Inverse Problems and Imaging (IPI)

Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations

Pages: 1 - 17, Volume 5, Issue 1, February 2011      doi:10.3934/ipi.2011.5.1

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Adriano De Cezaro - Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil (email)
Johann Baumeister - Fachbereich Mathematik, Johann Wolfgang Goethe Universität, Robert–Mayer–Str. 6–10, 60054 Frankfurt am Main, Germany (email)
Antonio Leitão - Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis, Brazil (email)

Abstract: We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.

Keywords:  Nonlinear systems, Ill-posed equations, Regularization, iterated Tikhonov method, Kaczmarz method.
Mathematics Subject Classification:  Primary: 65J20; Secondary: 47J06.

Received: October 2009;      Revised: September 2010;      Published: February 2011.