On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Yang, Ai-Li; Li, Xu; Wu, Yue

doi:10.1016/j.amc.2014.11.100

Cited by 25 publications

(5 citation statements)

References 32 publications

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“…We use left preconditioning with GMRES as a Krylov subspace method. We compare the elapsed CPU time (s) (CPU) and the number of iterations (IT) of the NMSS preconditioner with GMRES without preconditioning and GMRES method with GMSS preconditioner [17], Uzawa-HSS and PU-STS preconditioners [19,24,25]. In these examples, all of the optimal parameters are provided experimentally.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Let A ∈ R n×n be positive definite, B ∈ R n×m , rank(B) = r < m < n and α, β > 0 be given constants. Assume that A = P + S is a dominant positive define and skew-Hermitian splitting of A. Letã,b, andc are defined as in Theorem 4.2 and α, β > 0 satisfy(24) or(25). Then all eigenvalues of the NMSS-preconditioned matrix P −1 NMSS A are located in a circle centered at (1, 0) with radius 1.Theorem 5.4 Under the hypotheses of Lemma 5.3.…”

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confidence: 97%

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New modified shift-splitting preconditioners for non-symmetric saddle point problems

2019

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Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shiftsplitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

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Section: Numerical Resultsmentioning

confidence: 99%

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confidence: 97%

New modified shift-splitting preconditioners for non-symmetric saddle point problems

2019

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“…From (21), we get the following corollary. [38] proposed the Uzawa-HSS method and applied it for singular saddle point problems [39]. Recently, Miao and Cao [40] studied the semiconvergence of the generalized local HSS method established by Zhu [41] for singular saddle point problems; Zhou and Zhang [42] discussed the semiconvergence of the GMSSOR method which derived by Zhang et al in [43] for singular saddle point problems.…”

Section: Convergence Of the Gasor Methodsmentioning

confidence: 99%

Generalized ASOR and Modified ASOR Methods for Saddle Point Problems

Huang

Wang

Zhao

et al. 2016

Mathematical Problems in Engineering

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Recently, the accelerated successive overrelaxation-(SOR-) like (ASOR) method was proposed for saddle point problems. In this paper, we establish a generalized accelerated SOR-like (GASOR) method and a modified accelerated SOR-like (MASOR) method, which are extension of the ASOR method, for solving both nonsingular and singular saddle point problems. The sufficient conditions of the convergence (semiconvergence) for solving nonsingular (singular) saddle point problems are derived. Finally, numerical examples are carried out, which show that the GASOR and MASOR methods have faster convergence rates than the SORlike, generalized SOR (GSOR), modified SOR-like (MSOR-like), modified symmetric SOR (MSSOR), generalized symmetric SOR (GSSOR), generalized modified symmetric SOR (GMSSOR), and ASOR methods with optimal or experimentally found optimal parameters when the iteration parameters are suitably chosen.

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“…This means that the MGSSP preconditioner outperforms the other five preconditioners for the GMRES method, which is congruous with the results of Table 2. (1) with the following coefficient sub-matrices [40]:…”

Section: Numerical Experimentsmentioning

confidence: 99%

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

et al. 2017

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For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.for the saddle point problem (1), where α is a positive constant and I is the identity matrix.where α ≥ 0, β > 0 and I is the identity matrix. It is easy to see that P SS is a special case of P GSS when α = β. Numerical results in [21,20] confirmed that the GSS preconditioner is superior to the SS preconditioner.

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On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Cited by 25 publications

References 32 publications

New modified shift-splitting preconditioners for non-symmetric saddle point problems

New modified shift-splitting preconditioners for non-symmetric saddle point problems

Generalized ASOR and Modified ASOR Methods for Saddle Point Problems

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

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