Browse > Article
http://dx.doi.org/10.4134/BKMS.2015.52.5.1669

SEMI-CONVERGENCE OF THE PARAMETERIZED INEXACT UZAWA METHOD FOR SINGULAR SADDLE POINT PROBLEMS  

YUN, JAE HEON (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES CHUNGBUK NATIONAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 1669-1681 More about this Journal
Abstract
In this paper, we provide semi-convergence results of the parameterized inexact Uzawa method with singular preconditioners for solving singular saddle point problems. We also provide numerical experiments to examine the effectiveness of the parameterized inexact Uzawa method with singular preconditioners.
Keywords
singular saddle point problem; iterative method; Uzawa method; semi-convergence; Moore-Penrose inverse;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Arioli, I. S. Duff, and P. P. M. de Rijk, On the augmented system approach to sparse least squares problems, Numer. Math. 55 (1989), no. 6, 667-684.   DOI
2 Z. Z. Bai, B. N. Parlett, and Z. Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005), no. 1, 1-38.   DOI
3 Z. Z. Bai and Z. Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008), no. 11-12, 2900-2932.   DOI   ScienceOn
4 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
5 Z. H. Cao, On the convergence of general stationary linear iterative methods for singular linear systems, SIAM J. Matrix Anal. Appl. 29 (2007), 1382-1388.
6 Z. Chao and G. Chen, Semi-convergence analysis of the Uzawa-SOR methods for sin-gular saddle point problems, Appl. Math. Lett. 35 (2014), 52-57.   DOI   ScienceOn
7 Z. Chao, N. Zhang, and Y. Lu, Optimal parameters of the generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 266 (2014), 52-60.   DOI   ScienceOn
8 M. T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput. 183 (2006), no. 1, 409-415.   DOI   ScienceOn
9 H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput. 20 (1999), no. 4, 1299-1316.   DOI
10 H. C. Elman and D. J. Silvester, Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 17 (1996), no. 1, 33-46.   DOI
11 G. H. Golub, X. Wu, and J. Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), no. 1, 71-85.   DOI
12 F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incom-pressible flow of fluid with free surface, Phys. Fluids 8 (1965), 2182-2189.   DOI
13 J. H. Yun, Variants of the Uzawa method for saddle point problem, Comput. Math. Appl. 65 (2013), no. 7, 1037-1046.   DOI   ScienceOn
14 J. I. Li and T. Z. Huang, The semi-convergence of generalized SSOR method for singular augmented systems, High Performance Computing and Applications, Lecture Notes in Computer Science 5938 (2010), 230-235.
15 S. Wright, Stability of augmented system factorization in interior point methods, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 191-222.   DOI   ScienceOn
16 J. Y. Yuan and A. N. Iusem, Preconditioned conjugate gradient methods for generalized least squares problem, J. Comput. Appl. Math. 71 (1996), no. 2, 287-297.   DOI   ScienceOn
17 J. H. Yun, Convergence of relaxation iterative methods for saddle point problem, Appl. Math. Comput. 251 (2015), 65-80.   DOI   ScienceOn
18 G. F. Zhang and Q. H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 219 (2008), no. 1, 51-58.   DOI   ScienceOn
19 G. F. Zhang and S. S. Wang, A generalization of parameterized inexact Uzawa method for singular saddle point problems, Appl. Math. Comput. 219 (2013), no. 9, 4225-4231.   DOI   ScienceOn
20 N. Zhang and Y. Wei, On the convergence of general stationary iterative methods for range-Hermitian singular linear systems, Numer. Linear Algebra Appl. 17 (2010), no. 1, 139-154.   DOI   ScienceOn
21 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), no. 5-7, 808-817.   DOI   ScienceOn
22 L. Zhou and N. Zhang, Semi-convergence analysis of GMSSOR methods for singular saddle point problems, Comput. Math. Appl. 68 (2014), no. 5, 596-605.   DOI   ScienceOn