On semi-convergence of parameterized Uzawa methods for singular saddle point problems

Zheng, Bing; Bai, Zhong‐Zhi; Yang, Xi

doi:10.1016/j.laa.2009.03.033

Cited by 115 publications

(49 citation statements)

References 15 publications

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“…According to Theorem 3.1, similar to the analysis in [14,19], we can obtain semi-contraction factor of the GSOR method for the singular but consistent augmented linear system. …”

Section: Which Gives Thatmentioning

confidence: 92%

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On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

Chen¹,

Jiang

Zheng

2010

JCM

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The generalized successive overrelaxation (GSOR) method was presented and studied by Bai, Parlett and Wang [Numer. Math. 102(2005), pp.1-38] for solving the augmented system of linear equations, and the optimal iteration parameters and the corresponding optimal convergence factor were exactly obtained. In this paper, we further estimate the contraction and the semi-contraction factors of the GSOR method. The motivation of the study is that the convergence speed of an iteration method is actually decided by the contraction factor but not by the spectral radius in finite-step iteration computations. For the nonsingular augmented linear system, under some restrictions we obtain the contraction domain of the parameters involved, which guarantees that the contraction factor of the GSOR method is less than one. For the singular but consistent augmented linear system, we also obtain the semi-contraction domain of the parameters in a similar fashion. Finally, we use two numerical examples to verify the theoretical results and the effectiveness of the GSOR method.Mathematics subject classification: 65F10, 65F50; CR: G1.3.

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“…According to Theorem 3.1, similar to the analysis in [14,19], we can obtain semi-contraction factor of the GSOR method for the singular but consistent augmented linear system. …”

Section: Which Gives Thatmentioning

confidence: 92%

“…For example, the Uzawa-type methods [13,15], the preconditioned Krylov subspace methods [7,9,18], the relaxation methods [8,10,14,17], and the Hermitian and skew-Hermitian splitting methods [3][4][5][6]12], etc. Moreover, the singular augmented linear system has been specially studied in [11,19].…”

Section: Introductionmentioning

confidence: 99%

On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

Chen¹,

Jiang

Zheng

2010

JCM

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“…Several authors have presented semi-convergence analysis of relaxation iterative methods for solving the singular saddle point problem (1). Zheng et al [24] studied semi-convergence of the PU (Parameterized Uzawa) method, Li and Huang [12] examined semi-convergence of the GSSOR method, Zhang and Wang [21] studied semi-convergence of the GPIU method, Chao and Chen [5] provided semi-convergence analysis of the Uzawa-SOR method, and so on.…”

Section: Introductionmentioning

confidence: 99%

Acceleration of One-Parameter Relaxation Methods for Singular Saddle Point Problems

Yun¹

2016

Journal of the Korean Mathematical Society

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Abstract. In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.

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“…Zheng et al [21] studied semi-convergence of the PU (Parameterized Uzawa) method, Li and Huang [13] examined semi-convergence of the GSSOR method, Zhang and Wang [19] studied semi-convergence of the GPIU method, Chao and Chen [6] provided semi-convergence analysis of the Uzawa-SOR method, and Zhou and Zhang [22] studied semi-convergence of the GMSSOR (Generalized Modified SSOR) method.…”

Section: Introductionmentioning

confidence: 99%

Semi-Convergence of the Parameterized Inexact Uzawa Method for Singular Saddle Point Problems

Yun¹

2015

Bulletin of the Korean Mathematical Society

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On semi-convergence of parameterized Uzawa methods for singular saddle point problems

Cited by 115 publications

References 15 publications

On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

Acceleration of One-Parameter Relaxation Methods for Singular Saddle Point Problems

Semi-Convergence of the Parameterized Inexact Uzawa Method for Singular Saddle Point Problems

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