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Inverse problems : Tikhonov theory and algorithms /
Inverse problems arise in practical applications whenever one needs to deduce unknowns from observables. This monograph is a valuable contribution to the highly topical field of computational inverse problems. Both mathematical theory and numerical algorithms for model-based inverse problems are dis...
| Clasificación: | Libro Electrónico |
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| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
[Hackensack] New Jersey :
World Scientific,
2014.
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| Colección: | Series on applied mathematics.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Contents; 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 rules3.4.1 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 properties4.3 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 minimizers5.1.2 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 algorithm6.1.2 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.