Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-25T10:28:33.328Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Yan Dou ,
Ai-Li Yang  and
Yu-Jiang Wu
Yan Dou
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
Ai-Li Yang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
Yu-Jiang Wu*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
*
*Corresponding author. Email address:myjaw@lzu.edu.cn (Y.-J. Wu)
Get access

Abstract

Based on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.

Keywords

Saddle-point problemsUzawa methodpreconditioned shift-splittingconvergencepreconditioner

MSC classification

Secondary: 65F08: Preconditioners for iterative methods 65F10: Iterative methods for linear systems 65F20: Overdetermined systems, pseudoinverses
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 211 - 226
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arrow, K. J., Hurwicz, L., and Uzawa, H., Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, 1958.Google Scholar
[2] Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), pp. 791–815.CrossRefGoogle Scholar
[3] 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
[4] Bai, Z.-Z., Block preconditioners for elliptic PDE-constrained optimization problems, Computing, 91 (2011), pp. 379–395.Google Scholar
[5] Bai, Z.-Z., Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models, Numer. Linear Algebra Appl., 19 (2012), pp. 914–936.Google Scholar
[6] Bai, Z.-Z., and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), pp. 1–23.CrossRefGoogle 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., and Hadjidimos, A., Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix, Linear Algebra Appl., 463 (2014), pp. 322–339.Google Scholar
[9] Bai, Z.-Z., Ng, M. K., and Wang, Z.-Q., Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410–433.Google Scholar
[10] Bai, Z.-Z., Parlett, B. N., and Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 1–38.CrossRefGoogle Scholar
[11] Bai, Z.-Z., and Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), pp. 2900–2932.CrossRefGoogle Scholar
[12] Bai, Z.-Z., Yin, J.-F., and Su, Y.-F., A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24 (2006), pp. 539–552.Google Scholar
[13] Benzi, M., Golub, G. H., and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1–137.Google Scholar
[14] 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.CrossRefGoogle Scholar
[15] Brezzi, F., and Fortin, M., Mixed and hybrid finite elements methods, Springer Series in Computational Mathematics, Vol. 15, Springer-Verlag, New York, 2012.Google Scholar
[16] Cao, Y., Du, J., and Niu, Q., Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 272 (2014), pp. 239–250.CrossRefGoogle Scholar
[17] Cao, Y., Yao, L.-Q., Jiang, M.-Q., and Niu, Q., A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math., 31 (2013), pp. 398–421.CrossRefGoogle Scholar
[18] Cao, Y., and Yi, S.-C., A class of Uzawa–PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems, Appl. Math. Comput., 275 (2016), pp. 41–49.Google Scholar
[19] 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
[20] Fischer, B., Ramage, A., Silvester, D. J., and Wathen, A. J., Minimum residual methods for augmented systems, BIT Numer. Math., 38 (1998), pp. 527–543.Google Scholar
[21] Golub, G. H., and Wathen, A. J., An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19 (1998), pp. 530–539.Google Scholar
[22] Jiang, M.-Q., and Cao, Y., On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), pp. 973–982.Google Scholar
[23] Keller, C., Gould, N. I. M., and Wathen, A. J., Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300–1317.CrossRefGoogle Scholar
[24] Leem, K. H., Oliveira, S. P., and Stewart, D. E., Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations, Numer. Linear Algebra Appl., 11 (2004), pp. 293–308.CrossRefGoogle Scholar
[25] Miller, J. J. H., On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl., 8 (1971), pp. 397–406.Google Scholar
[26] Saad, Y., Iterative Methods for Sparse Linear Systems, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, 2003.Google Scholar
[27] Santos, C. H., Silva, B. P. B., and Yuan, J.-Y., Block SOR methods for rank-deficient least-squares problems, J. Comput. Appl. Math., 100 (1998), pp. 1–9.CrossRefGoogle Scholar
[28] Wu, X., Silva, B. P. B., and Yuan, J.-Y., Conjugate gradient method for rank deficient saddle point problems, Numer. Algor., 35 (2004), pp. 139–154.Google Scholar
[29] Yang, A.-L., Li, X., and Wu, Y.-J., On semi-convergence of the Uzawa–HSS method for singular saddle-point problems, Appl. Math. Comput., 252 (2015), pp. 88–98.Google Scholar
[30] Yang, A.-L., and Wu, Y.-J., The Uzawa–HSSmethod for saddle-point problems, Appl. Math. Lett., 38 (2014), pp. 38–42.Google Scholar
[31] Yun, J.-H., Variants of the Uzawa method for saddle point problems, Comput. Math. Appl., 65 (2013), pp. 1037–1046.Google Scholar
[32] Zhang, J.-J., and Shang, J.-J., A class of Uzawa–SOR methods for saddle point problems, Appl. Math. Comput., 216 (2010), pp. 2163–2168.Google Scholar