Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-30T22:12:15.620Z Has data issue: false hasContentIssue false
  • English
  • Français

A New GSOR Method for Generalised Saddle Point Problems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  27 January 2016

Na Huang  and
Chang-Feng Ma
Na Huang
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350117, China
Chang-Feng Ma*
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350117, China
*
*Corresponding author. Email address:[email protected] (C.-F. Ma)
Get access

Abstract

A novel generalised successive overrelaxation (GSOR) method for solving generalised saddle point problems is proposed, based on splitting the coefficient matrix. The proposed method is shown to converge under suitable restrictions on the iteration parameters, and we present some illustrative numerical results.

Keywords

Generalised saddle point problemgeneralised successive overrelaxation methodsplittingconvergence analysis

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 1 , February 2016 , pp. 23 - 41
Copyright
Copyright © Global-Science Press 2016 

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]Arioli, M., Manzini, G., A null space method for mixed finite-element approximations of Darcy's equation, Commun. Numer. Methods Engrg. 18, 645–657 (2002).Google Scholar
[2]Björck, A., Numerical Methods for Least Squares Problems, SIAM, Philadelphia (1996).Google Scholar
[3]Benzi, M., Solution of equality-constrained quadratic programming problems by a projection iterative method, Rend. Mat. Appl. 13, 275–296 (1993).Google Scholar
[4]Bai, Z.-Z., Benzi, M. and Chen, F., Modified HSS iteration methods for a class of complex symmetric linear systems, Computing 87, 93–111 (2010).Google Scholar
[5]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer, New York (1991).Google Scholar
[6]Benzi, M. and Golub, G.H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26, 20–41 (2004).Google Scholar
[7]Bai, Z.-Z. and Golub, G.H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle point problems, IMA J. Numer. Anal. 27, 1–23 (2007).Google Scholar
[8]Benzi, M., Golub, G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numer. 14, 1–137 (2005).Google Scholar
[9]Bai, Z.-Z., Golub, G.H. and Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput. 76, 287–298 (2007).Google Scholar
[10]Bai, Z.-Z., Golub, G.H., Lu, L.-Z. and Yin, J.-F., Block triangular and skew-Hermitian splitting method for positive-definite linear systems, SIAM J. Sci. Comput. 26, 844–863 (2005).Google Scholar
[11]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, 319–335 (2007).Google Scholar
[12]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, 603–626 (2003).Google Scholar
[13]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, 1–32 (2004).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, 1072–1092 (1997).Google Scholar
[15]Bai, Z.-Z., Parlett, B.N. and Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102, 1–38 (2005).Google Scholar
[16]Bai, Z.-Z. and Wang, Z.-Q., On parameterised inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428, 2900–2932 (2008).CrossRefGoogle Scholar
[17]Bank, R.E., Welfert, B.D. and Yserentant, H., A class of iterative methods for solving saddle point problems, Numer. Math. 56, 645–666 (1990).CrossRefGoogle Scholar
[18]Dyn, N. and Ferguson, W.E., The numerical solution of equality constrained quadratic programming problems, Math. Comput. 41, 165–170 (1983).Google Scholar
[19]Elman, H.C., Preconditioners for saddle point problems arising in computational fluid dynamics, Appl. Numer. Math. 43, 75–89 (2002).Google Scholar
[20]Elman, H.C. and Golub, G.H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31, 1645–1661 (1994).Google Scholar
[21]Elman, H.C., Ramage, A. and Silvester, D.J., Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow, ACM Trans. Math. Software 33, 1–18 (2007).CrossRefGoogle Scholar
[22]Elman, H.C., Silvester, D.J. and Wathen, A.J., Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math. 90, 665–688 (2002).Google Scholar
[23]Fortin, M. and Glowinski, R., Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems, Stud. Math. Appl. 15, North-Holland, Amsterdam (1983).Google Scholar
