1 Two Central Questions of This Book and an Introduction to
the Theories of Ill-posed and Coefficient Inverse Problems ... 1
1.1 Two Central Questions of This Book ...................... 2
1.1.1 Why the Above Two Questions Are the Central
Ones for Computations of CIPs .................... 4
1.1.2 Approximate Global Convergence ................... 6
1.1.3 Some Notations and Definitions .................. 11
1.2 Some Examples of Ill-posed Problems .................... 14
1.3 The Foundational Theorem of A.N. Tikhonov .............. 21
1.4 Classical Correctness and Conditional Correctness ...... 23
1.5 Quasi-solution ......................................... 25
1.6 Regularization ......................................... 27
1.7 The Tikhonov Regularization Functional ................. 31
1.7.1 The Tikhonov Functional ......................... 32
1.7.2 Regularized Solution ............................ 34
1.8 The Accuracy of the Regularized Solution for a Single
Value of α ............................................. 35
1.9 Global Convergence in Terms of Definition 1.1.2.4 ...... 39
1.9.1 The Local Strong Convexity ...................... 40
1.9.2 The Global Convergence .......................... 45
1.10 Uniqueness Theorems for Some Coefficient Inverse
Problems ............................................... 46
1.10.1 Introduction .................................... 46
1.10.2 Carleman Estimate for a Hyperbolic Operator ..... 48
1.10.3 Estimating an Integral .......................... 56
1.10.4 Cauchy Problem with the Lateral Data for a
Hyperbolic Inequality with Volterra-Like
Integrals ....................................... 57
1.10.5 Coefficient Inverse Problem for a Hyperbolic
Equation ........................................ 62
1.10.6 The First Coefficient Inverse Problem for a
Parabolic Equation .............................. 68
1.10.7 The Second Coefficient Inverse Problem for a
Parabolic Equation .............................. 70
1.10.8 The Third Coefficient Inverse Problem for a
Parabolic Equation .............................. 76
1.10.9 A Coefficient Inverse Problem for an Elliptic
Equation ........................................ 78
1.11 Uniqueness for the Case of an Incident Plane Wave
in Partial Finite Differences .......................... 79
1.11.1 Results ......................................... 81
1.11.2 Proof of Theorem 1.11.1.1 ....................... 83
1.11.3 The Carleman Estimate ........................... 85
1.11.4 Proof of Theorem 1.11.1.2 ....................... 90
2 Approximately Globally Convergent Numerical Method .......... 95
2.1 Statements of Forward and Inverse Problems ............. 97
2.2 Parabolic Equation with Application in Medical
Optics ................................................. 98
2.3 The Transformation Procedure for the Hyperbolic
Case .................................................. 100
2.4 The Transformation Procedure for the Parabolic Case ... 103
2.5 The Layer Stripping with Respect to the Pseudo
Frequency ............................................. 106
2.6 The Approximately Globally Convergent Algorithm ....... 109
2.6.1 The First Version of the Algorithm ............. 1ll
2.6.2 A Simplified Version of the Algorithm .......... 112
2.7 Some Properties of the Laplace Transform of the
Solution oftheCauchy Problem (2.1) and (2.2) .......... 115
2.7.1 The Study of the Limit (2.12) .................. 115
2.7.2 Some Additional Properties of the Solution
of the Problem (2.11) and (2.12) ............... 118
2.8 The First Approximate Global Convergence Theorem ...... 122
2.8.1 Exact Solution ................................. 123
2.8.2 The First Approximate Global Convergence
Theorem ........................................ 125
2.8.3 Informal Discussion of Theorem 2.8.2 ........... 137
2.8.4 The First Approximate Mathematical Model ....... 138
2.9 The Second Approximate Global Convergence Theorem ..... 140
2.9.1 Estimates of the Tail Function ................. 142
2.9.2 The Second Approximate Mathematical Model ...... 151
2.9.3 Preliminaries .................................. 155
2.9.4 The Second Approximate Global Convergence
Theorem ........................................ 157
2.10 Summary ............................................... 166
3 Numerical Implementation of the Approximately Globally
Convergent Method .......................................... 169
3.1 Numerical Study in 2D ................................. 170
3.1.1 The Forward Problem ............................ 171
3.1.2 Main Discrepancies Between the Theory and the
Numerical Implementation ....................... 173
3.1.3 Results of the Reconstruction .................. 174
3.2 Numerical Study in 3D ................................. 186
3.2.1 Computations of the Forward Problem ............ 186
3.2.2 Result of the Reconstruction ................... 188
3.3 Summary of Numerical Studies ......................... 191
4 The Adaptive Finite Element Technique and Its Synthesis
with the Approximately Globally Convergent Numerical
Method ..................................................... 193
4.1 Introduction .......................................... 193
4.1.1 The Idea of the Two-Stage Numerical
Procedure ...................................... 193
4.1.2 The Concept of the Adaptivity for CIPs ......... 194
4.2 Some Assumptions ...................................... 196
4.3 State and Adjoint Problems ............................ 198
4.4 The Lagrangian ........................................ 199
4.5 A Posteriori Error Estimate for the Lagrangian ........ 202
4.6 Some Estimates of the Solution an Initial Boundary
Value Problem for Hyperbolic Equation (4.9) ........... 210
