scholarly journals A general multi-step matrix splitting iteration method for computing PageRank

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 679-706
Author(s):  
Zhaolu Tian ◽  
Xiaojing Li ◽  
Zhongyun Liu
Keyword(s):  
Iteration Method ◽  
Krylov Subspace ◽  
Convergence Property ◽  
Krylov Subspace Methods ◽  
Gmres Method ◽  
Matrix Splitting ◽  
Subspace Methods ◽  
Numerical Examples ◽  
Preconditioning Technique ◽  
Splitting Iteration Method

Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework [6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

On Preconditioners Based on HSS for the Space Fractional CNLS Equations

2017 ◽  
Vol 7 (1) ◽  
pp. 70-81 ◽  
Author(s):  
Yu-Hong Ran ◽  
Jun-Gang Wang ◽  
Dong-Ling Wang
Keyword(s):  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Krylov Subspace Methods ◽  
Identity Matrix ◽  
Subspace Methods ◽  
Unconditionally Stable ◽  
Numerical Examples ◽  
Difference Formula ◽  
Conservative Difference ◽  
Centered Difference

AbstractThe space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

scholarly journals The modified matrix splitting iteration method for computing PageRank problem

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 725-740 ◽  
Author(s):  
Zhaolu Tian ◽  
Xiaoyan Liu ◽  
Yudong Wang ◽  
P.H. Wen
Keyword(s):  
Iteration Method ◽  
Matrix Splitting ◽  
Power Method ◽  
Numerical Examples ◽  
Comparison Results ◽  
Iteration Methods ◽  
Splitting Iteration Method ◽  
Outer Iteration

In this paper, based on the iteration methods [3,10], we propose a modified multi-step power-inner-outer (MMPIO) iteration method for solving the PageRank problem. In the MMPIO iteration method, we use the multi-step matrix splitting iterations instead of the power method, and combine with the inner-outer iteration [24]. The convergence of the MMPIO iteration method is analyzed in detail, and some comparison results are also given. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithm.

Regularized HSS iteration methods for stabilized saddle-point problems

2018 ◽  
Vol 39 (4) ◽  
pp. 1888-1923 ◽  
Author(s):  
Zhong-Zhi Bai
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Iterative Solvers ◽  
Convergence Theory ◽  
Unconditional Convergence ◽  
Saddle Point Problems ◽  
Subspace Methods ◽  
Iteration Methods

Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

Acceleration Technique Using Krylov Subspace Methods for 2D Arbitrary Geometry Characteristics Solver

2010 ◽  
Author(s):  
Hongbo Zhang ◽  
Hongchun Wu ◽  
Liangzhi Cao
Keyword(s):  
Krylov Subspace ◽  
Computing Time ◽  
Equation System ◽  
Krylov Subspace Methods ◽  
Gmres Method ◽  
Subspace Methods ◽  
Arbitrary Geometry ◽  
Acceleration Technique ◽  
Inner Iteration ◽  
Minimal Residual

The Generalized Minimal RESidual (GMRES) method, which is a widely-used version of Krylov subspace methods for solving large sparse non-symmetric linear systems, is adopted to accelerate the 2D arbitrary geometry characteristics solver AutoMOC. In this technique, a formulism of linear algebraic equation system for angular flux moments and boundary fluxes is derived as an alternative to traditional characteristics sweep (i.e. inner iteration) formalism, and then the GMRES method is implemented as an efficient linear system solver. Several numerical results demonstrate that the acceleration technique based on Krylov subspace methods can be applied to arbitrary geometry MOC solver successfully, and may obtain higher efficiency than the original characteristics solver does because of its spectacular effect on reducing both the number of outer iterations and the total computing time. Moreover, the results could be improved by Lyusternik-Wagner extrapolation technique in some cases.

A Modified GHSS Iteration Method for Continuous Sylvester Equation AX+XB=C

2020 ◽  
Author(s):  
Yuye Feng ◽  
Qingbiao Wu
Keyword(s):  
Unique Solution ◽  
Iteration Method ◽  
Convergence Property ◽  
Sylvester Equation ◽  
Numerical Examples ◽  
Splitting Iteration Method

This paper is concerned with the modification of a generalization of the Hermitian and skew-Hermitian splitting iteration method for solving large sparse continuous Sylvester equations. The analysis shows that the MGHSS iteration method converges unconditionally to the unique solution of AX+XB=C. An inexact variant of the GHSS iteration method (IMGHSS) has been presented and the analysis of its convergence property in detail has been discussed. Numerical examples are reported to confirm the efficiency of the proposed methods.

scholarly journals Kaczmarz method for saddle point systems

2021 ◽  
Vol 293 ◽  
pp. 02003
Author(s):  
Jinmei Wang ◽  
Lizi Yin ◽  
Ke Wang
Keyword(s):  
Convergence Rate ◽  
Saddle Point ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Subspace Methods ◽  
Numerical Examples ◽  
Kaczmarz Algorithm ◽  
Cpu Time ◽  
Kaczmarz Method

The Kaczmarz method is presented for solving saddle point systems. The convergence is analyzed. Numerical examples, compared with classical Krylov subspace methods, SOR-like method (2001) and recent modified SOR-like method (2014), show that the Kaczmarz algorithm is efficient in convergence rate and CPU time.

scholarly journals Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaces

2021 ◽  
Author(s):  
Yuka Hashimoto ◽  
Takashi Nodera
Keyword(s):  
Krylov Subspace ◽  
Time Series Data ◽  
Linear Equations ◽  
Transfer Operator ◽  
Krylov Subspace Methods ◽  
Krylov Subspace Method ◽  
Linear Operators ◽  
Series Data ◽  
Subspace Method ◽  
Subspace Methods

AbstractThe Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications.

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