Convergence rates for Kaczmarz-type regularization methods

Kindermann, Stefan; Leitão, A.

doi:10.3934/ipi.2014.8.149

Cited by 14 publications

(12 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The convergence of Algorithm BRI was studied in [21] in an infinite dimensional Hilbert space setting. Several well known iterative methods arise as special cases.…”

Section: Endmentioning

confidence: 99%

See 1 more Smart Citation

Convergence analysis for column-action methods in image reconstruction

2016

View full text Add to dashboard Cite

Column-oriented versions of algebraic iterative methods are interesting alternatives to their row-version counterparts: they converge to a least squares solution, and they provide a basis for saving computational work by skipping small updates. In this paper we consider the case of noise-free data. We present a convergence analysis of the column algorithms, we discuss two techniques (loping and flagging) for reducing the work, and we establish some convergence results for methods that utilize these techniques. The performance of the algorithms is illustrated with numerical examples from computed tomography.

show abstract

“…The convergence of Algorithm BRI was studied in [21] in an infinite dimensional Hilbert space setting. Several well known iterative methods arise as special cases.…”

Section: Endmentioning

confidence: 99%

“…Let q = 3 and ω 1 (k) > 0, ω 2 (k) = 0, ω 3 (k) > 0. This would produce the following iterates in (21)…”

Section: Convergence Analysis Of Loping/flaggingmentioning

confidence: 99%

Convergence analysis for column-action methods in image reconstruction

2016

View full text Add to dashboard Cite

show abstract

“…This variant enjoys both good preasymptotic and asymptotic convergence behavior, as indicated by the numerical experiments.Last, we note that in the context of inverse problems, Kaczmarz method has received much recent attention, and has demonstrated very encouraging results in a number of applications. The regularizing property and convergence rates in various settings have been analyzed for both linear and nonlinear inverse problems (see [23,2,12,18,4,22,24,17] for an incomplete list). However, these interesting works all focus on a fixed ordering of the linear system, instead of the randomized variant under consideration here, and thus they do not cover RKM.The rest of the paper is organized as follows.…”

mentioning

confidence: 99%

Preasymptotic convergence of randomized Kaczmarz method

Jiao

Jin

2017

Inverse Problems

View full text Add to dashboard Cite

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the initial error is smooth (e.g., sourcewise representation), the results allow explaining the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.

show abstract

“…Because of the semi‐convergence phenomenon, we can stop the iteration process early or far from a proper iteration index. There have been published some investigations on finding the relaxation parameters for postponing the semi‐convergence phenomenon (see Elfving, Nikazad, and Hansen and Kindermann and Leitao and the references therein for more details).…”

Section: Two Families Of Iterative Regularization Methodsmentioning

confidence: 99%

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

Farahani

Hadizadeh

Bulatov

et al. 2019

Math Methods in App Sciences

View full text Add to dashboard Cite

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill-posed problems that are obtained by a system of linear two-dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill-posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.

show abstract

Convergence rates for Kaczmarz-type regularization methods

Cited by 14 publications

References 29 publications

Convergence analysis for column-action methods in image reconstruction

Convergence analysis for column-action methods in image reconstruction

Preasymptotic convergence of randomized Kaczmarz method

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

Contact Info

Product

Resources

About