Morozov's discrepancy principle under general source conditions
type: Item
instance of: scholarly article
title: Morozov's discrepancy principle under general source conditions
zbMATH Open document ID: 1037.65057
DOI: 10.4171/ZAA/1140
author: M. Thamban Nair
author: Eberhard Schock
author: Ulrich Tautenhahn
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2003-01-01T00:00:00Z
review text: The authors study ill-posed problems \(Ax=b\) in a Hilbert space setting where instead of the exact data \(y\) some noisy data \(y^\delta\) are given such that \(\| y-y^\delta|\leq \delta\). They are interested in order optimality results of the regularized approximation of the minimum norm solution of the problem under the general source condition \[ x^+=[\varphi (A^*A)]^{1/2}v \quad {\text{with}}\quad \| v\| \leq E. \] Earlier \textit{U. Tautenhahn} [Numer. Funct. Anal. Optimization 19, 377--398 (1998; Zbl 0907.65049)] has shown that the classical Tikhonov regularization method combined with a special choice of the regularization parameter \(\alpha\) defined by the function \(\phi\) is optimal. Here the authors investigate regularization methods represented in the general form \[ x^{\delta}_{\alpha}=g_{\alpha}(A^*A)A^{*}y^{\delta} \] and combined with Morozov's discrepancy principle (independent of the function \(\varphi\)). They have found some analytical properties of the functions \(\varphi\) and \(d_{\alpha}\) that provide the order optimality of this simple regularization technique.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A52
zbMATH DE Number: 1895122
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: general source condition
zbMATH Keywords: order optimality
zbMATH Keywords: Morozov's discrepancy principle
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Hilbert space
zbMATH Keywords: minimum norm solution
reviewed by: Vladimir Gorbunov
MaRDI profile type: Publication
cites work: Q3334886
cites work: Regularization methods for large-scale problems
cites work: Q3745181
cites work: Regularization of exponentially ill-posed problems
cites work: Q3994510
full work available at URL: https://doi.org/10.4171/zaa/1140
OpenAlex ID: W2059325722
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description: scientific article; zbMATH DE number 1895122