Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis

Markus Haltmeier; Antonio Leitão; Otmar Scherzer

doi:10.3934/ipi.2007.1.289

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2007, Volume 1, Issue 2: 289-298. Doi: 10.3934/ipi.2007.1.289

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Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis

1.
Department of Computer Science,University of Innsbruck, Technikerstrasse 21a, A-6020 Innsbruck
2.
Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis

Received: January 2007

Published: May 2007

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Abstract

In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill-posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.

Keywords:

Ill-posed systems; Landweber--Kaczmarz; Regularization.

Mathematics Subject Classification: Primary: 65J20, 65J15; Secondary: 47J06.

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