Beilina / Klibanov Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

1. Auflage 2012
ISBN: 978-1-4419-7805-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 408 Seiten

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity).
Two central questions for CIPs are addressed: How to obtain a good approximations for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation.
The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real world problem of imaging of shallow explosives.

Autoren/Hrsg.

Beilina, Larisa

Klibanov, Michael Victor

Inhaltsverzeichnis

1;Preface;8
2;Contents;12
3;Chapter1 Two Central Questions of This Book and an Introduction to the Theories of Ill-posed and Coefficient Inverse Problems;18
3.1;1.1 Two Central Questions of This Book;19
3.1.1;1.1.1 Why the Above Two Questions Are the Central Ones for Computations of CIPs;21
3.1.2;1.1.2 Approximate Global Convergence;23
3.1.3;1.1.3 Some Notations and Definitions;28
3.2;1.2 Some Examples of Ill-posed Problems;31
3.3;1.3 The Foundational Theorem of A.N. Tikhonov;38
3.4;1.4 Classical Correctness and Conditional Correctness;40
3.5;1.5 Quasi-solution;42
3.6;1.6 Regularization;44
3.7;1.7 The Tikhonov Regularization Functional;48
3.7.1;1.7.1 The Tikhonov Functional;49
3.7.2;1.7.2 Regularized Solution;51
3.8;1.8 The Accuracy of the Regularized Solution for a Single Value of 0=x"010B;52
3.9;1.9 Global Convergence in Terms of Definition 1.1.2.4;56
3.9.1;1.9.1 The Local Strong Convexity;57
3.9.2;1.9.2 The Global Convergence;62
3.10;1.10 Uniqueness Theorems for Some Coefficient Inverse Problems;63
3.10.1;1.10.1 Introduction;63
3.10.2;1.10.2 Carleman Estimate for a Hyperbolic Operator;65
3.10.3;1.10.3 Estimating an Integral;73
3.10.4;1.10.4 Cauchy Problem with the Lateral Data for a Hyperbolic Inequality with Volterra-Like Integrals;74
3.10.5;1.10.5 Coefficient Inverse Problem for a Hyperbolic Equation;79
3.10.6;1.10.6 The First Coefficient Inverse Problem for a Parabolic Equation;85
3.10.7;1.10.7 The Second Coefficient Inverse Problem for a Parabolic Equation;87
3.10.8;1.10.8 The Third Coefficient Inverse Problem for a Parabolic Equation;93
3.10.9;1.10.9 A Coefficient Inverse Problem for an Elliptic Equation;95
3.11;1.11 Uniqueness for the Case of an Incident Plane Wave in Partial Finite Differences;96
3.11.1;1.11.1 Results;98
3.11.2;1.11.2 Proof of Theorem 1.11.1.1;100
3.11.3;1.11.3 The Carleman Estimate;102
3.11.4;1.11.4 Proof of Theorem 1.11.1.2;107
4;Chapter2 Approximately Globally Convergent Numerical Method;111
4.1;2.1 Statements of Forward and Inverse Problems;113
4.2;2.2 Parabolic Equation with Application in Medical Optics;114
4.3;2.3 The Transformation Procedure for the Hyperbolic Case;116
4.4;2.4 The Transformation Procedure for the Parabolic Case;119
4.5;2.5 The Layer Stripping with Respect to the Pseudo Frequency s;122
4.6;2.6 The Approximately Globally Convergent Algorithm;125
4.6.1;2.6.1 The First Version of the Algorithm;127
4.6.2;2.6.2 A Simplified Version of the Algorithm;128
