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### Open Access Journals

IPI

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.

IPI

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].

IPI

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 $T$*$T$.

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