Inverse problems and related topics, 2018

Preface......Page 6
Contents......Page 8
Part I Inverse Problems and Related Topics for Partial Differential Equations......Page 10
1 An Inverse Conductivity Problem  in Multifrequency Electric Impedance Tomography......Page 11
 1.1 Introduction......Page 12
 1.2 An Intermediate Estimate......Page 16
 1.3 Proof of the Main Result......Page 20
 1.4 Uniqueness......Page 23
 1.5 Expansion in the Eigenfunction Basis of the NP Operator......Page 25
 References......Page 37
 2.1 Introduction......Page 39
 2.2 Preliminaries......Page 42
  2.2.1 Weak Degeneracy......Page 43
  2.2.2 Strong Degeneracy......Page 44
 2.3 Main Result......Page 47
 References......Page 52
 3.1 Introduction......Page 54
  3.2.2 Unknown Functions......Page 56
  3.2.3 Main Results......Page 57
 3.3 Numerical Simulations......Page 58
 3.4 Proof of Theorem 3.1......Page 60
 References......Page 63
 4.1 Introduction......Page 65
 4.2 Uniqueness in Inverse Source Problems......Page 67
 4.3 Increasing Stability for Sources......Page 72
 4.4 Numerical Tests......Page 78
  4.4.1 2D Numerical Examples......Page 79
  4.4.2 3D Numerical Examples......Page 81
 4.5 Conclusion......Page 84
 References......Page 85
5 Inverse Moving Source Problem for Fractional Diffusion(-Wave) Equations: Determination of Orbits......Page 87
 5.1 Introduction......Page 88
 5.2 Preliminaries and Main Results......Page 89
 5.3 Proofs of Lemmas 5.1–5.3......Page 95
 5.4 Proofs of the Main Results......Page 100
 References......Page 105
6 Inverse Problems for a Compressible Fluid System......Page 107
  6.1.1 Review for the Main Methodology and the Results......Page 108
  6.1.2 Main Results for Inverse Problems for Compressible Fluids......Page 112
 6.2 Global Carleman Estimate......Page 115
 6.3 Proof of Theorem 6.1......Page 119
 6.4 Proof of Theorem 6.2 by Local Carleman Estimate......Page 131
 6.5 Proof of Theorem 6.2 by Global Carleman Estimate......Page 143
 References......Page 150
  7.1.1 Introduction......Page 155
  7.1.2 Main Theorems......Page 157
  7.2.1 Setting and Start of Carleman Estimate......Page 159
  7.2.2 Completion of the Proofs......Page 164
 7.3 Realizations of the Weight Functions......Page 169
 References......Page 171
Part II Regularization Theory of Inverse Problems......Page 172
 8.1 Introduction......Page 173
 8.2 Assumptions and Main Result......Page 175
 8.3 Proofs of Propositions8.1 and 8.2......Page 176
 8.4 Conclusions......Page 179
 References......Page 180
9 Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability......Page 181
 9.1 Introduction......Page 182
 9.2 Convergence......Page 184
 9.3 Convergence Rate Results......Page 186
 9.4 Examples......Page 190
  9.5.1 Numerical Studies for Model Problem 9.1......Page 195
  9.5.2 Numerical Studies for Model Problem 9.2......Page 198
  9.5.3 Numerical Studies for Model Problem 9.3......Page 201
 References......Page 206
10 Regularized Reconstruction of the Order in Semilinear Subdiffusion with Memory......Page 208
 10.1 Introduction......Page 209
 10.2 Functional Spaces and Preliminaries......Page 213
 10.3 The Main Result......Page 214
  10.4.1 Direct Problems: Solvability and Estimates......Page 217
  10.4.2 Solvability of IP, Explicit Formula to ν......Page 222
  10.4.3 Uniqueness in IP......Page 223
  10.4.4 Error Estimate for Noisy Data......Page 225
  10.5.1 Algorithm of Reconstruction......Page 227
  10.5.2 Numerical Experiments......Page 230
 10.6 Discussion and Conclusion......Page 236
 References......Page 238
11 On the Singular Value Decomposition  of n-Fold Integration Operators......Page 240
 11.1 Introduction......Page 241
 11.3 The Boundary Value Problem for the Singular Value Decomposition......Page 243
 11.4 The Onefold Integration Operator......Page 247
  11.5.1 General Assertions......Page 248
  11.5.2 On the Zeros of the Function f(z)=cos(z) cosh(z)+1 =0......Page 249
  11.5.3 Eigenfunctions......Page 251
 11.6 The n-fold Integration Operator......Page 252
 11.7 Distribution of the Eigenvalues λi......Page 256
 References......Page 259
 12.1 Introduction......Page 260
 12.2 Convergence Rate Theory for Convex Tikhonov Regularization......Page 262
 12.3 The Łojasiewicz-Inequality and Its Consequences......Page 265
 12.4 The KL-Regularity Condition......Page 267
 12.5 Relation to Conditional Stability Estimates......Page 269
 12.6 Example: Tikhonov Regularization......Page 272
 References......Page 275
 13.1 Introduction......Page 278
 13.2 Unconstrained Case......Page 281
 13.3 Bi-level Optimization......Page 288
  13.3.1 Constraint Bi-level Optimization......Page 293
 13.4 Implicit Function Calculus......Page 295
  13.4.1 Equality Constraint Optimization......Page 298
 13.5 Parameter-Dependent Constrained Optimization Problem......Page 299
  13.5.1 Optimal Control Problem......Page 304
  13.5.2 Shape Derivative......Page 306
 References......Page 308
Author Index......Page 310