[24]Feng, X.-L. and Shao, L., On the generalized SOR-like methods for saddle point problems, J. Appl. Math. Inform. 28, 663–677 (2010).Google Scholar
[25]Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York (1984).Google Scholar
[26]Gould, N.I.M., Hribar, M.E. and Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput. 23, 1376–1395 (2001).CrossRefGoogle Scholar
[27]Guo, P., Li, C.-X. and Wu, S.-L., A modified SOR-like methodfor the augmented systems, J. Comput. Appl. Math. 274, 58–69 (2015).Google Scholar
[28]Gill, P.E., Murray, W., Ponceleón, D.B. and Saunders, M.A., Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13, 292–311 (1992).Google Scholar
[29]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, 530–539 (1998).CrossRefGoogle Scholar
[30]Golub, G.H., Wu, X. and Yuan, J.-Y., SOR-like methods for augmented systems, BIT Numer. Math. 41, 71–85 (2001).Google Scholar
[31]Haws, J.C., Preconditioning KKT Systems Ph.D. thesis, Department of Mathematics, North Carolina State University, Raleigh (2002).Google Scholar
[32]Haber, E., Ascher, U.M. and Oldenburg, D., On optimization techniques for solving nonlinear inverse problems, Inverse Probl. 16, 1263–1280 (2000).Google Scholar
[33]Huang, N. and Ma, C.-F., The BGS-Uzawa and BJ-Uzawa iterative methods for solving the saddle point problem Appl. Math. Comput. 256, 94–108 (2015).Google Scholar
[34]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, 291–311 (2015).Google Scholar
[35]Klawonn, A., Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput. 19, 172–184 (1998).Google Scholar
[36]Krukier, L.A., Chikina, L.G. and Belokon, T.V., Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations, Appl. Numer. Math. 41, 89–105 (2002).Google Scholar
[37]Keller, C., Gould, N.I.M. and Wathen, A.J., Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl. 21, 1300–1317 (2000).CrossRefGoogle Scholar
[38]Krukier, L.A., Krukier, B.L. and Ren, Z.-R., Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems, Numer. Linear Algebra Appl. 21, 152–170 (2014).Google Scholar
[39]Krukier, L.A., Martynova, T.S. and Bai, Z.-Z., Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems, J. Comput. Appl. Math. 232, 3–16 (2009).Google Scholar
[40]Murphy, M.F., Golub, G.H. and Wathen, A.J., A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput. 21, 1969–1972 (2000).Google Scholar
[41]Plemmons, R.J., A parallel block iterative scheme applied to computations in structural analysis, SIAM J. Alg. Disc. Meth. 7, 337–347 (1986).Google Scholar
[42]Perugia, I. and Simoncini, V., Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl. 7, 585–616 (2000).Google Scholar
[43]Rusten, T. and Winther, R., A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13, 887–904 (1992).Google Scholar
[44]Ren, W.-Q. and Zhao, J.-X., Iterative methods with preconditioners for indefinite systems, J. Comput. Math. 17, 89–96 (1999).Google Scholar
[45]Selberherr, S., Analysis and Simulation of Semiconductor Devices, Springer, New York (1984).Google Scholar
[46]Strikwerda, J.C., An iterative method for solving finite difference approximations to the Stokes equations, SIAM J. Numer. Anal. 21, 447–458 (1984).Google Scholar
[47]Sartoris, G.E., A 3D rectangular mixed finite element method to solve the stationary semiconductor equations, SIAM J. Sci. Comput. 19, 387–403 (1998).Google Scholar
[48]Sarin, V. and Sameh, A., An efficient iterative method for the generalized Stokes problem SIAM J. Sci. Comput. 19, 206–226 (1998).CrossRefGoogle Scholar
[49]Varga, R.S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs (1962).Google Scholar
[50]Wang, L. and Bai, Z.-Z., Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44, 363–386 (2004).Google Scholar
[51]Zulehner, W., Analysis ofiterative methods for saddle point problems: a unified approach, Math. Comput. 71, 479–505 (2002).CrossRefGoogle Scholar
[52]Zhang, G.-F. and Lu, Q.-H., On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 219, 51–58 (2008).Google Scholar
[53]Zhang, G.-F., Yang, J.-L. and Wang, S.-S., On generalized parameterised inexact Uzawa method for a block two-by-two linear system, J. Comput. Appl. Math. 255, 193–207 (2014).CrossRefGoogle Scholar