Skip to main content
×
×
Home

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

  • Quan-Yu Dou (a1), Jun-Feng Yin (a1) and Ze-Yu Liao (a1)
Abstract
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.

Copyright
Corresponding author
*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)
References
Hide All
[1] Arrow K. J., Hurwicz L. and Uzawa H., Studies in Non-Linear Programming, Stanford University Press: Stanford, CA, 1958.
[2] Bacuta C., A unified approach for Uzawa algorithms, SIAM J. Numer. Anal., 44(2006), pp. 26332649.
[3] Bacuta C., McCracken B. and Shu L., Residual reduction algorithms for nonsymmetric saddle point problems, J. Comput. Appl. Math., 235(2011), pp. 16141628.
[4] Bai Z.-Z., On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing., 89(2010), pp. 171197.
[5] Bai Z.-Z., Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16(2009), pp. 447479.
[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. 914936.
[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. 123.
[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. 583603.
[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. 603626.
[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. 138.
[11] Bai Z.-Z., Wang Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428(2008), pp. 29002932.
[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. 539552.
[13] Benzi M., Golub G. H. and Liesen J., Numerical solution of saddle point problems, Acta Numer., 14(2005), pp. 137.
[14] Berman A. and Plemmons R. J., Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994.
[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. 10721092.
[16] Bramble J. H., Pasciak J. E. and Vassilev A. T., Uzawa type algorithm for nonsymmetric saddle point problems, Math. Comput., 69(2000), pp. 667689.
[17] Cao Y., Du J. and Niu Q., Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 272(2014), pp. 239250.
[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. 2027.
[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. 15031511.
[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.
[21] Fischer B., Ramage R., Silvester D. J. and Wathen A. J., Minimum residual methods for augmented systems, BIT Numer. Math., 38(1998), pp. 527543.
[22] Golub G. H., Wu X. and Yuan J.-Y., SOR-like methods for augmented systems, BIT Numer. Math., 41(2001), pp. 7185.
[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. 973982.
[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. 152170.
[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. 316.
[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. 397406.
[27] Rubinov A. and Yang X., Lagrange-type functions in constrained non-convex optimization, J. Math. Sci., 115(2003), pp. 24372505.
[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. 5561.
[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. 632641.
[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. 363386.
[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. 139154.
[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. 8898.
[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. 334345.
[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. 808817.
[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. 109114.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 79 *
Loading metrics...

Abstract views

Total abstract views: 282 *
Loading metrics...

* Views captured on Cambridge Core between 31st January 2017 - 22nd January 2018. This data will be updated every 24 hours.