Lavrentiev regularization for linear ill-posed problems under general source conditions
type: Item
instance of: scholarly article
title: Lavrentiev regularization for linear ill-posed problems under general source conditions
zbMATH Open document ID: 1063.65041
DOI: 10.4171/ZAA/1192
author: M. Thamban Nair
author: Ulrich Tautenhahn
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-10-28T00:00:00Z
review text: The Lavrentiev regularization \( A x_\alpha^\delta + \alpha x_\alpha^\delta = y^\delta \) for the approximate solution of ill-posed problems (1) \( Ax = y \) is considered. Here, \( \alpha > 0 \) denotes a regularization parameter and \( A: X \to X \) is a linear bounded injective selfadjoint nonnegative operator in a Hilbert space \( X \). In addition, \( y^\delta \in X \) is an approximation to the right-hand side \( y \in R(A) \) of equation (1) satisfying \( \| y^\delta - y \| \leq \delta \), where \( \| \cdot \| \) denotes the norm of the Hilbert space \( X \). For general source representations of the form \( x^\dagger = \varphi(A)z, \, \| z \| \leq E \), and under some additional conditions, the authors prove estimates of the form \( \| x_\alpha^\delta - x^\dagger \| \leq cE \rho^{-1}(\delta/E) \) for a priori- as well as a posteriori parameter choices of \( \alpha \), respectively. Here, \( x^\dagger \) denotes the solution of equation (1), and \( \rho(\lambda)= \lambda \varphi^{-1}(\lambda) \) holds. In addition, \( c \) is a nonnegative constant. The presented results then are extended to an iterated version of the method of Lavrentiev as well as to infinitely smoothing operators, respectively. Finally the results are applied to an inverse heat conduction problem.
reviewed by: Robert Plato
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65M32
zbMATH DE Number: 2111233
zbMATH Keywords: linear ill-posed problems
zbMATH Keywords: Lavrentiev regularization
zbMATH Keywords: regularization methods
zbMATH Keywords: general source conditions
zbMATH Keywords: a priori parameter choices
zbMATH Keywords: a posteriori parameter choices
zbMATH Keywords: inverse heat conduction problem
zbMATH Keywords: Hilbert space
MaRDI profile type: Publication
cites work: Moment theory and some inverse problems in potential theory and heat conduction
cites work: An a posteriori parameter choice for simplified regularization of ill- posed problems
cites work: A class of discrepancy principles for the simplified regularization of ill-posed problems
cites work: Error bounds and parameter choice strategies for simplified regularization in Hilbert scales
cites work: An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales
cites work: Arcangeli's Method for Fredholm Equations of the First Kind
cites work: Q5751877
cites work: Regularization of exponentially ill-posed problems
cites work: Q5529214
cites work: Q3994510
cites work: Tikhonov Regularization for Finitely and Infinitely Smoothing Operators
cites work: Geometry of linear ill-posed problems in variable Hilbert scales
cites work: Q5578503
cites work: On Morozov's method for Tikhonov regularization as an optimal order yielding algorithm
cites work: Q4506148
cites work: Q5311855
cites work: Morozov's discrepancy principle under general source conditions
cites work: Morozov's discrepancy principle for tikhonov
cites work: Resolvent estimates for Abel integral operators and the regularization of associated first kind integral equations
cites work: Q3687611
cites work: Ritz-regularization versus least-square-regularization. Solution methods for integral equations of the first kind
cites work: Optimality for ill-posed problems under general source conditions
cites work: On the method of Lavrentiev regularization for nonlinear ill-posed problems
full work available at URL: https://doi.org/10.4171/zaa/1192
OpenAlex ID: W2036725977
description: scientific article; zbMATH DE number 2111233