JCM
Journal of Computational Mathematics ›› 2014, Vol. 32 ›› Issue (3): 312-331.DOI: 10.4208/jcm.1401-CR6

• Original Articles • Previous Articles     Next Articles

THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS

Hongtao Fan, Bing Zheng   

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • Received:2013-08-28 Revised:2014-01-15 Online:2014-05-15 Published:2014-05-22
  • Supported by:

    We would like to express our sincere gratitude to the anonymous referees whose constructive comments have the presentation of this paper greatly improved. The work was supported by the National Natural Science Foundation (No.11171371 and No.11101195).

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Hongtao Fan, Bing Zheng. THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS[J]. Journal of Computational Mathematics, 2014, 32(3): 312-331.

For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 (2012) 8816–8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the convergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.
Key words: Generalized saddle point problems, Hermitian and skew-Hermitian matrix splitting, Iteration method, Convergence

CLC Number: 

[1] 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), 1–23.

[2] 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), 603–626.

[3] Z.-Z. Bai and Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), 2900–2932.

[4] Z.-Z. Bai, G.H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), 583–603.

[5] 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), 1–32.

[6] Z.-Z. Bai, B.N. Parlett and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), 1–38.

[7] L. Bergamaschi, J. Gondzio and G. Zilli, Preconditioning indefinite systems in interior point methods for optimization, Comput. Optim. Appl., 28 (2004), 149–171.

[8] M. Benzi and G.H. Golub, A precoditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20–41.

[9] M. Benzi, G.H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1–137.

[10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991.

[11] X.-J. Chen, On preconditioned Uzawa methods and SOR methods for saddle-point problems, J. Comput. Appl. Math., 100 (1998), 207–224.

[12] F. Chen and Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput., 206 (2008), 765–771.

[13] M.-R. Cui, Analysis of iterative algorithms of Uzawa type for saddle point problems, Appl. Numer. Math., 50 (2004), 133–146.

[14] J.H. Bramble and J.E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comput., 50 (1988), 1–17.

[15] Z.-H. Cao, Fast Uzawa algorithms for solving non-symmetric stabilized saddle point problems, Numer. Linear Algebra Appl., 11 (2004), 1–24.

[16] 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), 808–817.

[17] H.C. Elman, D.J. Silvester and A.J.Wathen, Finite Elements and Fast Iterative Solvers, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2005.

[18] T. Rusten and R. Winther, A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl., 13 (1992), 887–904.

[19] J.H. Bramble, J.E. Pasciak and A.T. Vassilev, Uzawa type algorithms for nonsymmetric saddle point problems, Math. Comput., 69 (1999), 667–689.

[20] G.H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math., 41 (2001), 71–85.

[21] 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), 973–982.

[22] X.-F. Ling and X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput., 178 (2006), 372–379.

[23] Y.-Q. Lin and Y.-M. Wei, Corrected Uzawa methods for solving large nonsymmetric saddle point problems, Appl. Math. Comput., 183 (2006), 1108–1120.

[24] Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle point problems, Numer. Linear Agebra Appl., 16 (2009), 447–479.

[25] Z.-H. Huang and T.-Z. Huang, Sepectral properties of the preconditioned AHSS iteration method for generalized saddle point problems, Comput. Appl. Math., 29 (2010), 269–295.

[26] Z.-Z. Bai and G.-Q. Li, Restrictively preconditioned conjugate grdient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561–580.

[27] 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. Comput., 76 (2007), 287–298.

[28] J.-F. Yin and Z.-Z. Bai, The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems, J. Comput. Math., 26 (2008), 240–249.

[29] D.M. Yong, Iterative Solution for Large Linear Sysytems, Academic Press, New York, 1971.

[30] J.-L. Li, D. Luo and Z.-J. Zhang, A generalization of local symmetric and skew-symmetric iteration methods for generalized saddle point problems, J. Appl. Math. Inform., 29 (2011), 1167–1178.

[31] Y.-Y. Zhou and G.-F. Zhang, A generalization of parameterized inexact Uzawa method for generalized saddle point problems, Appl. Math. Comput., 215 (2009), 599–607.

[32] M.-Z. Zhu, A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-Hermitian saddle point problem, Appl. Math. Comput., 218 (2012), 8816–8824.
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