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SOR-like Methods with Optimization Model for Augmented Linear Systems

Part of: Basic linear algebra Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Rui-Ping Wen ,
Su-Dan Li  and
Guo-Yan Meng
Rui-Ping Wen*
Affiliation:
Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, China
Su-Dan Li*
Affiliation:
Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, China
Guo-Yan Meng*
Affiliation:
Department of Mathematics, Xinzhou Teachers College, Xinzhou, 034000, China
*
*Corresponding author. Email addresses:[email protected] (R.-P. Wen), [email protected] (S.-D. Li), [email protected] (G.-Y. Meng)
*Corresponding author. Email addresses:[email protected] (R.-P. Wen), [email protected] (S.-D. Li), [email protected] (G.-Y. Meng)
*Corresponding author. Email addresses:[email protected] (R.-P. Wen), [email protected] (S.-D. Li), [email protected] (G.-Y. Meng)
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Abstract

There has been a lot of study on the SOR-like methods for solving the augmented system of linear equations since the outstanding work of Golub, Wu and Yuan (BIT 41(2001)71-85) was presented fifteen years ago. Based on the SOR-like methods, we establish a class of accelerated SOR-like methods for large sparse augmented linear systems by making use of optimization technique, which will find the optimal relaxation parameter ω by optimization models. We demonstrate the convergence theory of the new methods under suitable restrictions. The numerical examples show these methods are effective.

Keywords

SOR-like methodoptimizationaugmented linear systemsconvergence

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices 15A06: Linear equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 101 - 115
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Arrow, K., Hurwicz, L., and Uzawa, H., Studies in Nonlinear Programming, Stanford University Press, Stanford, 1958.Google Scholar
[2] 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
[3] Bai, Z. Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), pp. 791–815.Google Scholar
[4] Bai, Z. Z., and Chi, X. B., Asymptotically optimal successive overrelaxation methods for systems of linear equations, J. Comput. Math., 21(5) (2003), pp. 603–612.Google Scholar
[5] Bai, Z. Z., and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27(1) (2007), pp. 1–23.Google Scholar
[6] Bai, Z. Z., Golub, G. H., and Ng, M. K., On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), pp. 319–335.Google Scholar
[7] Bai, Z. Z., Golub, G. H., and Pan, J. Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), pp. 1–32.Google Scholar
[8] Bai, Z. Z., Parlett, B. N., and Wang, Z. Q., On generalized successive overrelaxatioon methods for augmented linear systems, Numer. Math., 102 (2005), pp. 1–38.Google Scholar
[9] Benzi, M., and Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 20–41.CrossRefGoogle Scholar
[10] Benzi, M., Golub, G. H., and Liesen, J., Numerical solution of saddle point problems, Acta Numerica, 14 (2005), pp. 1–137.Google Scholar
[11] Björck, A., Numerical stablity of methods for solving augmented systems. in Proceedings from Recent Developments in Optimization Theory and Nonlinear Analysis, Jerusalem, 1995, Censor, Y. and Reich, S., eds., Contemp. Math., 204, Amer. Math. Soc., Providence, RI, (1997), pp. 51–60.Google Scholar
[12] 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
[13] Chao, Z., and Chen, G. L., Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems, Appl. Math. Letter, 35 (2014), pp. 52–57.Google Scholar
[14] Chao, Z., Zhang, N. M., and Lu, Y. Z., Optimal parameters of the generalized symmetric SOR method for augmented system, J. Comput. Appl. Math., 266 (2014), pp. 52–60.Google Scholar
[15] Darvishi, M. T., and Hessari, P., Symmetric SOR method for augumented systems, Appl. Math. Comput., 173(1) (2006), pp. 404–420.Google Scholar
[16] Elman, H. C., and Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645–1661.Google Scholar
[17] Elman, H., and Silvester, D., Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput., 17 (1996), pp. 33–46.Google Scholar
[18] Fischer, , Ramage, A., Silvester, D. J., and Wathen, A. J., Minimum residual methods for augmented systems, BIT, 38 (1998), pp. 527–543.CrossRefGoogle Scholar
[19] Fortin, M., and Glowinski, R., Augmented Lagrangian Methods, Applications to the Numerical Solution of Boundary Value Problems, Amsterdam: North-Holland, 1983.Google Scholar
[20] Golub, G. H., Wu, X., and Yuan, J. Y., SOR-like methods for augumented systems, BIT, 41 (2001), pp. 71–85.Google Scholar
[21] Guo, P., Li, C. X., and Wu, S. L.. A modified SOR-like method for the augmented systems, J. Comput. Appl. Math., 274 (2015), pp. 58–69.Google Scholar
[22] Li, Z., Li, C.J., and Evans, D. J., Chebyshev acceleration for SOR-like method, Inter. J. Comput. Math., 82(5) (2005), pp. 583–593.CrossRefGoogle Scholar
[23] Liang, Z. Z., and Zhang, G. F., Modified unsymmetric SOR method for augmented systems, Appl. Math. Comput., 234 (2014), pp. 584–598.Google Scholar
[24] Nelder, J. A., and Mead, R., A simplex method for function minimization, Comput. J., 7 (1965), pp. 308–313.Google Scholar
[25] Shao, X. H., Li, Z., and Li, C. J., Modified SOR-like methods for the augmented systems, Inter. J. Comput. Math., 84 (2007), pp. 1653–1662.Google Scholar
[26] Sun, J. G., Structured backward errors for KKT systems, Linear Algebra Appl., 288 (1999), pp. 75–88.Google Scholar
[27] Wen, R. P., Wang, C. L., and Yan, X. H., Generalization of the nonstationary multisplitting iterative method for symmetric positive definite linear systems, Appl. Math. Comput., 216 (2010), pp. 1707–1714.Google Scholar
[28] Wen, R. P., Meng, G. Y., and Wang, C. L., Quasi-Chebyshev accelerated iteration methods based on optimization for linear systems, Comput. Math. Appl., 66 (2013), pp. 934–942.Google Scholar
[29] Wright, S., Stablity of augmented system factorizations in interior-point methods, SIAM J. Matrix Anal. Appl., 18 (1997) 191–222.Google Scholar
[30] Yuan, J. Y., Numerical methods for generalized least squares problems, J. Comput. Appl. Math., 66 (1996), pp. 571–584.Google Scholar
[31] Zhang, G. F., and Lu, Q. H., On generalized symmetric SOR method for augumented systems, J. Comput. Appl. Math., 219 (2008), pp. 51–58.Google Scholar
[32] Zhang, J. J., and Shang, J. J., A class of Uzawa-SOR methods for saddle point problem, Appl. Math. Comput., 216 (2010), pp. 2163–2168.Google Scholar
[33] Zheng, B., Wang, K., and Wu, Y. J., SSOR-like methods for saddle point problem, Inter. J. Comput. Math., 86 (2009), pp. 1405–1423.Google Scholar
[34] Zheng, Q. Q., and Ma, C. F., A new SOR-Like method for the saddle point problems, Appl. Math. Comput., 233 (2014), pp. 421–429.Google Scholar
[35] Zheng, Q. Q., and Ma, C. F., A class of triangular splitting methods for saddle point problems, J. Comput. Appl. Math., 298 (2016), pp. 13–23.Google Scholar