What do we learn from the discrepancy principle?
type: Item
instance of: scholarly article
title: What do we learn from the discrepancy principle?
zbMATH Open document ID: 1120.65071
DOI: 10.4171/ZAA/1298
author: Peter Mathé
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2007-03-20T00:00:00Z
review text: The problem under consideration is an operator equation \(y_\delta=Ax+\delta\xi\) with a linear, compact and injective operator \(A\) acting between Hilbert spaces \(X\) and \(Y\) and with bounded deterministic noise \(\xi\), \(\| \xi\| \leq 1\). The author analyzes the regularization of the equation's solution in the form of \(g_\alpha : (0, \| A^*A\| ] \rightarrow \mathbb R^+\), i.e., the regularized solution is \(x_{\alpha, \delta}:=g_\alpha (A^*A)A^*y_\delta\), \(\alpha>0\). He assumes that the sought solution has a sourcewise representation \(x=\varphi(A^*A)v\) for some \(v\in X\) with \(\| v\| \leq R\) where \(\varphi: (0,\| A^*A\| ] \rightarrow \mathbb{R}^+\) is the index function. The regularizing parameter \(\alpha\) is determined by the discrepancy principle: \(\alpha_*:= \sup\{\alpha\leq\| A^*A\| \), \(\| Ax_{\alpha, \delta}-y_\delta\| \leq\tau\delta\}\). The main results of the paper are upper and lower estimations of a deviation between the exact and the regularized solutions \(\| x_{\alpha_*, \delta}-x\| \) expressed in terms of a qualification of the regularization method, and the index function \(\varphi\).
reviewed by: Vladimir Gorbunov
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A52
zbMATH DE Number: 5135095
zbMATH Keywords: linear operator equations
zbMATH Keywords: regularization
zbMATH Keywords: source condition
zbMATH Keywords: error estimations
zbMATH Keywords: saturation
zbMATH Keywords: Hilbert spaces
MaRDI profile type: Publication
cites work: Q4895893
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full work available at URL: https://doi.org/10.4171/zaa/1298
OpenAlex ID: W2054712927
description: scientific article; zbMATH DE number 5135095