A Modified Block Flexible GMRES Method with Deflation at Each Iteration for the Solution of Non-Hermitian Linear Systems with Multiple Right-Hand Sides

2013 ◽  
Vol 35 (5) ◽  
pp. S345-S367 ◽  
Author(s):  
Henri Calandra ◽  
Serge Gratton ◽  
Rafael Lago ◽  
Xavier Vasseur ◽  
Luiz Mariano Carvalho
Keyword(s):  
Linear Systems ◽  
Gmres Method ◽  
Right Hand ◽  
Modified Block

scholarly journals A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right-hand sides

2018 ◽  
Vol 25 (5) ◽  
pp. e2148 ◽  
Author(s):  
Dong-Lin Sun ◽  
Ting-Zhu Huang ◽  
Yan-Fei Jing ◽  
Bruno Carpentieri
Keyword(s):  
Linear Systems ◽  
Gmres Method ◽  
Right Hand ◽  
Deflated Restarting

Enlarged GMRES for solving linear systems with one or multiple right-hand sides

2018 ◽  
Vol 39 (4) ◽  
pp. 1924-1956 ◽  
Author(s):  
Hussam Al Daas ◽  
Laura Grigori ◽  
Pascal Hénon ◽  
Philippe Ricoux
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Real Life ◽  
Original Method ◽  
Gmres Method ◽  
Convection Diffusion ◽  
Right Hand ◽  
Test Matrices ◽  
Linear Systems Of Equations ◽  
Diffusion Problems

Abstract We propose a variant of the generalized minimal residual (GMRES) method for solving linear systems of equations with one or multiple right-hand sides. Our method is based on the idea of the enlarged Krylov subspace to reduce communication. It can be interpreted as a block GMRES method. Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection. Furthermore, we propose a technique for deflating eigenvalues that has two benefits. The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version. The second is to have very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of a constrained pressure residual preconditioner). We test our method with these deflation techniques on academic test matrices arising from solving linear elasticity and convection–diffusion problems as well as matrices arising from two real-life applications, seismic imaging and simulations of reservoirs. With the same memory cost we obtain a saving of up to $50 \%$ in the number of iterations required to reach convergence with respect to the original method.

scholarly journals A Weighted Block GMRES method for solving linear systems with multiple right-hand sides

JSIAM Letters ◽  
2013 ◽  
Vol 5 (0) ◽  
pp. 65-68 ◽  
Author(s):  
Akira Imakura ◽  
Lei Du ◽  
Hiroto Tadano
Keyword(s):  
Linear Systems ◽  
Gmres Method ◽  
Right Hand

scholarly journals Residual-Based Simpler Block GMRES for Nonsymmetric Linear Systems with Multiple Right-Hand Sides

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Qinghua Wu ◽  
Liang Bao ◽  
Yiqin Lin
Keyword(s):  
Linear Systems ◽  
Linear Dependence ◽  
Numerical Experiments ◽  
Gmres Method ◽  
Algebraic Equations ◽  
Rank Deficiency ◽  
Nonsymmetric Linear Systems ◽  
Right Hand ◽  
Linear Algebraic Equations ◽  
Block Arnoldi

We propose in this paper a residual-based simpler block GMRES method for solving a system of linear algebraic equations with multiple right-hand sides. We show that this method is mathematically equivalent to the block GMRES method and thus equivalent to the simpler block GMRES method. Moreover, it is shown that the residual-based method is numerically more stable than the simpler block GMRES method. Based on the deflation strategy proposed by Calandra et al. (2013), we derive a deflation strategy to detect the possible linear dependence of the residuals and a near rank deficiency occurring in the block Arnoldi procedure. Numerical experiments are conducted to illustrate the performance of the new method.

scholarly journals A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems

2018 ◽  
Vol 78 (2) ◽  
pp. 746-769 ◽  
Author(s):  
Dong-Lin Sun ◽  
Ting-Zhu Huang ◽  
Bruno Carpentieri ◽  
Yan-Fei Jing
Keyword(s):  
Linear Systems ◽  
Gmres Method ◽  
Right Hand

The Linear System Solver of Programs Named CORTH and KYLIN2

2017 ◽  
Author(s):  
Pingzhou Ming ◽  
Junjie Pan ◽  
Xiaolan Tu ◽  
Dong Liu ◽  
Hongxing Yu
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Nuclear Power ◽  
Experimental Results ◽  
Gmres Method ◽  
Thermal Hydraulics ◽  
Lattice Calculation ◽  
Minimal Residual

Sub-channel thermal-hydraulics program named CORTH and assembly lattice calculation program named KYLIN2 have been developed in Nuclear Power Institute of China (NPIC). For the sake of promoting the computing efficiency of these programs and achieving the better description on fined parameters of reactor, the programs’ structure and details are interpreted. Then the characteristics of linear systems of these programs are analyzed. Based on the Generalized Minimal Residual (GMRES) method, different parallel schemes and implementations are considered. The experimental results show that calculation efficiencies of them are improved greatly compared with the serial situation.

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