Hasanov Hasanoglu / Romanov Introduction to Inverse Problems for Differential Equations

1. Auflage 2017
ISBN: 978-3-319-62797-7
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 264 Seiten

This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering.The book's content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations.
In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here are based on the basic and commonly used mathematical models governed by PDEs. These chapters describe not only these inverse problems, but also main inversion methods and techniques. Since the most distinctive features of any inverse problems related to PDEs are hidden in the properties of the corresponding solutions to direct problems, special attention is paid to the investigation of these properties.

Autoren/Hrsg.

Hasanov Hasanoglu, Alemdar

Romanov, Vladimir G.

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;1 Introduction Ill-Posedness of Inverse Problems for Differential and Integral Equations;13
3.1;1.1 Some Basic Definitions and Examples;13
3.2;1.2 Continuity with Respect to Coefficients and Source: Sturm-Liouville Equation;21
3.3;1.3 Why a Fredholm Integral Equation of the First Kind Is an Ill-Posed Problem?;25
4;Part I Introduction to Inverse Problems;32
5;2 Functional Analysis Background of Ill-Posed Problems;33
5.1;2.1 Best Approximation and Orthogonal Projection;34
5.2;2.2 Range and Null-Space of Adjoint Operators;41
5.3;2.3 Moore-Penrose Generalized Inverse;43
5.4;2.4 Singular Value Decomposition;48
5.5;2.5 Regularization Strategy. Tikhonov Regularization;55
5.6;2.6 Morozov's Discrepancy Principle;68
6;3 Inverse Source Problems with Final Overdetermination;73
6.1;3.1 Inverse Source Problem for Heat Equation;74
6.1.1;3.1.1 Compactness of Input-Output Operator and Fréchet Gradient;77
6.1.2;3.1.2 Singular Value Decomposition of Input-Output Operator;82
6.1.3;3.1.3 Picard Criterion and Regularity of Input/Output Data;89
6.1.4;3.1.4 The Regularization Strategy by SVD. Truncated SVD;94
6.2;3.2 Inverse Source Problems for Wave Equation;100
6.2.1;3.2.1 Non-uniqueness of a Solution;103
6.3;3.3 Backward Parabolic Problem;106
6.4;3.4 Computational Issues in Inverse Source Problems;114
6.4.1;3.4.1 Galerkin FEM for Numerical Solution of Forward Problems;115
6.4.2;3.4.2 The Conjugate Gradient Algorithm;117
6.4.3;3.4.3 Convergence of Gradient Algorithms for Functionals with Lipschitz Continuous Fréchet Gradient;122
6.4.4;3.4.4 Numerical Examples;126
7;Part II Inverse Problems for Differential Equations;130
8;4 Inverse Problems for Hyperbolic Equations;131
8.1;4.1 Inverse Source Problems;131
8.1.1;4.1.1 Recovering a Time Dependent Function;132
8.1.2;4.1.2 Recovering a Spacewise Dependent Function;134
8.2;4.2 Problem of Recovering the Potential for the String Equation;136
8.2.1;4.2.1 Some Properties of the Direct Problem;137
8.2.2;4.2.2 Existence of the Local Solution to the Inverse Problem;141
8.2.3;4.2.3 Global Stability and Uniqueness;146
8.3;4.3 Inverse Coefficient Problems for Layered Media;149
9;5 One-Dimensional Inverse Problems for Electrodynamic Equations;152
9.1;5.1 Formulation of Inverse Electrodynamic Problems;152
9.2;5.2 The Direct Problem: Existence and Uniqueness of a Solution;153
9.3;5.3 One-Dimensional Inverse Problems;162
9.3.1;5.3.1 Problem of Finding a Permittivity Coefficient;162
9.3.2;5.3.2 Problem of Finding a Conductivity Coefficient;167
10;6 Inverse Problems for Parabolic Equations;170
10.1;6.1 Relationships Between Solutions of Direct Problems for Parabolic and Hyperbolic Equations;170
10.2;6.2 Problem of Recovering the Potential for Heat Equation;173
10.3;6.3 Uniqueness Theorems for Inverse Problems Related to Parabolic Equations;175
10.4;6.4 Relationship Between the Inverse Problem and Inverse Spectral Problems for Sturm-Liouville Operator;178
10.5;6.5 Identification of a Leading Coefficient in Heat Equation: Dirichlet Type Measured Output;181
10.5.1;6.5.1 Some Properties of the Direct Problem Solution;182
10.5.2;6.5.2 Compactness and Lipschitz Continuity of the Input-Output Operator. Regularization;184
10.5.3;6.5.3 Integral Relationship and Gradient Formula;190
10.5.4;6.5.4 Reconstruction of an Unknown Coefficient;193
10.6;6.6 Identification of a Leading Coefficient in Heat Equation: Neumann Type Measured Output;198
10.6.1;6.6.1 Compactness of the Input-Output Operator;200
10.6.2;6.6.2 Lipschitz Continuity of the Input-Output Operator and Solvability of the Inverse Problem;204
10.6.3;6.6.3 Integral Relationship and Gradient Formula;207
11;7 Inverse Problems for Elliptic Equations;211
11.1;7.1 The Inverse Scattering Problem at a Fixed Energy;211
11.2;7.2 The Inverse Scattering Problem with Point Sources;214
11.3;7.3 Dirichlet to Neumann Map;219
12;8 Inverse Problems for the Stationary Transport Equations;225
12.1;8.1 The Transport Equation Without Scattering;225
12.2;8.2 Uniqueness and a Stability Estimate in the Tomography Problem;228
12.3;8.3 Inversion Formula;229
13;9 The Inverse Kinematic Problem;232
13.1;9.1 The Problem Formulation;232
13.2;9.2 Rays and Fronts;233
13.3;9.3 The One-Dimensional Problem;236
13.4;9.4 The Two-Dimensional Problem;239
14;Appendix A Invertibility of Linear Operators;243
15;Appendix B Some Estimates For One-Dimensional Parabolic Equation;251
16;References;257
17;Index;262