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ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

Part of: Numerical analysis in abstract spaces General theory of linear operators Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  19 April 2010

P. MAHALE  and
M. T. NAIR
P. MAHALE
Affiliation:
Department of Mathematics, I.I.T. Madras, Chennai - 600036, India (email: [email protected], [email protected])
M. T. NAIR*
Affiliation:
Department of Mathematics, I.I.T. Madras, Chennai - 600036, India (email: [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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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→X is a nonlinear operator and X 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 resulting in the convergence of the method as δ→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.

Keywords

Lavrentiev regularizationnonlinear ill-posed problemsmonotone operatorFréchet derivativeregularization parameterstopping indexGauss–Newton methodsource functiongeneral source conditionorder optimal result

MSC classification

Secondary: 47A52: Ill-posed problems, regularization 65J15: Equations with nonlinear operators (do not use ) 65J22: Inverse problems 65M30: Improperly posed problems
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 2 , October 2009 , pp. 191 - 217
Copyright
Copyright © Australian Mathematical Society 2010

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