Convergence of string-averaging method for a class of operators

Nikazad, Touraj; Abbasi, Mojgan; Mirzapour, Mahdi

doi:10.1080/10556788.2016.1209500

Cited by 10 publications

(9 citation statements)

References 30 publications

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“…Still, one step in AVEK has computational costs similar to the Kaczmarz method (if evaluating the forward operators and their adjoints are the computationally most expensive parts). Note that (1.2) and (1.3) might be called simultaneous and sequential, respectively [15,23,38,39]. Further, instead of using the average in (1.2) one might also consider convex combinations of F ′ i (x δ k ) * F i (x δ k ) − y δ i to define the iterative updates in simultaneous schemes (as is Cimmino's method [11]).…”

Section: Iterative Regularization Methodsmentioning

confidence: 99%

“…We further investigate the performance of the proposed AVEK method by comparing it with state-of-the-art accelerated versions of the Landweber and the Kaczmarz method proposed in [38]; compare also [13,37,39]. These accelerated methods take the same forms as the basic Landweber and the Kaczmarz method, with the only difference lying in the choice of step sizes; they select step sizes at each iteration via error minimizing relaxation (EMR) strategies.…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

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The Averaged Kaczmarz Iteration for Solving Inverse Problems

Li¹,

Haltmeier²

2018

SIAM J. Imaging Sci.

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We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or non-linear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz method. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint sub-problem per iterative update. On the other, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

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Section: Iterative Regularization Methodsmentioning

confidence: 99%

Section: Comparison With Other Methodsmentioning

confidence: 99%

The Averaged Kaczmarz Iteration for Solving Inverse Problems

Li¹,

Haltmeier²

2018

SIAM J. Imaging Sci.

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“…Most of the operators used for solving CFPs belong to a new introduced class of operators namely Φ-class [17]. A subclass of Φ-class has been defined in [19].…”

Section: 2mentioning

confidence: 99%

Superiorization of block accelerated cyclic subgradient methods

2020

J. Appl. Numer. Optim.

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In this paper, we introduce a block version and a perturbed block version of the accelerated cyclic subgradient projections method with constraints and give their convergence analyses. The performance of the algorithm is illustrated with a numerical example from the computed tomography and six standard nonlinear test problems. We compare our algorithms with the accelerated cyclic subgradient projections method. Our algorithms produce better results than accelerated cyclic subgradient projections method and have ability to reduce the value of an objective function. Furthermore, the perturbed block version is able to control semiconvergence phenomenon comparing two other methods.

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“…These kinds of algorithms are particularly suitable for parallel computing and therefore have the ability to handle huge-size problems such as deblurring problems, microscopy, medical and astronomic imaging, geophysical applications, digital tomosynthesis, and other areas. However, the string averaging process was used and analyzed in many research works as [19,26,7,25,47,51,3]. They used a relaxed version of the projection operators in the string averaging procedure whereas we extend it to generalized relaxation of sQNE operators.…”

Section: Introductionmentioning

confidence: 99%

A string averaging method based on strictly quasi-nonexpansive operators with generalized relaxation

Nikazad¹,

Mirzapour²

2021

Preprint

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We study a fixed point iterative method based on generalized relaxation of strictly quasi-nonexpansive operators. The iterative method is assembled by averaging of strings, and each string is composed of finitely many strictly quasi-nonexpansive operators. To evaluate the study, we examine a wide class of iterative methods for solving linear systems of equations (inequalities) and the subgradient projection method for solving nonlinear convex feasibility problems. The mathematical analysis is complemented by some experiments in image reconstruction from projections and classical examples, which illustrate the performance using generalized relaxation.

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Convergence of string-averaging method for a class of operators

Cited by 10 publications

References 30 publications

The Averaged Kaczmarz Iteration for Solving Inverse Problems

The Averaged Kaczmarz Iteration for Solving Inverse Problems

Superiorization of block accelerated cyclic subgradient methods

A string averaging method based on strictly quasi-nonexpansive operators with generalized relaxation

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