In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data yδ are given satisfying ||y - yδ|| ≤ δ with known noise level δ. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximation provides 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.

Thamban Nair M

Department of Mathematics

Fulltext

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Morozov's Discrepancy Principle under General Source Conditions

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

Problem of solving integral equations of the first kind, ∫ a b k(s, t) x(t) dt = y(s), s ∈ [a, b] arises in many of the inverse problems that appears in applications. The above problem is a prototype of an ill-posed problem. Thus, for obtaining stable approximate solutions based on noisy data, one has to rely on regularization methods. In practice, the noisy data may be based on a finite number of observations of y, say y(τ 1), y(τ 2),..., y(τ n) for some τ 1,..., τ n in [a, b]. In this paper, we consider approximations based on a collocation method when the nodes τ 1,..., τ n are associated with a convergent quadrature rule. We shall also consider further regularization of the procedure and show that the derived error estimates are of the same order as in the case of Tikhonov regularization when there is no approximation of the integral operator is involved. © 2011 Springer Science+Business Media, LLC.

Advances in Computational Mathematics

Quadrature based collocation methods for integral equations of the first kind

Morozovs discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozovs discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions. Copyright © Australian Mathematical Society 2009.

Bulletin of the Australian Mathematical Society

On Morozovs discrepancy principle for nonlinear ill-posed equations

In order to derive error estimates for regularized approximate solutions of ill-posed equations F(x)=y, where F : D(F)XY is a nonlinear operator between Hilbert spaces X and Y, certain source conditions are to be imposed on the unknown solution [image omitted]. For linear ill-posed equations of the form Ax=y, where A:XY is a bounded linear operator, Nair, Schock, and Tautenhahn considered a general source condition, namely, [image omitted]M, :={[(A*A)]1/2v: v}, where the function is general enough to include some of the well-studied special cases. We extend such procedure to the nonlinear ill-posed equations F(x)=y, so that it also includes the results of Tautenhahn where the special case of ():=/2 is considered.

Numerical Functional Analysis and Optimization

General source conditions for nonlinear ill-posed equations

Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well. © de Gruyter 2007.

Journal of Inverse and Ill-Posed Problems

Tikhonov regularization of nonlinear ill-posed equations under general source condition

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.

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