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Convergence rates for Kaczmarz-type regularization methods
Stefan Kindermann,  Antonio Leitão
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 ,  regularization. ,  convergence rates ,  Landweber--Kaczmarz
IPI
Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications
Markus Haltmeier,  Richard Kowar,  Antonio Leitão,  Otmar Scherzer
In part I we introduced modified Landweber--Kaczmarz methods and established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.
keywords: Ill-posed systems; Landweber--Kaczmarz; Regularization.
IPI
On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations
Johann Baumeister,  Barbara Kaltenbacher,  Antonio Leitão
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a non-linear inverse doping problem based on a bipolar model.
keywords: Kaczmarz method. ,  Levenberg-Marquardt method ,  Regularization ,  Nonlinear systems ,  Ill-posed equations
IPI
Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis
Markus Haltmeier,  Antonio Leitão,  Otmar Scherzer
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.
IPI
Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations
Adriano De Cezaro,  Johann Baumeister,  Antonio Leitão
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: Ill-posed equations ,  iterated Tikhonov method ,  Nonlinear systems ,  Regularization ,  Kaczmarz method.

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