Convergence of the gradient method for ill-posed problems
Stefan Kindermann
Inverse Problems & Imaging 2017, 11(4): 703-720 doi: 10.3934/ipi.2017033

We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for nonlinear operator equations is a special instance of this framework, and the gradient method then reduces to Landweber iteration. The main result of this article is a proof of weak and strong convergence under new nonlinearity conditions that generalize the classical tangential cone conditions.

keywords: Inverse problems iterative Regularization Landweber iteration tangential cone condition quasiconvexity
Convergence rates for Kaczmarz-type regularization methods
Stefan Kindermann Antonio Leitão
Inverse Problems & Imaging 2014, 8(1): 149-172 doi: 10.3934/ipi.2014.8.149
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
On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization
Stefan Kindermann Andreas Neubauer
Inverse Problems & Imaging 2008, 2(2): 291-299 doi: 10.3934/ipi.2008.2.291
In this paper we derive convergence and convergence rates results of the quasioptimality criterion for (iterated) Tikhonov regularization. We prove convergence and suboptimal rates under a qualitative condition on the decay of the noise with respect to the spectral family of $T$$T$*. Moreover, optimal rates are obtained if the exact solution satisfies a decay condition with respect to the spectral family of $T$*$T$.
keywords: heuristic parameter selection. Quasioptimality criterion

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