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.

Thamban Nair M

Department of Mathematics

Mathematical operators

Problem solving

DISCREPANCY PRINCIPLE

GELFAND TRIPLE

HILBERT SCALE

ILL POSED PROBLEM

ILL-POSED EQUATIONS

ORDER OPTIMAL GELFAND TRIPLE

Regularization

SOBOLEV

SOURCE SETS

TIKHONOV REGULARIZATION

Algebra

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IIT Madras

Role of Hilbert Scales in Regularization Theory

Springer Proceedings in Mathematics and Statistics

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

Many problems in science and engineering have their mathematical formulation as an operator equation Tx=y, where T is a linear or nonlinear operator between certain function spaces. In practice, such equations are solved approximately using numerical methods, as their exact solution may not often be possible or may not be worth looking for due to physical constraints. In such situations, it is desirable to know how the so-called approximate solution approximates the exact solution, and what the error involved in such procedures would be. This book is concerned with the investigation of the above theoretical issues related to approximately solving linear operator equations. The main tools used for this purpose are basic results from functional analysis and some rudimentary ideas from numerical analysis. To make this book more accessible to readers, no in-depth knowledge on these disciplines is assumed for reading this book. © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

Fulltext

Linear Operator Equations: Approximation and Regularization

Linear operator equations: Approximation and regularization

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high- and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience. © 2005 IOP Publishing Ltd.

Inverse Problems

Regularization in Hilbert scales under general smoothing conditions

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.

Zeitschrift fur Analysis und ihre Anwendung

Morozov's discrepancy principle under general source conditions

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.

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

Role of Hilbert scales in regularization theory

Journal	Data powered by TypesetSpringer Proceedings in Mathematics and Statistics
Publisher	Data powered by TypesetSpringer New York LLC
ISSN	21941009
Open Access	No