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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].
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.
On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations
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.
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.
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.
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