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Author Lu, Shuai, author

Title Regularization theory for ill-posed problems : selected topics / by Shuai Lu, Sergei V. Pereverzev
Published Berlin ; Boston : Walter de Gruyter GmbH & Co. KG, [2013]


Description 1 online resource (xiv, 289 pages) : illustrations
Series Inverse and Ill-Posed Problems Series
Inverse and ill-posed problems series.
Contents 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
Summary 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 innovative aspects. He demonstrates new, differentiatedviewpoints, and important examples for applications. The bookdemontrates thecurrent developments inthe field of regularization theory, such as multiparameter regularization and regularization in learning theory. The book is written for graduate and PhDs
Bibliography Includes bibliographical references and index
Notes 3.2.6 Generalization in the case of more than two regularization parameters
Print version record
Subject Numerical differentiation.
Numerical analysis -- Improperly posed problems.
MATHEMATICS -- Numerical Analysis.
Numerical analysis -- Improperly posed problems.
Numerical differentiation.
Form Electronic book
Author Pereverzev, Sergei V., author.
LC no. 2013014716
ISBN 9783110286496