Description 
1 online resource (xiv, 289 pages) : illustrations 
Series 
Inverse and IllPosed Problems Series 

Inverse and illposed 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 Finitedifference formulae; 1.1.2 Finitedifference 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 illposed linear operator equations. Leastsquares 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 finitedimensional approximation 

2.8 Model selection based on indirect observation in Gaussian white noise2.8.1 Linear models given by leastsquares 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 SavitzkyGolay 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 illposed problems. The author is an internationally outstanding and acceptedmathematicianin this field. In his book he offers a wellbalanced 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 

English 

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 

3110286491 