4.7;2.7 Some Properties of the Laplace Transform of the Solution of the Cauchy Problem (2.1) and (2.2);131
4.7.1;2.7.1 The Study of the Limit (2.12);131
4.7.2;2.7.2 Some Additional Properties of the Solution of the Problem (2.11) and (2.12);134
4.8;2.8 The First Approximate Global Convergence Theorem;138
4.8.1;2.8.1 Exact Solution;139
4.8.2;2.8.2 The First Approximate Global Convergence Theorem;141
4.8.3;2.8.3 Informal Discussion of Theorem 2.8.2;153
4.8.4;2.8.4 The First Approximate Mathematical Model;154
4.9;2.9 The Second Approximate Global Convergence Theorem;156
4.9.1;2.9.1 Estimates of the Tail Function;158
4.9.2;2.9.2 The Second Approximate Mathematical Model;167
4.9.3;2.9.3 Preliminaries;171
4.9.4;2.9.4 The Second Approximate Global Convergence Theorem;173
4.10;2.10 Summary;182
5;Chapter3 Numerical Implementation of the Approximately Globally Convergent Method;184
5.1;3.1 Numerical Study in 2D;185
5.1.1;3.1.1 The Forward Problem;186
5.1.2;3.1.2 Main Discrepancies Between the Theory and the Numerical Implementation;188
5.1.3;3.1.3 Results of the Reconstruction;189
5.2;3.2 Numerical Study in 3D;201
5.2.1;3.2.1 Computations of the Forward Problem;201
5.2.2;3.2.2 Result of the Reconstruction;203
5.3;3.3 Summary of Numerical Studies;206
6;Chapter4 The Adaptive Finite Element Technique and Its Synthesis with the Approximately Globally Convergent Numerical Method;208
6.1;4.1 Introduction;208
6.1.1;4.1.1 The Idea of the Two-Stage Numerical Procedure;208
6.1.2;4.1.2 The Concept of the Adaptivity for CIPs;209
6.2;4.2 Some Assumptions;211
6.3;4.3 State and Adjoint Problems;213
6.4;4.4 The Lagrangian;214
6.5;4.5 A Posteriori Error Estimate for the Lagrangian;217
6.6;4.6 Some Estimates of the Solution an Initial Boundary Value Problem for Hyperbolic Equation (4.9);225
6.7;4.7 Fréchet Derivatives of Solutions of State and Adjoint Problems;231
6.8;4.8 The Fréchet Derivative of the Tikhonov Functional;237
6.9;4.9 Relaxation with Mesh Refinements;240
6.9.1;4.9.1 The Space of Finite Elements;241
6.9.2;4.9.2 Minimizers on Subspaces;244
6.9.3;4.9.3 Relaxation;248
6.10;4.10 From the Abstract Scheme to the Coefficient Inverse Problem 2.1;250
6.11;4.11 A Posteriori Error Estimates for the Regularized Coefficient and the Relaxation Property of Mesh Refinements;252
6.12;4.12 Mesh Refinement Recommendations;256
6.13;4.13 The Adaptive Algorithm;259
6.13.1;4.13.1 The Algorithm In Brief;259
6.13.2;4.13.2 The Algorithm;259
6.14;4.14 Numerical Studies of the Adaptivity Technique;260
6.14.1;4.14.1 Reconstruction of a Single Cube;261
6.14.2;4.14.2 Scanning Acoustic Microscope;263
6.14.2.1;4.14.2.1 Ultrasound Microscopy;263
6.14.2.2;4.14.2.2 The Adaptivity Method for an Inverse Problem of Scanning Acoustic Microscopy;265
6.15;4.15 Performance of the Two-Stage Numerical Procedure in 2D;273
6.15.1;4.15.1 Computations of the Forward Problem;273
6.15.2;4.15.2 The First Stage;276
6.15.3;4.15.3 The Second Stage;279
6.16;4.16 Performance of the Two-Stage Numerical Procedure in 3D;281
6.16.1;4.16.1 The First Stage;290
6.16.2;4.16.2 The Second Stage;292
