A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems
zbMATH Open document ID: 1248.65056
author: Chuan-Gang Kang
author: Zhitao Wu
published in: Applied Mathematics and Computation
publication date: 2012-06-11T00:00:00Z
reviewed by: Werner H. Schmidt
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47J06
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 65L08
zbMATH DE Number: 6045292
zbMATH Keywords: nonlinear ill-posed problem
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: nonlinear implicite iterative method
zbMATH Keywords: Hanke criterion
zbMATH Keywords: discrepancy principle
zbMATH Keywords: Landweber scheme
zbMATH Keywords: numerical examples
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: convergence
zbMATH Keywords: stability
zbMATH Keywords: two-point boundary value problem
MaRDI profile type: Publication
full work available at URL: j.amc.2011.11.011
OpenAlex ID: W2061942528
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DOI: 10.1016/J.AMC.2011.11.011
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review text: Assume, a problem is modeled by a nonlinear operator equation \(F(x)= y\) in Hilbert spaces, and that the problem is ill-posed: The available data stems from a measurement process. Due to measurement errors, we have to deal with noise data \(y^\delta\) and we have to solve an equation \(F(x)= y^\delta\). The minimizer \(x^\delta_\alpha\) of the Tikhonov function NEWLINE\[NEWLINEJ_\alpha(x,\overline x)=\| y^\delta- F(x)\|^2+ \alpha\| x-\overline x\|^2NEWLINE\]NEWLINE is regarded as a regularized solution of the disturbed equation. In this paper, under certain assumptions a nonlinear implicit iterative scheme is proposed: NEWLINE\[NEWLINEx^\delta_{k+1}= x^\delta_k+{1\over\alpha_k} F'(x^\delta_k)^*(y^\delta- F(x^\delta_k)).NEWLINE\]NEWLINE This is the classical Landweber iterative method as \(\alpha_k= 1\). Under the restriction that \(\alpha_k\) is appropriate large, the monotonicity of iterative errors and the convergence and stability of the iterative sequence is proved.NEWLINENEWLINE The convergence and stability of the proposed method is also analyzed when the numbers \(\alpha_k\) are determined by the Hanke criterion. Numerical tests (for a two-point boundary value problem) show, that the method under consideration for nonlinear ill-posed problems is efficient.
description: scientific article; zbMATH DE number 6045292