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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)
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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 

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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