On Morozov's method for Tikhonov regularization as an optimal order yielding algorithm
type: Item
instance of: scholarly article
title: On Morozov's method for Tikhonov regularization as an optimal order yielding algorithm
zbMATH Open document ID: 0969.65042
DOI: 10.4171/ZAA/868
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2001-10-04T00:00:00Z
full work available at URL: https://eudml.org/doc/48907
review text: Many problems in science and engineering have their mathematical formulations as an operator equation \[ Kx= y,\tag{1} \] where \(K:X\to Y\) is a bounded linear operator between Hilbert spaces \(X\) and \(Y\) with its range \(R(K)\) not closed in \(Y\). It is well known that if \(R(K)\) is not closed, the equation (1) or the problem of solving (1) is ill-posed. Regularization procedures are employed for obtaining stable approximate solutions of ill-posed equations of the type (1). These procedures are especially useful when the data available is inexact. That is, we may have an approximation \(\widetilde y\) of \(y\) with a known error level \(\delta> 0\), \(\|y-\widetilde y\|\leq\delta\). In this paper the author shows that Tikhonov regularization for an ill-posed operator equation \(Kx= y\) using a possibly unbounded regularizing operator \(L\) yields an order-optimal algorithm with respect to certain stability set when the regularization parameter is chosen according to Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordinary Tikhonov regularization and also estimates available under the Hilbert scales approach.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47A52
zbMATH DE Number: 1278885
zbMATH Keywords: interpolation inequalities
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: ill-posed operator equation
zbMATH Keywords: order-optimal algorithm
zbMATH Keywords: stability
zbMATH Keywords: Morozov's discrepancy principle
zbMATH Keywords: error estimate
zbMATH Keywords: Hilbert scale
author: M. Thamban Nair
reviewed by: G. Dimitriu
MaRDI profile type: Publication
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OpenAlex ID: W2020305632
description: scientific article; zbMATH DE number 1278885