Regularization theory for ill-posed problems. Selected topics
type: Item
instance of: scholarly article
title: Regularization theory for ill-posed problems. Selected topics
zbMATH Open document ID: 1282.47001
DOI: 10.1515/9783110286496
author: Shuai Lu
author: Sergei V. Pereverzyev
published in: Inverse and Ill-Posed Problems Series
publication date: 2013-08-14T00:00:00Z
reviewed by: Robert Plato
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 65J22
Mathematics Subject Classification ID: 93E35
zbMATH DE Number: 6198405
zbMATH Keywords: inverse problem
zbMATH Keywords: ill-posed problem
zbMATH Keywords: regularization method
zbMATH Keywords: elliptic Cauchy problem
zbMATH Keywords: balancing principle
zbMATH Keywords: discrepancy principle
zbMATH Keywords: summation method
zbMATH Keywords: \( \lambda \)-method
zbMATH Keywords: Féjer method
zbMATH Keywords: stochastic noise model
zbMATH Keywords: deterministic noise model
zbMATH Keywords: projection method
zbMATH Keywords: source condition
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Picard criterion
zbMATH Keywords: Moore-Penrose generalized inverse
zbMATH Keywords: single parameter regularization
zbMATH Keywords: Tikhonov-Phillips regularization
zbMATH Keywords: iterated Tikhonov regularization
zbMATH Keywords: Landweber iteration
zbMATH Keywords: spectral cut-off method
zbMATH Keywords: qualification of a regularization scheme
zbMATH Keywords: Hilbert scale
zbMATH Keywords: deconvolution
zbMATH Keywords: Abel integral equation of the first kind
zbMATH Keywords: multiparameter regularization
zbMATH Keywords: meta-learning
zbMATH Keywords: learning theory
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: blood glucose prediction
describes a project that uses: Regularization tools
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110286496
OpenAlex ID: W4247295697
review text: This text provides a clear treatment of the fundamental regularization methods for linear ill-posed problems in Hilbert spaces. It contains a number of topics that have been developed rather recently and that are not yet covered by other introductory books on ill-posed problems. This includes stochastic noise models, the balancing principle as a generic a posteriori parameter choice strategy, multiparameter regularization, the adaptive choice of the regularization space, and a meta-learning based approach to regularization.NEWLINENEWLINEThe text consists of five chapters. Chapter 1 provides an introduction to some basic concepts for the regularization of inverse problems using some classical examples. It starts with a section on finite difference schemes for numerical differentiation, with the stepsize used as regularization parameter. Special emphasis is put here on the balancing principle as an efficient a posteriori choice strategy which in fact is also introduced in a general framework. The next section is devoted to the regularized summation of orthogonal series based on \( \lambda \)-methods. Here, a priori parameter choices are considered both in a deterministic and in a stochastic setting. The third section deals with the Cauchy problem for elliptic partial differential equations. After a reformulation as an operator equation for determining the unknown Dirichlet data at the inaccessible part of the boundary, regularization by discretization based on projection schemes is considered here, with a selection of the discretization level by the balancing principle. An application to detection of corrosion is then considered.NEWLINENEWLINEChapter 2 is concerned with single parameter regularization methods. After summarizing all necessary mathematical tools like the Moore-Penrose generalized inverse and the singular value decomposition, the authors address examples for single parameter regularization methods, e.g., the spectral cut-off method, Tikhonov-Phillips reg…
description: scientific article; zbMATH DE number 6198405