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On a New SSOR-Like Method with Four Parameters for the Augmented Systems

  • Hui-Di Wang (a1) and Zheng-Da Huang (a1)
Abstract
Abstract

In this paper, we propose a new SSOR-like method with four parameters to solve the augmented system. And we analyze the convergence of the method and get the optimal convergence factor under suitable conditions. It is proved that the optimal convergence factor is the same as the GMPSD method [M.A. Louka and N.M. Missirlis, A comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented systems, Numer. Math. 131(2015) 517-540] with five parameters under the same assumption.

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*Corresponding author. Email addresses: hdwang@zju.edu.cn (H.-D. Wang), zdhuang@zju.edu.cn (Z.- D. Huang)
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[1] F. Brezzi and M. Fortin , Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.

[3] J. H. Bramble , J. E. Pasciak , and A. T. Vassilev , Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 10721092.

[5] M. Benzi and G. H. Golub , A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 2041.

[6] Z.-Z. Bai and M. K. Ng , On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput., 26 (2005), pp. 17101724.

[7] Z.-Z. Bai , G. H. Golub , and C.-K. Li , Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comp., 76 (2007), pp. 287298.

[8] Z.-Z. Bai and Z.-Q. Wang , On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), pp. 29002932.

[9] Z.-Z. Bai , M. K. Ng , and Z.-Q. Wang , Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410433.

[10] Z.-Z. Bai , B. N. Parlett , and Z.-Q. Wang , On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 138.

[11] Z.-Z. Bai , G. H. Golub , and M. K. Ng , Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603626.

[12] Z.-Z. Bai , G. H. Golub , and J.-Y. Pan , Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), pp. 132.

[13] Z.-Z. Bai and G. H. Golub , Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle point problems, IMA J. Numer. Anal., 27 (2007), pp. 123.

[14] Z.-Z. Bai , G. H. Golub , and M. K. Ng , On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), pp. 413440.

[15] M. Benzi , A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 360374.

[18] M. T. Darvishi and P. Hessari , A modified symmetric successive overrelaxation method for augmented systems, Comput. Math. Appl., 61 (2011), pp. 31283135.

[19] N. Dyn and W. E. Ferguson , The numerical solution of equality constrained quadratic programming problems, Math. Comp., 41 (1983), pp. 165170.

[20] H. C. Elman and G. H. Golub , Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 16451661.

[21] H. C. Elman , D. J. Silvester , and A. J. Wathen , Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90 (2002), pp. 665688.

[24] N. I. M. Gould , M. E. Hribar , and J. Nocedal , On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23 (2001), pp. 13761395.

[26] Q.-Y. Hu and J. Zou , An iterative method with variable relaxation parameters for saddle point problems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 317338.

[27] Q.-Y. Hu and J. Zou , Two new variants of nonlinear inexact Uzawa algorithms for saddle point problems, Numer. Math., 93 (2002), pp. 333359.

[28] Z.-D. Huang and X.-Y. Zhou , On the minimum convergence factor of a class of GSOR-like methods for augmented systems, Numer. Algor., 70 (2014), pp. 113132.

[29] M.-Q. Jiang and Y. Cao , On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), pp. 973982.

[30] C. Keller , N. I. M. Gould , and A. J. Wathen , Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 13001317.

[31] C.-J. Li , Z. Li , Y.-Y. Nie , and D. J. Evans , Generalized AOR method for the augmented system, Int. J. Comput. Math., 81 (2004), pp. 495504.

[32] Z. Li , C.-J. Li , D. J. Evans , and T. Zhang , Two parameter GSOR method for the augmented systems, Int. J. Comput. Math., 82 (2005), pp. 10331042.

[33] Y.-Q. Lin and Y.-M. Wei , Fast corrected Uzawa methods for solving symmetric saddle point problems, Calcolo, 43 (2006), pp. 6582.

[34] Y.-Q. Lin and Y.-M. Wei , A note on constraint preconditioners for nonsymmetric saddle point problems, Numer. Linear Algebra Appl., 14 (2007), pp. 659664.

[36] J.-F. Lu and Z.-Y. Zhang , A modified nonlinear inexact Uzawa algorithm with a variable relaxation parameter for the stabilized saddle point problem, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 19341957.

[37] X. Li , A.-L. Yang , and Y.-J. Wu , Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for saddle-point problems, Int. J. Comput. Math., 91 (2014), pp. 12241283.

[38] M. A. Louka and N. M. Missirlis , A comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented linear systems, Numer. Math., 131 (2015), pp. 517540.

[39] M. M. Martins , W. Yousif , and J. L. Santos , A variant of the AOR method for augmented systems, Math. Comput., 81 (2012), pp. 399417.

[40] H. S. Najafi and S. A. Edalatpanah , A new modified SSOR iteration method for solving augmented linear systems, Int. J. Comput. Math., 91 (2014), pp. 539552.

[41] H. S. Najafi and S. A. Edalatpanah , On the modified symmetric successive over-relaxation method for augmented systems, Comput. Appl. Math., 34 (2015), pp. 607617.

[42] H. N. Pour and H. S. Goughery , New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. Algor., 69 (2015), pp. 207225.

[43] E. D. Sturier and J. Liesen , Block-diagonal preconditioners for indefinite linear algebraic systems, SIAM J. Sci. Comput., 26 (2005), pp. 15981619.

[44] X.-H. Shao , Z. Li , and C.-J. Li , Modified SOR-like method for the augmented system, Int. J. Comput. Math., 84 (2007), pp. 16531662.

[45] C. Wen and T.-Z. Huang , Modified SSOR-like method for augmented system, Math. Model. Anal., 16 (2011), pp. 475487.

[46] S.-L. Wu , T.-Z. Huang , and X.-L. Zhao , A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math., 228 (2009), pp. 424433.

[47] K. Wang , J.-J. Di , and D. Liu , Improved PHSS iterative methods for solving saddle point problems, Numer. Algor., 71 (2016), pp. 753773.

[49] B. Zheng , K. Wang , and Y.-J. Wu , SSOR-like methods for saddle point problems, Int. J. Comput. Math., 86 (2009), pp. 14051423.

[50] B. Zheng , Z.-Z. Bai , and X. Yang , On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 431 (2009), pp. 808817.

[51] G.-F. Zhang and Q.-H. Lu , On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 219 (2008), pp. 5158.

[53] L.-T. Zhang , T.-Z. Huang , S.-H. Cheng , and Y.-P. Wang , Convergence of a generalized MSSOR method for augmented systems, J. Comput. Appl. Math., 236 (2012), pp. 18411850.

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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