A Fast Averaged Kaczmarz Iteration with Convex Penalty for Inverse Problems in Hilbert Spaces

Xia, Yuxin; Wang, Wei; Han, Bo

doi:10.1137/21m1445181

Cited by 2 publications

(2 citation statements)

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“…The typical interior character of problem ( 1) is its ill-posedness, which means that the solution is highly sensitive to the slight change of the right hand side data y. In order to find a stable and rational solution, many regularization methods have been proposed during the past few decades (see, for example, [1][2][3][4][5]). Among these methods, iterative regularization methods show their activeness and are intensively focused for their stability and robust for inverse problems.…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

Xia¹,

Han²,

Fu³

2022

Inverse Problems

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In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton−Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

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Section: Introductionmentioning

confidence: 99%

“…., where µ n,k is the step size, ξ n,0 = ξ n and x n,0 = x n . The next iterate is then defined as ξ n+1 = ξ n,kn and x n+1 = x n,kn with the integer k n ⩾ 1 determined in the same way as the method (5). The convergence of the method ( 7) is given in [17].…”

Section: Introductionmentioning

confidence: 99%