DOI QR코드

DOI QR Code

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

  • YUN, JAE HEON
  • Received : 2014.11.11
  • Published : 2015.09.30

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

References

  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. https://doi.org/10.1007/BF01389335
  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. https://doi.org/10.1007/s00211-005-0643-0
  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. https://doi.org/10.1016/j.laa.2008.01.018
  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. https://doi.org/10.1016/j.aml.2014.04.014
  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. https://doi.org/10.1016/j.cam.2014.01.023
  8. M. T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput. 183 (2006), no. 1, 409-415. https://doi.org/10.1016/j.amc.2006.05.094
  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. https://doi.org/10.1137/S1064827596312547
  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. https://doi.org/10.1137/0917004
  11. G. H. Golub, X. Wu, and J. Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), no. 1, 71-85. https://doi.org/10.1023/A:1021965717530
  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. https://doi.org/10.1063/1.1761178
  13. 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.
  14. S. Wright, Stability of augmented system factorization in interior point methods, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 191-222. https://doi.org/10.1137/S0895479894271093
  15. 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. https://doi.org/10.1016/0377-0427(95)00239-1
  16. J. H. Yun, Variants of the Uzawa method for saddle point problem, Comput. Math. Appl. 65 (2013), no. 7, 1037-1046. https://doi.org/10.1016/j.camwa.2013.01.037
  17. J. H. Yun, Convergence of relaxation iterative methods for saddle point problem, Appl. Math. Comput. 251 (2015), 65-80. https://doi.org/10.1016/j.amc.2014.11.047
  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. https://doi.org/10.1016/j.cam.2007.07.001
  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. https://doi.org/10.1016/j.amc.2012.10.116
  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. https://doi.org/10.1002/nla.663
  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. https://doi.org/10.1016/j.laa.2009.03.033
  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. https://doi.org/10.1016/j.camwa.2014.07.003

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)