Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

Thamban Nair M

Department of Mathematics

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IIT Madras

Optimal order yielding discrepancy principle for simplified regularization in hilbert scales: finite-dimensional realizations

International Journal of Mathematics and Mathematical Sciences

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equation Tx=y, where T is a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, when T is a positive and selfadjoint operator. When the data y is known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997). © 2003 Hindawi Publishing Corporation. All rights reserved.

An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in hilbert scales

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) With an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.

Integral Equations and Operator Theory

Optimal order results for a class of regularization methods using unbounded operators

It is shown 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 ordininary Tikhonov regularization and also estimates available under the Hilbert scales approach. © Heldermann Verlag.

Zeitschrift fur Analysis und ihre Anwendung

On Morozov's method for tikhonov regularization as an optimal order yielding algorithm

Simplified regularization in the setting of Hilbert scales has been considered for obtaining stable approximate solutions for ill-posed operator equations. The derived error estimates using an a posteriori as well as an a priori parameter choice strategy are shown to be of optimal order with respect to certain natural assumptions on the ill-posedness of the equation.

Journal	International Journal of Mathematics and Mathematical Sciences
Publisher	Hindawi Publishing Corporation
ISSN	01611712
Open Access	Yes