Numerical methods for inverse problems 1st Edition by Michel Kern ISBN 978-1848218185 1848218184
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ISBN 10: 1848218184
ISBN 13: 978-1848218185
Author: Michel Kern
This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system.
The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications.
This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.
Table of contents:
Part 1 Introduction and Examples
Chapter 1 Overview of Inverse Problems
1.1. Direct and inverse problems
1.2. Well-posed and ill-posed problems
Chapter 2 Examples of Inverse Problems
2.1. Inverse problems in heat transfer
2.2. Inverse problems in hydrogeology
2.3. Inverse problems in seismic exploration
2.4. Medical imaging
2.5. Other examples
Part 2 Linear Inverse Problems
Chapter 3 Integral Operators and Integral Equations
3.1. Definition and first properties
3.2. Discretization of integral equations
3.2.1. Discretization by quadrature–collocation
3.2.2. Discretization by the Galerkin method
3.3. Exercises
Chapter 4 Linear Least Squares Problems – Singular Value Decomposition
4.1. Mathematical properties of least squares problems
4.1.1. Finite dimensional case
4.2. Singular value decomposition for matrices
4.3. Singular value expansion for compact operators
4.4. Applications of the SVD to least squares problems
4.4.1. The matrix case
4.4.2. The operator case
4.5. Exercises
Chapter 5 Regularization of Linear Inverse Problems
5.1. Tikhonov’s method
5.1.1. Presentation
5.1.2. Convergence
5.1.3. The L-curve
5.2. Applications of the SVE
5.2.1. SVE and Tikhonov’s method
5.2.2. Regularization by truncated SVE
5.3. Choice of the regularization parameter
5.3.1. Morozov’s discrepancy principle
5.3.2. The L-curve
5.3.3. Numerical methods
5.4. Iterative methods
5.5. Exercises
Part 3 Nonlinear Inverse Problems
Chapter 6 Nonlinear Inverse Problems – Generalities
6.1. The three fundamental spaces
6.2. Least squares formulation
6.2.1. Difficulties of inverse problems
6.2.2. Optimization, parametrization, discretization
6.3. Methods for computing the gradient – the adjoint state method
6.3.1. The finite difference method
6.3.2. Sensitivity functions
6.3.3. The adjoint state method
6.3.4. Computation of the adjoint state by the Lagrangian
6.3.5. The inner product test
6.4. Parametrization and general organization
6.5. Exercises
Chapter 7 Some Parameter Estimation Examples
7.1. Elliptic equation in one dimension
7.1.1. Computation of the gradient
7.2. Stationary diffusion: elliptic equation in two dimensions
7.2.1. Computation of the gradient: application of the general method
7.2.2. Computation of the gradient by the Lagrangian
7.2.3. The inner product test
7.2.4. Multiscale parametrization
7.2.5. Example
7.3. Ordinary differential equations
7.3.1. An application example
7.4. Transient diffusion: heat equation
7.5. Exercises
Chapter 8 Further Information
8.1. Regularization in other norms
8.1.1. Sobolev semi-norms
8.1.2. Bounded variation regularization norm
8.2. Statistical approach: Bayesian inversion
8.2.1. Least squares and statistics
8.2.2. Bayesian inversion
8.3. Other topics
8.3.1. Theoretical aspects: identifiability
8.3.2. Algorithmic differentiation
8.3.3. Iterative methods and large-scale problems
8.3.4. Software
Appendices
Appendix 1
Appendix 2
Appendix 3
Bibliography
Index
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