4.7 Frechet Derivatives of Solutions of State and
Adjoint Problems ...................................... 216
4.8 The Frechet Derivative of the Tikhonov Functional ..... 222
4.9 Relaxation with Mesh Refinements ...................... 225
4.9.1 The Space of Finite Elements ................... 226
4.9.2 Minimizers on Subspaces ........................ 229
4.9.3 Relaxation ..................................... 233
4.10 From the Abstract Scheme to the Coefficient Inverse
Problem 2.1 ........................................... 235
4.11 A Posteriori Error Estimates for the Regularized
Coefficient and the Relaxation Property of Mesh
Refinements ........................................... 237
4.12 Mesh Refinement Recommendations ....................... 241
4.13 The Adaptive Algorithm ................................ 244
4.13.1 The Algorithm In Brief ......................... 244
4.13.2 The Algorithm .................................. 244
4.14 Numerical Studies of the Adaptivity Technique ......... 245
4.14.1 Reconstruction of a Single Cube ................ 246
4.14.2 Scanning Acoustic Microscope ................... 248
4.15 Performance of the Two-Stage Numerical Procedure in
2D .................................................... 258
4.15.1 Computations of the Forward Problem ............ 258
4.15.2 The First Stage ................................ 261
4.15.3 The Second Stage ............................... 264
4.16 Performance of the Two-Stage Numerical Procedure in
3D .................................................... 266
4.16.1 The First Stage ................................ 275
4.16.2 The Second Stage ............................... 277
4.17 Numerical Study of the Adaptive Approximately
Globally Convergent Algorithm ......................... 279
4.17.1 Computations of the Forward Problem ............ 285
4.17.2 Reconstruction by the Approximately Globally
Convergent Algorithm ........................... 288
4.17.3 The Adaptive Part .............................. 289
4.18 Summary of Numerical Studies of Chapter 4 ............. 291
5 Blind Experimental Data .................................... 295
5.1 Introduction .......................................... 295
5.2 The Mathematical Model ................................ 297
5.3 The Experimental Setup ................................ 298
5.4 Data Simulations ...................................... 301
5.5 State and Adjoint Problems for Experimental Data ...... 302
5.6 Data Pre-Processing ................................... 304
5.6.1 The First Stage of Data Immersing .............. 304
5.6.2 The Second Stage of Data Immersing ............. 307
5.7 Some Details of the Numerical Implementation of the
Approximately ......................................... 309
5.7.1 Stopping Rule for ............................... 311
5.8 Reconstruction by the Approximately Globally
Convergent Numerical Method ........................... 311
5.8.1 Dielectric Inclusions and Their Positions ...... 311
5.8.2 Tables and Images .............................. 312
5.8.3 Accuracy of the Blind Imaging .................. 314
5.8.4 Performance of a Modified Gradient Method ...... 316
5.9 Performance of the Two-Stage Numerical Procedure ...... 319
5.9.1 The First Stage ................................ 319
5.9.2 The Third Stage of Data Immersing .............. 320
5.9.3 Some Details of the Numerical Implementation
of the Adaptivity .............................. 323
5.9.4 Reconstruction Results for Cube Number 1 ....... 323
5.9.5 Reconstruction Results for the Cube Number 2 ... 325
5.9.6 Sensitivity to the Parameters α and β .......... 327
5.9.7 Additional Effort for Cube Number 1 ............ 327
5.10 Summary ............................................... 332
6 Backscattering Data ........................................ 335
6.1 Introduction .......................................... 335
6.2 Forward and Inverse Problems .......................... 337
6.3 Laplace Transform ..................................... 339
6.4 The Algorithm ......................................... 340
6.4.1 Preliminaries .................................. 340
6.4.2 The Sequence of Elliptic Equations ............. 342
6.4.3 The Iterative Process .......................... 344
6.4.4 The Quasi-Reversibility Method ................. 345
6.5 Estimates for the QRM ................................. 346
6.6 The Third Approximate Mathematical Model .............. 354
6.6.1 Exact Solution ................................. 354
6.6.2 The Third Approximate Mathematical Model ....... 356
6.7 The Third Approximate Global Convergence Theorem ...... 358
6.8 Numerical Studies ..................................... 367
6.8.1 Main Discrepancies Between Convergence
Analysis and Numerical Implementation .......... 367
6.8.2 A Simplified Mathematical Model of Imaging
of Plastic Land Mines .......................... 368
6.8.3 Some Details of the Numerical Implementation ... 369
6.8.4 Numerical Results .............................. 372
6.8.5 Backscattering Without the QRM ................. 374
6.9 Blind Experimental Data Collected in the Field ........ 376
6.9.1 Introduction ................................... 378
6.9.2 Data Collection and Imaging Goal ............... 379
6.9.3 The Mathematical Model and the Approximately
Globally Convergent Algorithm .................. 381
6.9.4 Uncertainties .................................. 385
6.9.5 Data Pre-processing ............................ 388
6.9.6 Results of Blind Imaging ....................... 391
6.9.7 Summary of Blind Imaging ....................... 392
References .................................................... 393
Index ......................................................... 401
| 
