A new family of global methods for linear systems with multiple right-hand sides

Zhang, Jianhua; Dai, Hua; Zhao, Jing

doi:10.1016/j.cam.2011.09.020

Cited by 19 publications

(9 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first global methods are global full orthogonalization (Gl-FOM) and global generalized minimal residual (Gl-GMRES), introduced in [20]. Gl-BiCG and Gl-BiCGStab were suggested in [21], and global variants of less well-known Krylov subspace methods were subsequently proposed in [18] (Gl-CMRH), [34] (Gl-CGS), [15] (Gl-SCD) and [35] (Gl-BiCR and its variants).…”

Section: Discussion Of the Literaturementioning

confidence: 99%

On short recurrence Krylov type methods for linear systems with many right-hand sides

Rashedi

Ebadi

Birk

et al. 2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Block and global Krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides. These methods are advantageous, since they allow to cast the major part of the arithmetic in terms of matrix-block vector products, and since, in the block case, they take their iterates from a potentially richer subspace. In this paper we consider the most established Krylov subspace methods which rely on short recurrencies, i.e. BiCG, QMR and BiCGStab. We propose modifications of their block variants which increase numerical stability, thus at least partly curing a problem previously observed by several authors. Moreover, we develop modifications of the "global" variants which almost halve the number of matrix-vector multiplications. We present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial, and that the new "economic" versions of the "global" BiCG and QMR method can be considered as good alternatives to the BiCGStab variants.

show abstract

Section: Discussion Of the Literaturementioning

confidence: 99%

On short recurrence Krylov type methods for linear systems with many right-hand sides

Rashedi

Ebadi

Birk

et al. 2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…The COCR method is a special case of the BiCR method. Additionally, since the BiCR method was proposed, this method was also improved (or modified) for various systems of linear equations involving non-Hermitian coefficient matrices, e.g., refer to [42][43][44] for details. In previous work, we have proved that it is mathematically equivalent to the BiCGCR algorithm proposed by Clemens in [41], the difference lying only in the choice of the scalar factors α k and β k within the inner iteration loop; refer to [7,16] for details.…”

Section: The Derivation Of the Bicgcrmethodsmentioning

confidence: 99%

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

Huang

Carpentieri

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods.

show abstract

“…Note that the residual norms from Rb-GlSGMRES stagnate at iteration steps near 600 with ||B − A X k || F ∕||R 0 || F ≈ 10 −10 ; it can still be as accurate as GlGMRES and can attain a higher maximum attainable accuracy. This can be illustrated by (12), although a small k ( ♯ k ) might lead to a large F ( k ); it can sometimes be damped by small residual norm ||R k || F , making the whole upper bound in (12) small. Test 3.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…6,7 Compared to solving an ms-dimensional l 2 -least squares problem at the mth step of the block GMRES method, the global GMRES (GlGMRES) method presented by Jbilou et al 8 solves an m-dimensional l 2 -least squares problem instead, leading to considerable savings and lower computational costs. Since then, some variants and properties of GlGMRES have been investigated in many papers; see for example the works [9][10][11][12][13] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Making global simpler GMRES more stable

Liu

Shen

Jia

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary The global GMRES method is well known for the solution of nonsymmetric linear systems with multiple right hands. In this paper, the condition number for evaluating the stability of the global simpler GMRES method is defined. With this condition number, it is shown that Zong et al.'s global simpler method with a simple basis of the Krylov matrix subspace might lead to instability when there is a large decrease in the residual norm. An adaptive global simpler method is then presented to avoid the potential instability. For this adaptive method, theoretical results deliver the lower and upper bounds for the condition number of the basis, and when the linear system is of single right hand, the estimate is proven to be tighter than Jiránek et al.'s upper bound by numerical tests. Numerical results also show that the adaptive strategy leads to a well‐conditioned basis, and in most cases, the global simpler GMRES with normalized residuals as the basis can also be reliable.

show abstract

A new family of global methods for linear systems with multiple right-hand sides

Cited by 19 publications

References 32 publications

On short recurrence Krylov type methods for linear systems with many right-hand sides

On short recurrence Krylov type methods for linear systems with many right-hand sides

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

Making global simpler GMRES more stable

Contact Info

Product

Resources

About