- Description
- 1 online resource
- Additional Authors
- Romanov, Vladimir G.
- Notes
- 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.Includes bibliographical references and index.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.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.--