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Introduction to Inverse Problems for Differential Equations

:	Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
:	Introduction to Inverse Problems for Differential Equations
:	Springer-Verlag
:	9783319627977
:	1
:	CHF 85.40
:

:	Analysis
:	English

:	264
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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