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Preface; Contents; 1 Introduction Ill-Posedness of Inverse Problems for Differential and Integral Equations; 1.1 Some Basic Definitions and Examples; 1.2 Continuity with Respect to Coefficients and Source: Sturm-Liouville Equation; 1.3 Why a Fredholm Integral Equation of the First Kind Is an Ill-Posed Problem?; Part I Introduction to Inverse Problems; 2 Functional Analysis Background of Ill-Posed Problems; 2.1 Best Approximation and Orthogonal Projection; 2.2 Range and Null-Space of Adjoint Operators; 2.3 Moore-Penrose Generalized Inverse; 2.4 Singular Value Decomposition
2.5 Regularization Strategy. Tikhonov Regularization2.6 Morozov's Discrepancy Principle; 3 Inverse Source Problems with Final Overdetermination; 3.1 Inverse Source Problem for Heat Equation; 3.1.1 Compactness of Input-Output Operator and Fréchet Gradient; 3.1.2 Singular Value Decomposition of Input-Output Operator; 3.1.3 Picard Criterion and Regularity of Input/Output Data; 3.1.4 The Regularization Strategy by SVD. Truncated SVD; 3.2 Inverse Source Problems for Wave Equation; 3.2.1 Non-uniqueness of a Solution; 3.3 Backward Parabolic Problem
3.4 Computational Issues in Inverse Source Problems3.4.1 Galerkin FEM for Numerical Solution of Forward Problems; 3.4.2 The Conjugate Gradient Algorithm; 3.4.3 Convergence of Gradient Algorithms for Functionals with Lipschitz Continuous Fréchet Gradient; 3.4.4 Numerical Examples; Part II Inverse Problems for Differential Equations; 4 Inverse Problems for Hyperbolic Equations; 4.1 Inverse Source Problems; 4.1.1 Recovering a Time Dependent Function; 4.1.2 Recovering a Spacewise Dependent Function; 4.2 Problem of Recovering the Potential for the String Equation
4.2.1 Some Properties of the Direct Problem4.2.2 Existence of the Local Solution to the Inverse Problem; 4.2.3 Global Stability and Uniqueness; 4.3 Inverse Coefficient Problems for Layered Media; 5 One-Dimensional Inverse Problems for Electrodynamic Equations; 5.1 Formulation of Inverse Electrodynamic Problems; 5.2 The Direct Problem: Existence and Uniqueness of a Solution; 5.3 One-Dimensional Inverse Problems ; 5.3.1 Problem of Finding a Permittivity Coefficient; 5.3.2 Problem of Finding a Conductivity Coefficient; 6 Inverse Problems for Parabolic Equations
6.1 Relationships Between Solutions of Direct Problems for Parabolic and Hyperbolic Equations6.2 Problem of Recovering the Potential for Heat Equation; 6.3 Uniqueness Theorems for Inverse Problems Related to Parabolic Equations; 6.4 Relationship Between the Inverse Problem and Inverse Spectral Problems for Sturm-Liouville Operator ; 6.5 Identification of a Leading Coefficient in Heat Equation: Dirichlet Type Measured Output; 6.5.1 Some Properties of the Direct Problem Solution; 6.5.2 Compactness and Lipschitz Continuity of the Input-Output Operator. Regularization
2.5 Regularization Strategy. Tikhonov Regularization2.6 Morozov's Discrepancy Principle; 3 Inverse Source Problems with Final Overdetermination; 3.1 Inverse Source Problem for Heat Equation; 3.1.1 Compactness of Input-Output Operator and Fréchet Gradient; 3.1.2 Singular Value Decomposition of Input-Output Operator; 3.1.3 Picard Criterion and Regularity of Input/Output Data; 3.1.4 The Regularization Strategy by SVD. Truncated SVD; 3.2 Inverse Source Problems for Wave Equation; 3.2.1 Non-uniqueness of a Solution; 3.3 Backward Parabolic Problem
3.4 Computational Issues in Inverse Source Problems3.4.1 Galerkin FEM for Numerical Solution of Forward Problems; 3.4.2 The Conjugate Gradient Algorithm; 3.4.3 Convergence of Gradient Algorithms for Functionals with Lipschitz Continuous Fréchet Gradient; 3.4.4 Numerical Examples; Part II Inverse Problems for Differential Equations; 4 Inverse Problems for Hyperbolic Equations; 4.1 Inverse Source Problems; 4.1.1 Recovering a Time Dependent Function; 4.1.2 Recovering a Spacewise Dependent Function; 4.2 Problem of Recovering the Potential for the String Equation
4.2.1 Some Properties of the Direct Problem4.2.2 Existence of the Local Solution to the Inverse Problem; 4.2.3 Global Stability and Uniqueness; 4.3 Inverse Coefficient Problems for Layered Media; 5 One-Dimensional Inverse Problems for Electrodynamic Equations; 5.1 Formulation of Inverse Electrodynamic Problems; 5.2 The Direct Problem: Existence and Uniqueness of a Solution; 5.3 One-Dimensional Inverse Problems ; 5.3.1 Problem of Finding a Permittivity Coefficient; 5.3.2 Problem of Finding a Conductivity Coefficient; 6 Inverse Problems for Parabolic Equations
6.1 Relationships Between Solutions of Direct Problems for Parabolic and Hyperbolic Equations6.2 Problem of Recovering the Potential for Heat Equation; 6.3 Uniqueness Theorems for Inverse Problems Related to Parabolic Equations; 6.4 Relationship Between the Inverse Problem and Inverse Spectral Problems for Sturm-Liouville Operator ; 6.5 Identification of a Leading Coefficient in Heat Equation: Dirichlet Type Measured Output; 6.5.1 Some Properties of the Direct Problem Solution; 6.5.2 Compactness and Lipschitz Continuity of the Input-Output Operator. Regularization