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Regularization theory for ill-posed problems : selected topics /
Thismonograph is a valuable contribution to thehighly topical and extremly productive field ofregularisationmethods for inverse and ill-posed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a well-balanced mixtureof basic and innov...
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Berlin ; Boston :
Walter de Gruyter GmbH & Co. KG,
[2013]
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| Colección: | Inverse and ill-posed problems series.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; 1 An introduction using classical examples; 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle; 1.1.1 Finite-difference formulae; 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize; 1.1.3 A posteriori choice of the stepsize; 1.1.4 Numerical illustration; 1.1.5 The balancing principle in a general framework; 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties; 1.2.1 Summation methods.
- 1.2.2 Deterministic noise model1.2.3 Stochastic noise model; 1.2.4 Smoothness associated with a basis; 1.2.5 Approximation and stability properties of -methods; 1.2.6 Error bounds; 1.3 The elliptic Cauchy problem and regularization by discretization; 1.3.1 Natural linearization of the elliptic Cauchy problem; 1.3.2 Regularization by discretization; 1.3.3 Application in detecting corrosion; 2 Basics of single parameter regularization schemes; 2.1 Simple example for motivation; 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization.
- 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models2.3.1 The best possible accuracy for the deterministic noise model; 2.3.2 The best possible accuracy for the Gaussian white noise model; 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces; 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales; 2.6 Estimation of linear functionals from indirect noisy observations; 2.7 Regularization by finite-dimensional approximation.
- 2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by least-squares methods; 2.8.2 Operator monotone functions; 2.8.3 The problem of model selection (continuation); 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited); 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs; 2.9.2 Error bounds in L2; 2.9.3 Adaptation to the unknown bound of the approximation error; 2.9.4 Numerical differentiation in the space of continuous functions.
- 2.9.5 Relation to the Savitzky-Golay method. Numerical examples3 Multiparameter regularization; 3.1 When do we really need multiparameter regularization?; 3.2 Multiparameter discrepancy principle; 3.2.1 Model function based on the multiparameter discrepancy principle; 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle; 3.2.3 Properties of the model function approximation; 3.2.4 Discrepancy curve and the convergence analysis; 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle.