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

  • H. Noormohammadi Pour (a1) and H. Sadeghi Goughery (a1)
Abstract
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.

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*Corresponding author. Email address:hsadeghi31@yahoo.com (H. Sadeghi Goughery)
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[1] Z.-Z. Bai , Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), pp. 791815.

[2] Z.-Z. Bai and G.H. Golub , Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), pp. 123.

[3] Z.-Z. Bai , G.H. Golub and C.-K. Li , Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), pp. 583603.

[4] Z.-Z. Bai , G.H. Golub , L.-Z. Lu and J.-F. Yin , Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26 (2005), pp. 844863.

[5] Z.-Z. Bai , G.H. Golub and M.K. Ng , Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603626.

[6] Z.-Z. Bai , G.H. Golub and M.K. Ng , On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), pp. 319335.

[7] 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), pp. 132.

[8] Z.-Z. Bai , M.K. Ng and Z.-Q. Wang , Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410433.

[9] Z.-Z. Bai , B.N. Parlett and Z.-Q. Wang , On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 138.

[10] Z.-Z. Bai and Z.-Q. Wang , On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), pp. 29002932.

[11] M. Benzi and G.H. Golub , A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 2041.

[12] M. Benzi , G.H. Golub and J. Liesen , Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1137.

[13] J.H. Bramble , J.E. Pasciak and A.V. Vassilev , Analysis of the inexact Uzawa algorithm for saddle point problem, SIAM J. Numer. Anal., 34 (1997), pp. 10721092.

[14] H.C. Elman and G.H. Golub , Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Mumer. Anal., 31 (1994), pp. 16451661.

[15] G.H. Golub and D. Vanderstraeten , On the preconditioning of matrices with skew-symmetric splittings, Numer. Algor., 25 (2000), pp. 223239.

[17] G.H. Golub and A.J. Wathen , An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19 (1998), pp. 530539.

[22] Q. Hu and J. Zou , An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 317338.

[23] A. Klawonn , An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 540552.

[24] J.-F. Lu and Z.-Y. Zhang , 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. 19341957.

[25] W. Queck , The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type, SIAM J. Numer. Anal., 26 (1989), pp. 10161030.

[27] Y. Saad and M.H. Schultz , GMRES: A generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856869.

[28] V. Simoncini and M. Benzi , Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 377389.

[29] C.-L. Wang and Z.-Z. Bai , Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices, Linear Algebra Appl., 330 (2001), pp. 215218.

[30] G.-F. Zhang , J.-L. Yang and S.-S. Wang , On generalized parameterized inexact Uzawa type method for a block two-by-two linear system, J. Comput. Appl. Math., 255 (2014), pp. 193207.

[31] W. Zulehner , Analysis of iterative methods for saddle point problems: a unified approach, Math. Comput., 71 (2002), pp. 479505.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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