Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems
zbMATH Open document ID: 1156.65053
DOI: 10.1216/JIE-2008-20-2-201
published in: Journal of Integral Equations and Applications
publication date: 2008-08-05T00:00:00Z
review text: In a recent paper [Numer. Funct. Anal. Optimization 27, No. 5--6, 637--665 (2006; Zbl 1101.65054)], the author suggested a multilevel method for solving ill-posed operator equations governed by a compact, weakly sequentially closed mapping \(F\) between two Hilbert spaces. This stable and globally convergent method combines regularization by discretization with an iterative approach. In this paper, optimal convergence rates of Hölder- or logarithmic type are proven under suitable assumptions on a linearization of \(F\) and on the initial guess. Furthermore, numerical aspects of the minimization problems appearing on each level of discretization are studied, and it is shown that it suffices to consider Karush-Kuhn-Tucker points rather than global minimizers. Finally, numerical tests are carried out for a nonlinear integral equation with an analytic kernel.
reviewed by: Etienne Emmrich
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47J06
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5306881
zbMATH Keywords: nonlinear equation
zbMATH Keywords: ill-posed problem
zbMATH Keywords: regularization
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: multilevel approach
zbMATH Keywords: nonlinear integral equation
zbMATH Keywords: numerical examples
zbMATH Keywords: Hilbert spaces
author: Barbara Kaltenbacher
MaRDI profile type: Publication
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full work available at URL: jie-2008-20-2-201
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