Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T17:11:53.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Accelerated Hermitian and Skew-Hermitian Splitting Methods for Saddle-Point Problems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  20 February 2017

H. Noormohammadi Pour  and
H. Sadeghi Goughery
H. Noormohammadi Pour
Affiliation:
Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran
H. Sadeghi Goughery*
Affiliation:
Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran
*
*Corresponding author. Email address:hsadeghi31@yahoo.com (H. Sadeghi Goughery)
Get access

Abstract

We generalize the accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems. These methods involve four iteration parameters whose special choices can recover the preconditioned HSS and accelerated HSS iteration methods. Also a new efficient case is introduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem. Numerical experiments are used to further examine the effectiveness and robustness of iterations.

Keywords

saddle-point problemHermitian and skew-Hermitian splittingpreconditioning

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 167 - 185
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] Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), pp. 791–815.Google Scholar
[2] 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.Google Scholar
[3] Bai, Z.-Z., Golub, G.H. and Li, C.-K., Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), pp. 583–603.CrossRefGoogle Scholar
[4] Bai, Z.-Z., Golub, G.H., Lu, L.-Z. and Yin, J.-F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26 (2005), pp. 844–863.CrossRefGoogle Scholar
[5] Bai, Z.-Z., Golub, G.H. and Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603–626.CrossRefGoogle Scholar
[6] Bai, Z.-Z., Golub, G.H. and Ng, M.K., On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), pp. 319–335.Google 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., Ng, M.K. and Wang, Z.-Q., Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410–433.CrossRefGoogle Scholar
[9] 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
[10] 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
[11] Benzi, M. and Golub, G.H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 20–41.Google Scholar
[12] Benzi, M., Golub, G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1–137.Google Scholar
[13] Bramble, J.H., Pasciak, J.E. and Vassilev, A.V., Analysis of the inexact Uzawa algorithm for saddle point problem, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092.Google Scholar
[14] Elman, H.C. and Golub, G.H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Mumer. Anal., 31 (1994), pp. 1645–1661.Google Scholar
[15] Golub, G.H. and Vanderstraeten, D., On the preconditioning of matrices with skew-symmetric splittings, Numer. Algor., 25 (2000), pp. 223–239.CrossRefGoogle Scholar
[16] Golub, G.H. and Van Loan, C.F., Matrix Computations, 3rd Edition, The Johns Hopkins University Press, Baltimore, 1996.Google Scholar
[17] 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.CrossRefGoogle Scholar
[18] Golub, G.H., Wu, X. and Yuan, J.-Y., SOR-like methods for augmented systems, BIT, 41 (2001), pp. 71–85.CrossRefGoogle Scholar
[19] Huang, N. and Ma, C.-F., The BGS-Uzawa and BJ-Uzawa iterative methods for solving the saddle point problem, Appl. Math. Comput., 256 (2015), pp. 94–108.Google Scholar
[20] Huang, N. and Ma, C.-F., The nonlinear inexact Uzawa hybrid algorithms based on one-step Newton method for solving nonlinear saddle-point problems, Appl. Math. Comput., 270 (2015), pp. 291–311.Google Scholar
[21] Huang, N. and Ma, C.-F., An inexact relaxed DPSS preconditioner for saddle point problem, Appl. Math. Comput., 265 (2015), pp. 431–447.Google Scholar
[22] Hu, Q. and Zou, J., An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 317–338.Google Scholar
[23] Klawonn, A., An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 540–552.Google Scholar
[24] Lu, J.-F. and Zhang, Z.-Y., A modified nonlinear inexact Uzawa algorithm with a variable relaxation parameter for the stabilized saddle point problem, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1934–1957.CrossRefGoogle Scholar
[25] Queck, W., The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type, SIAM J. Numer. Anal., 26 (1989), pp. 1016–1030.CrossRefGoogle Scholar
[26] Saad, Y., Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996.Google Scholar
[27] Saad, Y. and Schultz, M.H., GMRES: A generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869.CrossRefGoogle Scholar
[28] Simoncini, V. and Benzi, M., Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 377–389.CrossRefGoogle Scholar
[29] Wang, C.-L. and Bai, Z.-Z., Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices, Linear Algebra Appl., 330 (2001), pp. 215–218.CrossRefGoogle Scholar
[30] Zhang, G.-F., Yang, J.-L. and Wang, S.-S., On generalized parameterized inexact Uzawa type method for a block two-by-two linear system, J. Comput. Appl. Math., 255 (2014), pp. 193–207.CrossRefGoogle Scholar
[31] Zulehner, W., Analysis of iterative methods for saddle point problems: a unified approach, Math. Comput., 71 (2002), pp. 479–505.CrossRefGoogle Scholar