Inverse Problems and Imaging (IPI)

Convergence rates for Kaczmarz-type regularization methods

Pages: 149 - 172, Volume 8, Issue 1, February 2014      doi:10.3934/ipi.2014.8.149

       Abstract        References        Full Text (451.8K)       Related Articles       

Stefan Kindermann - Industrial Mathematics Institute, Johannes Kepler University Linz, A-4040 Linz, Austria (email)
Antonio Leitão - Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis, Brazil (email)

Abstract: This article is devoted to the convergence analysis of a special family of iterative regularization methods for solving systems of ill--posed operator equations in Hilbert spaces, namely Kaczmarz-type methods. The analysis is focused on the Landweber--Kaczmarz (LK) explicit iteration and the iterated Tikhonov--Kaczmarz (iTK) implicit iteration. The corresponding symmetric versions of these iterative methods are also investigated (sLK and siTK). We prove convergence rates for the four methods above, extending and complementing the convergence analysis established originally in [22,13,12,8].

Keywords:  Ill-posed systems, Landweber--Kaczmarz, convergence rates, regularization.
Mathematics Subject Classification:  Primary: 65J20, 65J15; Secondary: 47J06.

Received: August 2012;      Revised: November 2013;      Available Online: March 2014.