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ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS

  • Yun, Jae Heon (Department of Mathematics, College of Natural Sciences Chungbuk National University)
  • Received : 2015.04.10
  • Published : 2016.05.01

Abstract

In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

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, G. H. Golub, and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004), no. 1, 1-32. https://doi.org/10.1007/s00211-004-0521-1
  3. 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
  4. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
  5. Z. Chao and G. Chen, Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems, Appl. Math. Lett. 35 (2014), 52-57. https://doi.org/10.1016/j.aml.2014.04.014
  6. Z. Chao, N.-M. Zhang, and Y.-Z. 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
  7. 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
  8. 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
  9. B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen, Minimum residual methods for augmented systems, BIT 38 (1998), 527-543. https://doi.org/10.1007/BF02510258
  10. G. H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), no. 3, 71-85. https://doi.org/10.1023/A:1021965717530
  11. 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
  12. 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.
  13. G. H. Santos, B. P. B. Silva, and J.-Y. Yuan, Block SOR methods for rank deficient least squares problems, J. Comput. Appl. Math. 100 (1998), no. 1, 1-9. https://doi.org/10.1016/S0377-0427(98)00114-9
  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. S.-L. Wu, T.-Z. Huang, and X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (2009), no. 1, 424-433. https://doi.org/10.1016/j.cam.2008.10.006
  16. D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. N. Zhang, T.-T. Lu, and Y. Wei, Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math. 255 (2014), 334-345. https://doi.org/10.1016/j.cam.2013.05.015
  23. 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
  24. 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