Introduction to Inverse Problems for Differential Equations

✍ Scribed by Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 521
• Edition: 2nd ed. 2021
• Category: Library

✦ Synopsis

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.

For the second edition, the authors have added two new chapters focusing on real-world applications of inverse problems arising in wave and vibration phenomena. They have also revised the whole text of the first edition.

✦ Table of Contents

Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction Ill-Posedness of Inverse Problems
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 Regularization
2.6 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 the Input and Output. Solvability and Stability Estimate
3.1.4 The Regularization Strategy by SVD. Truncated SVD
3.2 Inverse Source Problems for Wave Equation
3.2.1 Non-uniqueness and Uniqueness of a Solution
3.3 Backward Parabolic Problem
3.4 Computational Issues in Inverse Source Problems
3.4.1 The Galerkin Finite Element Method
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 Source Term
4.1.2 Recovering a Spacewise Dependent Source Term
4.2 Problem of Recovering the Potential in String Equation
4.2.1 Some Properties of the Direct Problem
4.2.2 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 in Electrodynamics
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 the Permittivity Coefficient
5.3.2 Problem of Finding the Conductivity Coefficient
6 Inverse Problems for Parabolic Equations
6.1 Relationship Between Solutions of Parabolic and Hyperbolic Direct Problems
6.2 Problem of Recovering the Potential in Heat Equation
6.3 Uniqueness Theorems for Inverse Problems
6.4 Relationship With Inverse Spectral Problems for Sturm-Liouville Operator
6.5 Identification of a Leading Coefficient From Dirichlet 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
6.5.3 Integral Relationship and Gradient Formula
6.5.4 Reconstruction of an Unknown Coefficient
6.6 Identification of a Leading Coefficient From Neumann Measured Output
6.6.1 Compactness of the Input-Output Operator
6.6.2 Lipschitz Continuity of the Input-Output Operator and Solvability of the Inverse Problem
6.6.3 Integral Relationship and Gradient Formula
7 Inverse Problems for Elliptic Equations
7.1 The Inverse Scattering Problem at a Fixed Energy
7.2 The Inverse Scattering Problem with Point Sources
7.3 Dirichlet-to-Neumann Map
8 Inverse Problems for Stationary Transport Equations
8.1 The Transport Equation Without Scattering
8.2 Uniqueness and a Stability Estimate in the Tomography Problem
8.3 Inversion Formula
9 The Inverse Kinematic Problems
9.1 The Problem Formulation
9.2 Rays and Fronts
9.3 The One-Dimensional Problem
9.4 The Two-Dimensional Problem
Part III Inverse Problems in Wave Phenomena and Vibration
10 Inverse Problems for Damped Wave Equations
10.1 Determination of Principal Coefficient from Dirichlet-to-Neumann Operator
10.1.1 Structure of Solutions of Wave Equation and Dependence on the Dirichlet Input
10.1.2 Necessary Estimates for the Direct Problem Solution
10.1.3 The Dirichlet-to-Neumann Operator
10.1.4 Existence of a Quasi-Solution of the Inverse Problem
10.1.5 Uniqueness of the Solution to the Inverse Problem
10.1.6 Fréchet Differentiability of the Tikhonov Functional
10.2 Recovering a Potential from Neumann-to-Dirichlet Operator
10.2.1 Regularity Properties of the Direct Problem Solution in Subdomains Defined by the Final Time and Characteristics
10.2.2 Existence and Uniqueness of the Inverse Problem Solution
10.2.3 Necessary Estimates for the Weak and Regular Weak Solutions of the Direct Problem
10.2.4 The Neumann-to-Dirichlet Operator
10.2.5 Existence of a Quasi-Solution. Gradient Formula for the Tikhonov Functional
10.3 Recovering a Potential From Dirichlet-to-Neumann Operator
10.3.1 Regularity Properties of the Direct Problem Solution. Formation of Discontinuities
10.3.2 Local Existence Theorem and Stability Estimate
10.3.3 Necessary Estimates for the Weak and Regular Weak Solutions of the Direct Problem
10.3.4 The Dirichlet-to-Neumann Operator
10.3.5 Fréchet Differentiability of the Tikhonov Functional. Gradient Formula
10.3.6 Lipschitz Continuity of the Fréchet Gradient
11 Inverse Problems for Euler-Bernoulli Beam and Kirchhoff Plate Equations
11.1 Initial Boundary Value Problems for Dynamic Euler-Bernoulli Equation
11.1.1 The Initial Boundary Value Problem for Simply Supported Beam
11.1.2 The Initial Boundary Value Problem for a Cantilever Beam Under Free-End Shear Force
11.2 Identification of a Temporal Load in a Simply Supported Beam From Boundary Measured Slope
11.2.1 The Input-Output Operator: Compactness and Lipschitz Continuity
11.2.2 Existence of a Quasi-Solution of the Inverse Problem
11.2.3 Fréchet Differentiability of the Tikhonov Functional. Gradient Formula
11.2.4 Lipschitz Continuity of the Fréchet Gradient
11.2.5 Reconstruction of an Unknown Temporal Load from Boundary Measured Slope
11.3 Determination of Unknown Shear Force in a Cantilever Beam From Boundary Measured Bending Moment
11.3.1 The Neumann-to-Neumann Operator and Existence of a Quasi-Solution
11.3.2 Fréchet Gradient of the Tikhonov Functional. Lipschitz Continuity of the Gradient
11.3.3 Numerical Reconstruction of Unknown Shear Force from Boundary Measured Moment
11.4 Unique Recovery of an Unknown Spatial Load in Damped Beam Equation from Final Time Output
11.4.1 The Singular Value Decomposition and Sufficient Condition for the Uniqueness
11.4.2 Application to Forced Vibration Under Pure Spatial Load
11.4.3 Application to Forced Vibration Under Harmonic Load
11.4.4 Adjoint Method Based on the Quasi-Solution Approach
11.4.5 Numerical Reconstruction of Unknown Spatial Load from Final Time Output
11.5 Determination of the Flexural Rigidity of a Simply Supported Damped Beam from Measured Boundary Slope
11.5.1 Properties of the Input-Output Operator and Existence of a Quasi-Solution
11.5.2 Fréchet Differentiability of the Tikhonov Functional and Gradient Formula
11.6 Determination of the Flexural Rigidity of a Cantilever Beam from Measured Boundary Bending Moment
11.6.1 The Neumann-to-Neumann Operator. Existence of a Quasi-Solution
11.6.2 Fréchet Differentiability of the Tikhonov Functional and Gradient Formula
11.7 Spatial Load Identification in a Vibrating Kirchhoff Plate from Measured Boundary Slope
11.7.1 Necessary Estimates for the Direct Problem Solution
11.7.2 The Input-Output Operator. Existence of a Quasi-Solution
11.7.3 Fréchet Differentiability of the Tikhonov Functional. Gradient Formula
A Invertibility of Linear Operators
A.1 Invertibility of Bounded Below Linear Operators
A.2 Invertibility of Linear Compact Operators
B Necessary Estimates For One-Dimensional Parabolic Equation
B.1 The Problem with Non-homogeneous Initial Data and Source
B.2 The Problem with Neumann Input
B.3 The Problem with Dirichlet Input
C Necessary Estimates For Euler-Bernoulli Beam Equation
C.1 Existence of Weak Solutions
C.1.1 Uniform Estimate in Finite Dimensional Subspace
C.1.2 Existence and Uniqueness Theorems
C.2 The Problem with Neumann Inputs (Bending Moments)
C.3 The Problem with Dirichlet Input (Slope)
C.4 The Problem with Non-homogeneous Initial Data
Bibliography
Index