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.

Thamban Nair M

Department of Mathematics

Fulltext

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IIT Madras

Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

Zeitschrift fur Analysis und ihre Anwendung

A generalization of a simplified form of the continuous regularized Gauss-Newton method has been considered for obtaining stable approximate solutions for ill-posed operator equations of the form F(x) = y, where F is a nonlinear operator defined on a subset of a Hilbert space ℋ<inf>1</inf> with values in another Hilbert space ℋ<inf>2</inf>. Convergence of the method for exact data is proved without assuming any specific source condition on the unknown solution. For the case of noisy data, order optimal error estimates based on an a posteriori as well as an a priori stopping rule are derived under a general source condition which includes the classical source conditions such as the Hölder-type and logarithmic type, and certain nonlinearity assumptions on the operator F. © de Gruyter 2011.

Journal of Inverse and Ill-Posed Problems

A generalization of continuous regularized Gauss-Newton method for ill-posed problems

We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F: D(F)≤X→ is a nonlinear operator andX is a Hilbert space. Recently, Bakushinsky and Smirnova [Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim. 28 (2007) 13-25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with ||y-δ ||≥δresulting in the convergence of the method as 0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova. © 2010 Australian Mathematical Society.

ANZIAM Journal

Iterated Lavrentiev regularization for nonlinear ILL-posed problems

Regularized versions of continuous analogues of Newton's method and modified Newton's method for obtaining approximate solutions to a nonlinear ill-posed operator equation of the form F(u)=f, where F is a monotone operator defined from a Hilbert space H into itself, have been studied in the literature. For such methods, error estimates are available only under Holder-type source conditions on the solution. In this paper, presenting the background materials systematically, we derive error estimates under a general source condition. For the special case of the regularized modified Newton's method under a Holder-type source condition, we also carry out error analysis by replacing the monotonicity of F by a weaker assumption. This analysis facilitates inclusion of certain examples of parameter identification problems, which was not possible otherwise. Moreover, an a priori stopping rule is considered when we have a noisy data f instead of f. This rule yields not only convergence of the regularized approximations to the exact solution as the noise level tends to zero but also provides convergence rates that are optimal under the source conditions considered.

Numerical Functional Analysis and Optimization

Regularized versions of continuous newton's method and continuous modified newton's method under general source conditions

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.

Computational Methods in Applied Mathematics

Convergence Rates for Lavrentiev-Type Regularization in Hilbert Scales

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.

International Journal of Mathematics and Mathematical Sciences

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

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.

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

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