IPI
Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications
Markus Haltmeier Richard Kowar Antonio Leitão Otmar Scherzer
Inverse Problems & Imaging 2007, 1(3): 507-523 doi: 10.3934/ipi.2007.1.507
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
Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis
Markus Haltmeier Antonio Leitão Otmar Scherzer
Inverse Problems & Imaging 2007, 1(2): 289-298 doi: 10.3934/ipi.2007.1.289
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.

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