6.17;4.17 Numerical Study of the Adaptive Approximately Globally Convergent Algorithm;294
6.17.1;4.17.1 Computations of the Forward Problem;300
6.17.2;4.17.2 Reconstruction by the Approximately Globally Convergent Algorithm;303
6.17.3;4.17.3 The Adaptive Part;304
6.18;4.18 Summary of Numerical Studies of Chapter 4;306
7;Chapter5 Blind Experimental Data;309
7.1;5.1 Introduction;309
7.2;5.2 The Mathematical Model;311
7.3;5.3 The Experimental Setup;312
7.4;5.4 Data Simulations;315
7.5;5.5 State and Adjoint Problems for Experimental Data;316
7.6;5.6 Data Pre-Processing;318
7.6.1;5.6.1 The First Stage of Data Immersing;318
7.6.2;5.6.2 The Second Stage of Data Immersing;321
7.7;5.7 Some Details of the Numerical Implementation of the Approximately Globally Convergent Algorithm;323
7.7.1;5.7.1 Stopping Rule for ;325
7.8;5.8 Reconstruction by the Approximately Globally Convergent Numerical Method;325
7.8.1;5.8.1 Dielectric Inclusions and Their Positions;325
7.8.2;5.8.2 Tables and Images;326
7.8.3;5.8.3 Accuracy of the Blind Imaging;328
7.8.4;5.8.4 Performance of a Modified Gradient Method;330
7.9;5.9 Performance of the Two-Stage Numerical Procedure;333
7.9.1;5.9.1 The First Stage;333
7.9.2;5.9.2 The Third Stage of Data Immersing;334
7.9.3;5.9.3 Some Details of the Numerical Implementation of the Adaptivity;337
7.9.4;5.9.4 Reconstruction Results for Cube Number 1;337
7.9.4.1;5.9.4.1 The First Stage of Mesh Refinements;338
7.9.4.2;5.9.4.2 The Second Stage of Mesh Refinements;338
7.9.5;5.9.5 Reconstruction Results for the Cube Number 2;339
7.9.5.1;5.9.5.1 The First Stage;340
7.9.5.2;5.9.5.2 The Second Stage;340
7.9.6;5.9.6 Sensitivity to the Parameters a and b;341
7.9.7;5.9.7 Additional Effort for Cube Number 1;341
7.10;5.10 Summary;346
8;Chapter6 Backscattering Data;349
8.1;6.1 Introduction;349
8.2;6.2 Forward and Inverse Problems;351
8.3;6.3 Laplace Transform;353
8.4;6.4 The Algorithm;354
8.4.1;6.4.1 Preliminaries;354
8.4.2;6.4.2 The Sequence of Elliptic Equations;356
8.4.3;6.4.3 The Iterative Process;358
8.4.4;6.4.4 The Quasi-Reversibility Method;359
8.5;6.5 Estimates for the QRM;360
8.6;6.6 The Third Approximate Mathematical Model;368
8.6.1;6.6.1 Exact Solution;368
8.6.2;6.6.2 The Third Approximate Mathematical Model;370
8.7;6.7 The Third Approximate Global Convergence Theorem;372
8.8;6.8 Numerical Studies;381
8.8.1;6.8.1 Main Discrepancies Between Convergence Analysis and Numerical Implementation;381
8.8.2;6.8.2 A Simplified Mathematical Model of Imaging of Plastic Land Mines;382
8.8.3;6.8.3 Some Details of the Numerical Implementation;383
8.8.4;6.8.4 Numerical Results;386
8.8.5;6.8.5 Backscattering Without the QRM;388
8.9;6.9 Blind Experimental Data Collected in the Field;390
8.9.1;6.9.1 Introduction;392
8.9.2;6.9.2 Data Collection and Imaging Goal;393
8.9.3;6.9.3 The Mathematical Model and the Approximately Globally Convergent Algorithm;395
8.9.4;6.9.4 Uncertainties;399
8.9.5;6.9.5 Data Pre-processing;402
8.9.6;6.9.6 Results of Blind Imaging;405
8.9.7;6.9.7 Summary of Blind Imaging;406
9;References;407
10;Index;414