Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T00:38:04.078Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  31 January 2017

Quan-Yu Dou ,
Jun-Feng Yin  and
Ze-Yu Liao
Quan-Yu Dou*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China
Jun-Feng Yin*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China
Ze-Yu Liao*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China
*
*Corresponding author. Email addresses:08douquanyu@tongji.edu.cn (Q.-Y. Dou), yinjf@tongji.edu.cn (J.-F. Yin), 103632@tongji.edu.cn (Z.-Y. Liao)
*Corresponding author. Email addresses:08douquanyu@tongji.edu.cn (Q.-Y. Dou), yinjf@tongji.edu.cn (J.-F. Yin), 103632@tongji.edu.cn (Z.-Y. Liao)
*Corresponding author. Email addresses:08douquanyu@tongji.edu.cn (Q.-Y. Dou), yinjf@tongji.edu.cn (J.-F. Yin), 103632@tongji.edu.cn (Z.-Y. Liao)
Get access

Abstract

Based on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

Keywords

Nonsymmetric saddle point problemsfast shift-splittingconvergencesemi-convergence

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 172 - 191
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arrow, K. J., Hurwicz, L. and Uzawa, H., Studies in Non-Linear Programming, Stanford University Press: Stanford, CA, 1958.Google Scholar
[2] Bacuta, C., A unified approach for Uzawa algorithms, SIAM J. Numer. Anal., 44(2006), pp. 2633–2649.Google Scholar
[3] Bacuta, C., McCracken, B. and Shu, L., Residual reduction algorithms for nonsymmetric saddle point problems, J. Comput. Appl. Math., 235(2011), pp. 1614–1628.Google Scholar
[4] Bai, Z.-Z., On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing., 89(2010), pp. 171–197.Google Scholar
[5] Bai, Z.-Z., Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16(2009), pp. 447–479.Google Scholar
[6] Bai, Z.-Z., Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models, Numer. Linear Algebra Appl., 19(2012), pp. 914–936.Google Scholar
[7] Bai, Z.-Z. and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27(2007), pp. 1–23.Google Scholar
[8] Bai, Z.-Z., Golub, G. H. and Li, C.-K., Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28(2006), pp. 583–603.Google Scholar
[9] Bai, Z.-Z., Golub, G. H. and Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24(2003), pp. 603–626.Google Scholar
[10] Bai, Z.-Z., Parlett, B. N. and Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102(2005), pp. 1–38.Google Scholar
[11] Bai, Z.-Z., Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428(2008), pp. 2900–2932.Google Scholar
[12] Bai, Z.-Z., Yin, J.-F. and Su, Y.-F., A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24(2006), pp. 539–552.Google Scholar
[13] Benzi, M., Golub, G. H. and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14(2005), pp. 1–37.Google Scholar
[14] Berman, A. and Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994.Google Scholar
[15] Bramble, J. H., Pasciak, J. E. and Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34(1997), pp. 1072–1092.Google Scholar
[16] Bramble, J. H., Pasciak, J. E. and Vassilev, A. T., Uzawa type algorithm for nonsymmetric saddle point problems, Math. Comput., 69(2000), pp. 667–689.Google Scholar
[17] Cao, Y., Du, J. and Niu, Q., Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 272(2014), pp. 239–250.Google Scholar
[18] Cao, Y., Li, S. and Yao, L.-Q., A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems, Appl. Math. Lett., 49(2015), pp. 20–27.Google Scholar
[19] Cao, Y. and Miao, S.-X., On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems, Comput. Math. Appl., 71(2016), pp. 1503–1511.Google Scholar
[20] Elman, H. C., Ramage, A. and Silvester, D. J., IFISS: A Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Soft., 14(2007), pp. 33.Google Scholar
[21] Fischer, B., Ramage, R., Silvester, D. J. and Wathen, A. J., Minimum residual methods for augmented systems, BIT Numer. Math., 38(1998), pp. 527–543.Google Scholar
[22] Golub, G. H., Wu, X. and Yuan, J.-Y., SOR-like methods for augmented systems, BIT Numer. Math., 41(2001), pp. 71–85.Google Scholar
[23] Jiang, M.-Q. and Cao, Y., On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231(2009), pp. 973–982.Google Scholar
[24] Krukier, L. A., Krukier, B. L. and Ren, Z.-R., Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems, Numer. Linear Algebra Appl., 21(2014), pp. 152–170.Google Scholar
[25] Krukier, L. A., Martynova, T. S. and Bai, Z.-Z., Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems, J. Comput. Appl. Math., 232(2009), pp. 3–16.Google Scholar
[26] Miller, J. H., On the location of zeros of certain classes of polynomials with applications to numerical analysis, IMA J. Appl. Math., 8(1971), pp. 397–406.Google Scholar
[27] Rubinov, A. and Yang, X., Lagrange-type functions in constrained non-convex optimization, J. Math. Sci., 115(2003), pp. 2437–2505.Google Scholar
[28] Salkuyeh, D. K., Masoudi, M. and Hezari, D., On the generalized shift-splitting preconditioner for saddle point problems, Appl. Math. Lett., 48(2015), pp. 55–61.Google Scholar
[29] Shen, Q.-Q. and Shi, Q., Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems, Comput. Math. Appl., 72(2016), pp. 632–641.Google Scholar
[30] Wang, L. and Bai, Z.-Z., Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math., 44(2004), pp. 363–386.Google Scholar
[31] Wu, X., Silva, B. P. B. and Yuan, J.-Y., Conjugate gradient method for rank deficient saddle point problems, Numer. Algor., 35(2004), pp. 139–154.Google Scholar
[32] Yang, A.-L., Li, X. and Wu, Y.-J., On semi-convergence of the Uzawa-HSS method for singular saddle-point problems, Appl. Math. Comput., 252(2015), pp. 88–98.Google Scholar
[33] Zhang, N.-M., Lu, T.-T. and Wei, Y.-M., Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math., 255(2014), pp. 334–345.Google Scholar
[34] Zheng, B., Bai, Z.-Z. and Yang, X., On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 431(2009), pp. 808–817.Google Scholar
[35] Zhou, S.-W., Yang, A.-L., Dou, Y. and Wu, Y.-J., The modified shift-splitting preconditioners for nonsymmetric saddle point problems, Appl. Math. Lett., 59(2016), pp. 109–114.Google Scholar