This graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering.
This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.
Author(s): Andreas Kirsch
Series: Applied Mathematical Sciences
Edition: 3
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 400
City: Cham
Tags: Inverse Problems, Regularization, Eigenvalue, Inverse Scattering
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction and Basic Concepts
1.1 Examples of Inverse Problems
1.2 Ill-Posed Problems
1.3 The Worst-Case Error
1.4 Problems
2 Regularization Theory for Equations of the First Kind
2.1 A General Regularization Theory
2.2 Tikhonov Regularization
2.3 Landweber Iteration
2.4 A Numerical Example
2.5 The Discrepancy Principle of Morozov
2.6 Landweber's Iteration Method with Stopping Rule
2.7 The Conjugate Gradient Method
2.8 Problems
3 Regularization by Discretization
3.1 Projection Methods
3.2 Galerkin Methods
3.2.1 The Least Squares Method
3.2.2 The Dual Least Squares Method
3.2.3 The Bubnov–Galerkin Method for Coercive Operators
3.3 Application to Symm's Integral Equation of the First Kind
3.4 Collocation Methods
3.4.1 Minimum Norm Collocation
3.4.2 Collocation of Symm's Equation
3.5 Numerical Experiments for Symm's Equation
3.6 The Backus–Gilbert Method
3.7 Problems
4 Nonlinear Inverse Problems
4.1 Local Illposedness
4.2 The Nonlinear Tikhonov Regularization
4.2.1 Existence of Solutions and Stability
4.2.2 Source Conditions And Convergence Rates
4.2.3 A Parameter-Identification Problem
4.2.4 A Glimpse on Extensions to Banach Spaces
4.3 The Nonlinear Landweber Iteration
4.4 Problems
5 Inverse Eigenvalue Problems
5.1 Introduction
5.2 Construction of a Fundamental System
5.3 Asymptotics of the Eigenvalues and Eigenfunctions
5.4 Some Hyperbolic Problems
5.5 The Inverse Problem
5.6 A Parameter Identification Problem
5.7 Numerical Reconstruction Techniques
5.8 Problems
6 An Inverse Problem in Electrical Impedance Tomography
6.1 Introduction
6.2 The Direct Problem and the Neumann–Dirichlet Operator
6.3 The Inverse Problem
6.4 The Factorization Method
6.5 Problems
7 An Inverse Scattering Problem
7.1 Introduction
7.2 The Direct Scattering Problem
7.3 Properties of the Far Field Patterns
7.4 Uniqueness of the Inverse Problem
7.5 The Factorization Method
7.6 The Interior Transmission Eigenvalue Problem
7.6.1 The Radially Symmetric Case
7.6.2 Discreteness And Existence in the General Case
7.6.3 The Inverse Spectral Problem for the Radially Symmetric Case
7.7 Numerical Methods
7.7.1 A Simplified Newton Method
7.7.2 A Modified Gradient Method
7.7.3 The Dual Space Method
7.8 Problems
A Basic Facts from Functional Analysis
A.1 Normed Spaces and Hilbert Spaces
A.2 Orthonormal Systems
A.3 Linear Bounded and Compact Operators
A.4 Sobolev Spaces of Periodic Functions
A.5 Sobolev Spaces on the Unit Disc
A.6 Spectral Theory for Compact Operators in Hilbert Spaces
A.7 The Fréchet Derivative
A.8 Convex Analysis
A.9 Weak Topologies
A.10 Problems
B Proofs of the Results of Section 2.7摥映數爠eflinksspscgsps22.72
