5.2 Completeness of Products of Solutions of PDE |
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159 | (6) |
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5.3 Uniqueness and Stability in Ω |
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165 | (7) |
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5.4 Recovery of Several Coefficients |
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172 | (6) |
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178 | (8) |
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186 | (7) |
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5.7 Transmission Problems |
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193 | (9) |
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5.8 Maxwell's and Elasticity Systems |
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202 | (6) |
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208 | (3) |
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Chapter 6 Scattering Problems and Stationary Waves |
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211 | (30) |
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211 | (3) |
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214 | (8) |
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6.2 The Inverse Source Problem with Many Wave Numbers |
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222 | (3) |
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6.3 Scattering by a Medium |
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225 | (6) |
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6.4 Scattering by Obstacles |
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231 | (7) |
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238 | (3) |
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Chapter 7 Integral Geometry and Tomography |
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241 | (28) |
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7.1 The Radon Transform and its Inverse |
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241 | (10) |
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7.2 The Energy Integral Methods |
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251 | (4) |
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7.3 Boman's Counterexample |
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255 | (4) |
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7.4 The Transport Equation |
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259 | (7) |
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266 | (3) |
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Chapter 8 Hyperbolic Problems |
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269 | (40) |
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269 | (4) |
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8.1 The One-Dimensional Case |
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273 | (8) |
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8.2 Single Boundary Measurements |
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281 | (9) |
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8.3 Many Measurements: The Use of Beam Solutions |
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290 | (6) |
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8.4 Many measurements: Methods of Boundary Control |
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296 | (7) |
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8.5 Recovery of Discontinuity of the Speed of Propagation |
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303 | (4) |
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307 | (2) |
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Chapter 9 Inverse Parabolic Problems |
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309 | (46) |
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309 | (4) |
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9.1 Final Overdetermination |
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313 | (6) |
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9.2 Lateral Overdetermination: Single Measurements |
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319 | (6) |
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9.3 The Inverse Problem of Option Pricing |
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325 | (6) |
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9.4 Lateral Overdetermination: Many Measurements |
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331 | (4) |
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9.5 Discontinuous Principal Coefficient and Recovery of a Domain |
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335 | (10) |
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345 | (5) |
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350 | (2) |
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352 | (3) |
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Chapter 10 Some Numerical Methods |
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355 | (26) |
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10.1 Numerics for the Cauchy Problem |
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356 | (6) |
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362 | (5) |
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367 | (3) |
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370 | (3) |
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10.5 Range Test Algorithms |
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373 | (4) |
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377 | (4) |
Appendix. Function Spaces |
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381 | (4) |
References |
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385 | (20) |
Index |
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405 | |