Numerical methods for inverse problems [electronic resource] / Michel Kern.

Bibliographic Details
Uniform Title:Mathematics and statistics series (ISTE)
Main Author: Kern, Michel (Author)
Language:English
Published: London, UK : Hoboken, NJ, USA : ISTE ; Wiley & Sons, 2016.
Series:Mathematics and statistics series (ISTE)
Subjects:
Inverse problems (Differential equations) > Numerical solutions.
Online Access:
Wiley Online Library Online Books: 2016 (Wiley Online Library)
Wiley Frontlist All Obook 2016: 2016 (Wiley Online Library)
ProQuest Ebook Central - Academic Complete: 2016 (Ebook Central @ Proquest)
Wiley Obooks 2011 to Present: 2016 (Wiley Online Library)
Format: Electronic eBook
Contents:
  • Machine generated contents note: pt. 1 Introduction and Examples
  • ch. 1 Overview of Inverse Problems
  • 1.1. Direct and inverse problems
  • 1.2. Well-posed and ill-posed problems
  • ch. 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
  • pt. 2 Linear Inverse Problems
  • ch. 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
  • ch. 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
  • ch. 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
  • pt. 3 Nonlinear Inverse Problems
  • ch. 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
  • ch. 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
  • ch. 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.