Kaczmarz-Type Inner-Iteration Preconditioned Flexible GMRES Methods for Consistent Linear Systems

Du, Yishu; Hayami, Ken; Zheng, Ning; Morikuni, Keiichi; Yin, Jingxue

doi:10.1137/20m1344937

Cited by 14 publications

(4 citation statements)

References 32 publications

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“…where • F denotes the Frobenius norm, σ min (•) denotes the smallest nonzero singular value and E denotes the expectation (over the choice of the rows). This elegant result has inspired a lot of subsequent works; see references therein [3,22,23,29,30,33,35,39,56,59,60,65].…”

Section: Introductionmentioning

confidence: 91%

Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency

Han¹,

Su²,

Xie³

2022

Preprint

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The Douglas-Rachford (DR) method is a widely used method for finding a point in the intersection of two closed convex sets (feasibility problem). However, the method converges weakly and the associated rate of convergence is hard to analyze in general. In addition, the direct extension of the DR method for solving more-than-two-sets feasibility problems, called the r-sets-DR method, is not necessarily convergent. Recently, the introduction of randomization and momentum techniques in optimization algorithms has attracted increasing attention as it makes the algorithms more efficient. In this paper, we show that randomization is powerful for the DR method and can make the divergent r-sets-DR method converge. Specifically, we propose the randomized r-sets-DR (RrDR) method for solving the linear system Ax = b and prove its linear convergence in expectation, and the convergence rate does not depend on the dimension of the matrix A. We also study RrDR with heavy ball momentum and show that it converges at an accelerated linear rate. Numerical experiments are provided to confirm our results and demonstrate the significant improvement in accuracy and efficiency of the DR method that randomization and momentum bring.C i where C i := {x : a i , x = b i }.

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Section: Introductionmentioning

confidence: 91%

Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency

Han¹,

Su²,

Xie³

2022

Preprint

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show abstract

“…Kaczmarz method is one of the well-known iterative projection method, which was firstly proposed in [11] and further extended to block and inconsistent case in [6,9]. Since the linear convergence of randomized Kaczmarz method was established by Strohmer and Vershynin [14], variants of randomized Kaczmarz method are presented and deeply studied [3,4,8,10]. On the other hand, iterative methods based on Householder orthogonal reflection also attract much attention from the community of numerical linear algebra.…”

Section: Introductionmentioning

confidence: 99%

Restarted randomized surrounding methods for solving large linear equations

Yin¹,

Li²

2022

Preprint

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A class of restarted randomized surrounding methods are presented to accelerate the surrounding algorithms by restarted techniques for solving the linear equations. Theoretical analysis prove that the proposed method converges under the randomized row selection rule and the expectation convergence rate is also addressed. Numerical experiments further demonstrate that the proposed algorithms are efficient and outperform the existing method for over-determined and under-determined linear equation, as well as in the application of image processing.

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“…where i k is chosen with probability proportional to a i 2 2 and λ min (A) denote the minimum eigenvalue of A. Since then, various randomized Kaczmarz method are studied and improved by greedy strategies [2], sampling techniques [5], extrapolating acceleration [17], block version [12,20], averaging [18] and other improvements [6,13,23].…”

Section: Introductionmentioning

confidence: 99%

The randomized circumcentered-reflection iteration method for solving consistent linear equations

Yin

2022

Preprint

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After randomly reflecting on two hyperplanes, a new iteration method is established by making use of the circumceter of the reflective points from the viewpoint of geometry. The linear combination could be non-convex when the angle between the hyperplances is small. Theoretical analysis show that the proposed method converges and the convergence rate in expectation is also addressed in detail. The relation between our method and block Kaczmarz method is well discussed. Numerical experiments further verify that the new algorithms is efficient, and outperform the existing randomized Kaczmarz methods and randomized reflection methods in terms of the number of iterations and CPU time, especially when the coefficient matrix has highly coherent rows.

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Kaczmarz-Type Inner-Iteration Preconditioned Flexible GMRES Methods for Consistent Linear Systems

Cited by 14 publications

References 32 publications

Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency

Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency

Restarted randomized surrounding methods for solving large linear equations

The randomized circumcentered-reflection iteration method for solving consistent linear equations

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