ON THE UPSS METHOD FOR NON-HERMITIAN SINGULAR SADDLE POINT PROBLEMS

Shuxin Miao; Jing Zhang

doi:10.11948/20190083

2019 Volume 9 Issue 6

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Shuxin Miao, Jing Zhang. ON THE UPSS METHOD FOR NON-HERMITIAN SINGULAR SADDLE POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2308-2317. doi: 10.11948/20190083

Citation:

Shuxin Miao, Jing Zhang. ON THE UPSS METHOD FOR NON-HERMITIAN SINGULAR SADDLE POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2308-2317. doi: 10.11948/20190083

ON THE UPSS METHOD FOR NON-HERMITIAN SINGULAR SADDLE POINT PROBLEMS

Shuxin Miao¹,
Jing Zhang^1, ,

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

Corresponding author: Email address:18419955028@163.com(J. Zhang)
Fund Project: This work was supported by National Natural Science Foundation of China (No. 11861059)

Abstract

Abstract

Recently, a new Uzawa-type method, referred as the UPSS method, is proposed for solving the non-Hermitian nonsingular saddle point problems, see Dou, Yang and Wu (2017). In this paper, we give the semi-convergence analysis of the UPSS method when it is used to solve non-Hermitian singular saddle point problems. An example is given to verify the effectiveness of this method for solving non-Hermitian singular saddle point problems.
- Singular saddle-point problem /
- UPSS method /
- semi-convergence
MSC: 65F10, 65F15

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References

[1]	K. J. Arrow, L. Hurwicz, H. Uzawa, Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, 1958. Google Scholar
[2]	Z. J. Bai, Z. Z. Bai, On nonsingularity of block two-by-two matrices, Linear Algebra Appl., 2013, 439, 2388-2404. doi: 10.1016/j.laa.2013.06.004 CrossRef Google Scholar
[3]	Z. Z. Bai, G. H. Golub, M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 2003, 24, 603-626. doi: 10.1137/S0895479801395458 CrossRef Google Scholar
[4]	Z. Z. Bai, B. N. Parlett, Z. Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 2005, 102, 1-38. doi: 10.1007/s00211-005-0643-0 CrossRef Google Scholar
[5]	Z. Z. Bai, J. F. Yin, Y. F. Su, A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 2006, 24, 539-552. Google Scholar
[6]	M. Benzi, G. H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta. Numer., 2005, 14, 1-137. doi: 10.1017/S0962492904000212 CrossRef Google Scholar
[7]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. Google Scholar
[8]	D. Bertaccini, G. H. Golub, S. S. Capizzano, C. T. Possio, Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math., 2005, 99, 441-484. doi: 10.1007/s00211-004-0574-1 CrossRef Google Scholar
[9]	Y. Cao, M. Q. Jiang, Y. L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl., 2011, 18, 875-895. doi: 10.1002/nla.772 CrossRef Google Scholar
[10]	Y. Cao, S. C. Yi, A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems, Appl. Math. Comput., 2016, 275, 41-49. Google Scholar
[11]	Z. Chao, G. Cheng, Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems, Appl. Math. Letters, 2014, 35, 52-57. doi: 10.1016/j.aml.2014.04.014 CrossRef Google Scholar
[12]	Y. Dou, A.-L. Yang, Y. J. Wu, A new Uzawa-type iteration method for non-Hermitian saddle-point problems, East J. Appl. Math., 2017, 7, 211-226. doi: 10.4208/eajam.290816.130117a CrossRef Google Scholar
[13]	G. H. Golub, X. Wu, J. Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math., 2001, 41, 71-85. doi: 10.1023/A:1021965717530 CrossRef Google Scholar
[14]	Z. G. Huang, L. G. Wang, Z. Xu, J. J. Cui, The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems, Comput. Appl. Math., 2018, 37, 1213-1231. doi: 10.1007/s40314-016-0390-0 CrossRef Google Scholar
[15]	M. Q. Jiang, Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 2009, 231, 973-982. doi: 10.1016/j.cam.2009.05.012 CrossRef Google Scholar
[16]	C. X. Li, S. L. Wu, A single-step HSS method for non-Hermitian positive definite linear systems, Appl. Math. Lett., 2015, 44, 26-29. doi: 10.1016/j.aml.2014.12.013 CrossRef Google Scholar
[17]	Z. Li, M. C. Lai, X. Peng, Z. Zhang, A least squares augmented immersed interface method for solving Navier-Stokes and Darcy coupling equations, Computers and Fluids, 2018, 167, 384-399. doi: 10.1016/j.compfluid.2018.03.032 CrossRef Google Scholar
[18]	S. X. Miao, A new Uzawa-type method for saddle point problems, Appl. Math. Comput., 2017, 300, 95-102. Google Scholar
[19]	J. J. H. Miller, On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl., 1971, 8, 397-406. doi: 10.1093/imamat/8.3.397 CrossRef Google Scholar
[20]	X. Wu, B. P. B. Silva, J. Y. Yuan, Conjugate gradient method for rank deficient saddle point problem, Numer. Algor., 2004, 35, 139-154. doi: 10.1023/B:NUMA.0000021758.65113.f5 CrossRef Google Scholar
[21]	J. S. Xiong, X. B. Gao, Semi-convergence analysis of Uzawa-AOR method for singular saddle point problems, Comp. Appl. Math., 2017, 36, 383-395. doi: 10.1007/s40314-015-0233-4 CrossRef Google Scholar
[22]	A. L. Yang, X. Li, Y. J. Wu, On semi-convergence of the Uzawa-HSS method for singular saddle-point problems, Appl. Math. Comput., 2015, 252, 88-98. Google Scholar
[23]	A. L. Yang, Y. J. Wu, The Uzawa-HSS method for saddle-point problems, Appl. Math. Lett., 2014, 38, 38-42. doi: 10.1016/j.aml.2014.06.018 CrossRef Google Scholar
[24]	J. Y. Yuan, Numerical methods for generalized least squares problem, J. Comput.Appl. Math., 1996, 66, 571-584. doi: 10.1016/0377-0427(95)00167-0 CrossRef Google Scholar
[25]	J. H. Yun, Variants of the Uzawa method for saddle point problem, Comput. Math. Appl., 2013, 65, 1037-1046. doi: 10.1016/j.camwa.2013.01.037 CrossRef Google Scholar
[26]	J. J. Zhang, J. J. Shang, A class of Uzawa-SOR methods for saddle point problems, Appl. Math. Comput., 2010, 216, 2163-2168. Google Scholar
[27]	N. M. Zhang, T. T. Lu, Y. M. Wei, Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math., 2014, 255, 334-345. doi: 10.1016/j.cam.2013.05.015 CrossRef Google Scholar
[28]	N. M. Zhang, Y. M. Wei, On the convergence of general stationary iterative methods for Range- Hermitian singular linear systems, Numer. Linear Algebra Appl., 2010, 17, 139-154. doi: 10.1002/nla.663 CrossRef Google Scholar
[29]	B. Zheng, Z. Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 2009, 431, 808-817. doi: 10.1016/j.laa.2009.03.033 CrossRef Google Scholar