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.

Thamban Nair M

Department of Mathematics

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

Zeitschrift fur Analysis und ihre Anwendung

Hilbert scales, which are generalizations of Sobolev scales, play crucial roles in the regularization theory. In this paper, it is intended to discuss some important properties of Hilbert scales with illustrations through examples constructed using the concept of Gelfand triples, and using them to describe source conditions and for deriving error estimates in the regularized solutions of ill-posed operator equations. We discuss the above with special emphasis on some of the recent work of the author. © Springer International Publishing Switzerland 2015.

Springer Proceedings in Mathematics and Statistics

Role of Hilbert scales in regularization theory

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y , one looks for a stable method for approximating x, a least-residual norm solution which minimizes a seminorm x → Lx, where L : D(L) X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition Tx2 + Lx2 γx2 for all x D(LL) for some constant γ 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation (TT + LL)xδ = Tyδ where yδ is a noisy data and 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T,L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle. © World Scientific Publishing Company.

Analysis and Applications

A unified treatment for Tikhonov regularization using a general stabilizing operator

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.

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	Zeitschrift fur Analysis und ihre Anwendung
Publisher	Heldermann Verlag
ISSN	02322064
Open Access	Yes