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.

Thamban Nair M

Department of Mathematics

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IIT Madras

Error bounds and parameter choice strategies for simplified regularization in Hilbert scales

Integral Equations and Operator Theory

For solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the operator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions. © 2008, Institute of Mathematics, NAS of Belarus. All rights reserved.

Fulltext

Computational Methods in Applied Mathematics

Convergence Rates for Lavrentiev-Type Regularization in Hilbert Scales

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.

International Journal of Mathematics and Mathematical Sciences

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

In this paper we study the problem of identifying the solution x† of linear ill-posed problems Ax = y with non-negative and self-adjoint operators A on a Hilbert space X where instead of exact data y noisy data y δ ∈ X are given satisfying ∥y -y δ∥ ≤ δ with known noise level δ. Regularized approximations xα δ are obtained by the method of Lavrentiev regularization, that is, xα δ is the solution of the singularly perturbed operator equation Ax + αx = yδ, and the regularization parameter α is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximations provide order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extend recent results for infinitely smoothing operators. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order m and discuss a special ill-posed problem arising in inverse heat conduction.

Zeitschrift fur Analysis und ihre Anwendung

Lavrentiev regularization for linear Ill-posed problems under general source conditions

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.

Journal	Integral Equations and Operator Theory
Publisher	Birkhauser Verlag Basel
ISSN	0378620X
Open Access	No