A parameter choice strategy for a multi-level augmentation method solving ill-posed operator equations
type: Item
instance of: scholarly article
title: A parameter choice strategy for a multi-level augmentation method solving ill-posed operator equations
zbMATH Open document ID: 1172.65025
DOI: 10.1216/JIE-2008-20-4-569
author: Zhongying Chen
author: Ying Jiang
author: Lihong Song
author: Hongqi Yang
published in: Journal of Integral Equations and Applications
publication date: 2009-02-19T00:00:00Z
review text: The problem of solving the first kind operator equation (1) \(K_x= y\), where \(K\) is a linear compact operator from a Hilbert space \(X\) to another Hilbert space \(X\) is considered. In practice, one has to solve the perturbed operator equation (2) \(K^\delta_x= y^\delta\) with \(\| y- y^\delta\|\leq\delta\), where \(\delta> 0\) is an error level. The above problem is ill-posed when the range of \(K\) is not closed and therefore regularization methods are needed to obtain a stable approximate solution. The purpose of this paper is to develop a new a posteriori regularization parameter choice strategy for the multi-level augmentation method based on a multiscale decomposition of the range space of the operator and the solution space of the equation for solving ill-posed operator equations of the first kind. An optimal order of convergence for the approximate solution obtained from the multilevel augmentation method using the a posteriori regularization parameter is established. Some numerical examples to illustrate the effectiveness of the algorithm proposed by the authors are presented.
reviewed by: Otu Vaarmann
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A52
zbMATH DE Number: 5509280
zbMATH Keywords: ill-posed linear operator equations
zbMATH Keywords: Tikhonov regularization method
zbMATH Keywords: multi-level augmentation method
zbMATH Keywords: a posteriori parameter choice strategy
zbMATH Keywords: Hilbert space
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
cites work: Q5314382
cites work: A multilevel augmentation method for solving ill-posed operator equations
cites work: Wavelet-Galerkin methods for ill-posed problems
cites work: Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates
cites work: Q4895893
cites work: Q4289802
cites work: Self-regularization by projection for noisy pseudodifferential equations of negative order
cites work: On the regularizing properties of a full multigrid method for ill-posed problems
cites work: Discretization strategy for linear ill-posed problems in variable Hilbert scales
cites work: On the regularization of projection methods for solving ill-posed problems
cites work: Q5432771
full work available at URL: jie-2008-20-4-569
OpenAlex ID: W2046951210
description: scientific article; zbMATH DE number 5509280