An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides

Simoncini, Valeria; Gallopoulos, Efstratios

doi:10.1137/0916053

Cited by 121 publications

(106 citation statements)

References 27 publications

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“…In this case, an alternative to block methods is the use of seed projection techniques for solving (1.1). The idea used in the seed Krylov methods is to select a single "seed" system and some Krylov method as a generator of approximations for multiple right-hand sides [2,19,21]. In the Krylov seed algorithm a single Krylov subspace -corresponding to the seed system-is generated, then the residuals of the non-seed systems are projected orthogonally onto this generated Krylov subspace in order to get the approximate solutions.…”

Section: The Weighted Seed Gmres Methodsmentioning

confidence: 99%

“…In [2], the authors show that a better convergence behaviour of the seed CG process when compared to the classical CG process. For unsymmetric systems, a seed GMRES method was proposed in [19] and a Morgan's Krylov subspace method augmented with eigenvectors was presented in [6].…”

Section: The Weighted Seed Gmres Methodsmentioning

confidence: 99%

“…This procedure is repeated with another seed system until all the systems are solved. Improvements of seed techniques were also applied to the GMRES method and the obtained seed GMRES (SGMRES) method was compared with block and classical solvers in [19]. Note that the efficiency of seed methods depends on how closely related the right-hand sides are [1,2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

Elbouyahyaoui¹,

Heyouni

2018

B Soc Paran Mat

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In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.

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Section: The Weighted Seed Gmres Methodsmentioning

confidence: 99%

Section: The Weighted Seed Gmres Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

Elbouyahyaoui¹,

Heyouni

2018

B Soc Paran Mat

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“…The first one is the so-called seed methods (see Simoncini & Gallopoulos 1995or Gutknecht 2007 and references therein) when a single system is chosen as a seed system, a corresponding Krylov subspace is obtained and thereafter initial residuals of all the other systems are projected to this subspace. These techniques can be useful in the case when not all right-hand sides are available at the same time.…”

Section: Page 8 Of 37 Geophysical Journal Internationalmentioning

confidence: 99%

A review of block Krylov subspace methods for multisource electromagnetic modelling

Puzyrev¹,

María

2015

Geophys. J. Int.

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SUMMARYPractical applications of controlled-source electromagnetic modeling require solutions for multiple sources at several frequencies, thus leading to a dramatic increase of the computational cost. In this paper we present an approach using block Krylov subspace solvers that are iterative methods especially designed for problems with multiple right-hand-sides. Their main advantage is the shared subspace for approximate solutions, hence, these methods are expected to converge in less iterations than the corresponding standard solver applied to each linear system. Block solvers also share the same preconditioner, which is constructed only once. Simultaneously computed block operations have better utilization of cache due to the less frequent access to the system matrix. In this paper we implement two different block solvers for sparse matrices resulting from the finitedifference and the finite-element discretizations, discuss the computational cost of the algorithms and study their dependence on the number of right-hand sides given at once. The effectiveness of the proposed methods is demonstrated on two electromagnetic survey scenarios, including a large marine model. As the results of the simulations show, when a powerful preconditioning is employed, block methods are faster than standard iterative techniques in terms of both iterations and time.

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“…Conventional block iterative methods (see [13,7] and references therein) cannot be applied to (1) if the b i are supplied successively (like in the MOR applications to be discussed), since these methods require that the right-hand sides are available simultaneously. A new method for solving systems A i x i = b i is proposed in [11], where the focus is on varying A i .…”

Section: Introductionmentioning

confidence: 99%

Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-hand Sides Arising in Model Reduction

Benner

Lihong

2011

Lecture Notes in Electrical Engineering

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We discuss the numerical solution of successive linear systems of equations Ax = b i , i = 1, 2, . . . m by iterative methods based on recycling Krylov subspaces. We propose various recycling algorithms which are based on the generalized conjugate residual (GCR) method. The recycling algorithms reuse the descent vectors computed while solving the previous linear systems Ax = b j , j = 1, 2, . . . , i − 1, such that a lot of computational work can be saved when solving the current system Ax = b i . The proposed algorithms are robust for solving sequences of linear systems arising in circuit simulation. Sequences of linear systems need to be solved, e.g., in model order reduction (MOR) for systems with many terminals. Numerical experiments illustrate the efficiency and robustness of the proposed method.

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An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides

Cited by 121 publications

References 27 publications

On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

A review of block Krylov subspace methods for multisource electromagnetic modelling

Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-hand Sides Arising in Model Reduction

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