Inverse Problems. Tikhonov Theory And Algorithms

✍ Scribed by Kazufumi Ito, Bangti Jin

• Publisher: World Scientific Publishing
• Year: 2015
• Tongue: English
• Leaves: 330
• Series: Series on Applied Mathematics, Volume 22
• Category: Library

✦ Table of Contents

Contents
Preface
1. Introduction
2. Models in Inverse Problems
2.1 Introduction
2.2 Elliptic inverse problems
2.2.1 Cauchy problem
2.2.2 Inverse source problem
2.2.3 Inverse scattering problem
2.2.4 Inverse spectral problem
2.3 Tomography
2.3.1 Computerized tomography
2.3.2 Emission tomography
2.3.3 Electrical impedance tomography
2.3.4 Optical tomography
2.3.5 Photoacoustic tomography
3. Tikhonov Theory for Linear Problems
3.1 Well-posedness
3.2 Value function calculus
3.3 Basic estimates
3.3.1 Classical source condition
3.3.2 Higher-order source condition
3.4 A posteriori parameter choice rules
3.4.1 Discrepancy principle
3.4.2 Hanke-Raus rule
3.4.3 Quasi-optimality criterion
3.5 Augmented Tikhonov regularization
3.5.1 Augmented Tikhonov regularization
3.5.2 Variational characterization
3.5.3 Fixed point algorithm
3.6 Multi-parameter Tikhonov regularization
3.6.1 Balancing principle
3.6.2 Error estimates
3.6.3 Numerical algorithms
Bibliographical notes
4. Tikhonov Theory for Nonlinear Inverse Problems
4.1 Well-posedness
4.2 Classical convergence rate analysis
4.2.1 A priori parameter choice
4.2.2 A posteriori parameter choice
4.2.3 Structural properties
4.3 A new convergence rate analysis
4.3.1 Necessary optimality condition
4.3.2 Source and nonlinearity conditions
4.3.3 Convergence rate analysis
4.4 A class of parameter identification problems
4.4.1 A general class of nonlinear inverse problems
4.4.2 Bilinear problems
4.4.3 Three elliptic examples
4.5 Convergence rate analysis in Banach spaces
4.5.1 Extensions of the classical approach
4.5.2 Variational inequalities
4.6 Conditional stability
Bibliographical notes
5. Nonsmooth Optimization
5.1 Existence and necessary optimality condition
5.1.1 Existence of minimizers
5.1.2 Necessary optimality
5.2 Nonsmooth optimization algorithms
5.2.1 Augmented Lagrangian method
5.2.2 Lagrange multiplier theory
5.2.3 Exact penalty method
5.2.4 Gauss-Newton method
5.2.5 Semismooth Newton Method
5.3
p sparsity optimization
5.3.1
0 optimization
5.3.2
p (0 < p < 1)-optimization
5.3.3 Primal-dual active set method
5.4 Nonsmooth nonconvex optimization
5.4.1 Biconjugate function and relaxation
5.4.2 Semismooth Newton method
5.4.3 Constrained optimization
6. Direct Inversion Methods
6.1 Inverse scattering methods
6.1.1 The MUSIC algorithm
6.1.2 Linear sampling method
6.1.3 Direct sampling method
6.2 Point source identification
6.3 Numerical unique continuation
6.4 Gel’fand-Levitan-Marchenko transformation
6.4.1 Gel’fand-Levitan-Marchenko transformation
6.4.2 Application to inverse Sturm-Liouville problem
Bibliographical notes
7. Bayesian Inference
7.1 Fundamentals of Bayesian inference
7.2 Model selection
7.3 Markov chain Monte Carlo
7.3.1 Monte Carlo simulation
7.3.2 MCMC algorithms
7.3.3 Convergence analysis
7.3.4 Accelerating MCMC algorithms
7.4 Approximate inference
7.4.1 Kullback-Leibler divergence
7.4.2 Approximate inference algorithms
Bibliographical notes
Appendix A Singular Value Decomposition
Appendix B Noise Models
Appendix C Exponential Families
Bibliography
Index