Introduction to Inverse Problems for Differential Equations (eBook)

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