An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems
zbMATH Open document ID: 1206.47060
DOI: 10.1216/JIE-2010-22-3-369
author: Radu I. Boţ
author: Bernd Hofmann
published in: Journal of Integral Equations and Applications
publication date: 2011-01-12T00:00:00Z
review text: The paper is concerned with nonlinear ill-posed operator equations \(F(x)=y\), where \(F\) is a mapping between real Banach spaces \(X\) and \(Y\). The exact element \(y\) is given by its approximation \(y^{\delta}\), \(\| y^{\delta}-y\|_Y \leq \delta\). The authors establish rate of convergence estimates for variational regularization methods associated with the Tikhonov type functional \(T_{\alpha}^{\delta}(x)=\psi(\| F(x)-y^{\delta}\|_Y)+\alpha \Omega(x)\). The estimates are given in terms of the Bregman distance and a generalized source condition presented as a variational inequality.
Mathematics Subject Classification ID: 47J06
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 49N45
zbMATH DE Number: 5835147
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Tikhonov regularisation
zbMATH Keywords: convergence rates
zbMATH Keywords: variational inequalities
zbMATH Keywords: source conditions
zbMATH Keywords: distance function
zbMATH Keywords: structure of nonlinearity
zbMATH Keywords: generalised Young inequality
reviewed by: Mikhail Yu. Kokurin
MaRDI profile type: Publication
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full work available at URL: jie-2010-22-3-369
OpenAlex ID: W2015551786
description: scientific article; zbMATH DE number 5835147