Equivalence of regularization and truncated iteration for general ill-posed problems
type: Item
instance of: scholarly article
title: Equivalence of regularization and truncated iteration for general ill-posed problems
zbMATH Open document ID: 0849.65021
DOI: 10.1016/0024-3795(94)00114-6
author: Reginaldo J. Santos
published in: Linear Algebra and its Applications
publication date: 1996-11-11T00:00:00Z
review text: The linear algebraic equations deduced from discretizing a Fredholm integral equation of the first kind is usually very ill-posed. One standard way to obtain stable solutions is to modify the problem with the Tikhonov regularization. Another is to apply an iterative method to the normal equations. \textit{H. E. Fleming} [ibid. 130, 133-150 (1990; Zbl 0714.65045)] has established an equivalence between the two types of methods if the coefficient matrix has full rank. It is proven that every direct regularization method of a very general type is equivalent to a truncated iterative method and vice versa. This paper extends these results to incomplete-rank matrices. By using a formula for general iterative methods, a simpler and unified proof is given for both the overdetermined and the underdetermined case.
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
zbMATH DE Number: 871254
zbMATH Keywords: ill-posed problem
zbMATH Keywords: Fredholm integral equation of the first kind
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: iterative method
zbMATH Keywords: normal equations
zbMATH Keywords: incomplete-rank matrices
reviewed by: Liu Xinguo
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00114-6
OpenAlex ID: W2040100614
cites work: Q3207128
cites work: The Convergence of Linear Stationary Iterative Processes for Solving Singular Unstructured Systems of Linear Equations
cites work: Equivalence of regularization and truncated iteration in the solution of ill-posed image reconstruction problems
cites work: Q3359644
cites work: Q3334886
cites work: Regularization methods for large-scale problems
cites work: An Iteration Formula for Fredholm Integral Equations of the First Kind
cites work: Theory and Methods Related to the Singular-Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind
cites work: Q4125663
cites work: Q3779677
cites work: Q5604166
description: scientific article; zbMATH DE number 871254