scholarly journals A Polynomial Preconditioned Global CMRH Method for Linear Systems with Multiple Right-Hand Sides

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ke Zhang ◽  
Chuanqing Gu
Keyword(s):  
Theoretical Result ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Polynomial Degree ◽  
Right Hand

The restarted global CMRH method (Gl-CMRH(m)) (Heyouni, 2001) is an attractive method for linear systems with multiple right-hand sides. However, Gl-CMRH(m) may converge slowly or even stagnate due to a limited Krylov subspace. To ameliorate this drawback, a polynomial preconditioned variant of Gl-CMRH(m) is presented. We give a theoretical result for the square case that assures that the number of restarts can be reduced with increasing values of the polynomial degree. Numerical experiments from real applications are used to validate the effectiveness of the proposed method.

scholarly journals Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Cui-Xia Li ◽  
Yan-Jun Liang ◽  
Shi-Liang Wu
Keyword(s):  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Point Of View ◽  
Subspace Methods ◽  
Practical Point ◽  
Complex Symmetric Linear Systems ◽  
Complex Symmetric

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

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.

An Efficient Variant of the GMRES(m) Method Based on the Error Equations

2012 ◽  
Vol 2 (1) ◽  
pp. 19-32
Author(s):  
Akira Imakura ◽  
Tomohiro Sogabe ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Initial Guess ◽  
Subspace Methods ◽  
Nonsymmetric Linear Systems ◽  
Mathematical Background

AbstractThe GMRES(m) method proposed by Saad and Schultz is one of the most successful Krylov subspace methods for solving nonsymmetric linear systems. In this paper, we investigate how to update the initial guess to make it converge faster, and in particular propose an efficient variant of the method that exploits an unfixed update. The mathematical background of the unfixed update variant is based on the error equations, and its potential for efficient convergence is explored in some numerical experiments.

scholarly journals Parallel GMRES with a multiplicative Schwarz preconditioner

2011 ◽  
Vol Volume 14 - 2011 - Special... ◽  
Author(s):  
Désiré Nuentsa Wakam ◽  
Guy-Antoine Atenekeng-Kahou
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Data Parallelism ◽  
Subspace Method ◽  
Pipeline Parallelism ◽  
Hybrid Solver ◽  
International Audience ◽  
Nous Utilisons ◽  
Schwarz Preconditioner

International audience This paper presents a robust hybrid solver for linear systems that combines a Krylov subspace method as accelerator with a Schwarz-based preconditioner. This preconditioner uses an explicit formulation associated to one iteration of the multiplicative Schwarz method. The Newtonbasis GMRES, which aim at expressing a good data parallelism between subdomains is used as accelerator. In the first part of this paper, we present the pipeline parallelism that is obtained when the multiplicative Schwarz preconditioner is used to build the Krylov basis for the GMRES method. This is referred as the first level of parallelism. In the second part, we introduce a second level of parallelism inside the subdomains. For Schwarz-based preconditioners, the number of subdomains are keeped small to provide a robust solver. Therefore, the linear systems associated to subdomains are solved efficiently with this approach. Numerical experiments are performed on several problems to demonstrate the benefits of using these two levels of parallelism in the solver, mainly in terms of numerical robustness and global efficiency. Cet article présente un solveur hybride robuste pour des systèmes linéaires. Ce solveur parallèle construit un préconditionneur de type Schwarz pour accélerer une méthode basée sur les sous-espaces de Krylov. Le préconditionneur est défini à partir d’une formulation explicite correspondant à une itération de Schwarz multiplicatif. Dans le but de réduire les communications et les dépendences entre les sous-domaines, nous utilisons la version de GMRES qui dissocie la construction de la base de Krylov et son orthogonalisation. Nous présentons dans un premier temps le parallélisme qui est obtenu lorsque ce préconditionneur Schwarz multiplicatif est utilisé dans la construction de la base de Krylov. C’est le premier niveau de parallélisme. Dans la deuxième partie de ce travail, nous introduisons un deuxième niveau de parallélisme à l’intérieur de chaque sous-domaine. Pour des décompositions de domaines avec recouvrement, le nombre de sous-domaines doit rester faible pour fournir un solveur robuste. De ce fait, les systèmes linéaires associés aux sous-domaines sont résolus de manière efficace avec ce deuxième niveau de parallélisme. Plusieurs tests numériques sont présentés à la fin du document pour valider l’efficacité de cette approche.

scholarly journals A relaxed block splitting preconditioner for complex symmetric indefinite linear systems

2018 ◽  
Vol 16 (1) ◽  
pp. 561-573
Author(s):  
Yunying Huang ◽  
Guoliang Chen
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Minimal Polynomial ◽  
Coefficient Matrix ◽  
Subspace Iteration ◽  
Preconditioned Matrix ◽  
Complex Symmetric ◽  
Indefinite Linear Systems

AbstractIn this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the original block two-by-two coefficient matrix. We study the spectral properties and the eigenvector distributions of the corresponding preconditioned matrix. In addition, the degree of the minimal polynomial of the preconditioned matrix is also derived. Finally, some numerical experiments are presented to illustrate the effectiveness of the relaxed splitting preconditioner.

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.

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