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A Block Diagonal Preconditioner for Generalised Saddle Point Problems

Part of: Basic linear algebra Nonlinear algebraic or transcendental equations Numerical linear algebra

Published online by Cambridge University Press:  20 July 2016

Zhong Zheng  and
Guo Feng Zhang
Zhong Zheng*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China
Guo Feng Zhang*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China
*
Corresponding author. Email addresses:[email protected] (Z. Zheng), [email protected] (G. F. Zhang)
Corresponding author. Email addresses:[email protected] (Z. Zheng), [email protected] (G. F. Zhang)
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Abstract

A lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.

Keywords

Generalised saddle point problemKrylov subspace methodsalternating direction iterationpreconditioningconvergence

MSC classification

Secondary: 15A06: Linear equations 65F10: Iterative methods for linear systems 65H10: Systems of equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 3 , August 2016 , pp. 235 - 252
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Axelsson, O. and Neytcheva, M., Eigenvalue estimates for preconditioned saddle point matrices, Numer. Linear Algebra Appl. 13, 339–360 (2006).CrossRefGoogle Scholar
[2]Bai, Z.Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75, 791–815 (2006).CrossRefGoogle Scholar
[3]Bai, Z.Z., Optimal parameters in the HSS–like methods for saddle–point problems, Numer. Linear Algebra Appl. 16, 447–479 (2009).CrossRefGoogle Scholar
[4]Bai, Z.Z., Motivations and realizations of Krylov subspace methods for large sparse linear systems, J. Comp. Appl. Math. 283, 71–78 (2015).CrossRefGoogle Scholar
[5]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).CrossRefGoogle Scholar
[6]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, 583–603 (2006).CrossRefGoogle Scholar
[7]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).CrossRefGoogle Scholar
[8]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).CrossRefGoogle Scholar
[9]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).CrossRefGoogle Scholar
[10]Bai, Z.Z., Golub, G.H. and Ng, M.K., On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl. 428, 413–440 (2008).CrossRefGoogle 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).CrossRefGoogle Scholar
[12]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).CrossRefGoogle Scholar
[13]Bai, Z.Z. and Wang, Z.Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428, 2900–2932 (2008).CrossRefGoogle Scholar
[14]Benzi, M., Golub, G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numer. 14, 1–137 (2005).CrossRefGoogle Scholar
[15]Benzi, M. and Liu, J., Block preconditioning for saddle point systems with indefinite (1, 1) block, Int. J. Comput. Math. 84, 1117–1129 (2007).CrossRefGoogle Scholar
[16]Benzi, M. and Simoncini, V., On the eigenvalues of a class of saddle point matrices, Numer. Math. 103, 173–196 (2006).CrossRefGoogle Scholar
[17]Bergamaschi, L., On eigenvalue distribution of constraint–preconditioned symmetric saddle point matrices, Numer. Linear Algebra Appl. 19, 754–772 (2012).CrossRefGoogle Scholar
[18]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).CrossRefGoogle Scholar
[19]Cao, Z.H., Augmentation block preconditioners for saddle point-type matrices with singular (1, 1) blocks, Numer. Linear Algebra Appl. 15, 515–533 (2008).CrossRefGoogle Scholar
[20]Cao, Y., Jiang, M.Q. and Zheng, Y.L., A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18, 875–895 (2011).CrossRefGoogle Scholar
[21]Chan, L.C., Ng, M.K. and Tsing, N.K., Spectral Analysis for HSS Preconditioners, Numer. Math. Theor. Meth. Appl. 1, 57–77 (2008).Google Scholar
[22]Chen, F. and Jiang, Y.L., A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206, 765–771 (2008).Google Scholar
[23]Elman, H.C., Silvester, D.J. and Wathen, A.J., Iterative methods for problems in computational fluid dynamics, in Iterative Methods in Scientific Computing, Chan, R. H., Chan, C. T., and Golub, G. H. (eds.), pp. 271–327, Springer, Singapore (1997).Google Scholar
[24]Elman, H.C., Ramage, A. and Silvester, D.J., Algorithm 866: IFISS: A Matlab toolbox for modelling incompressible flow, ACM Trans, Math. Software (TOMS) 33, 14 (2007).CrossRefGoogle Scholar
[25]Elman, H.C., Silvester, D.J. and Wathen, A.J., Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, (2005).CrossRefGoogle Scholar
[26]Harman, H.H. and Jones, W.H., Factor analysis by minimizing residuals, Psychometrika 30, 351–368 (1963).Google Scholar
[27]Li, C.J., Li, B.J. and Evans, D.J., A generalized successive overrelaxation method for least squares problems, BIT Numer. Math. 28, 347–355 (1998).CrossRefGoogle Scholar
[28]Ma, S.L. and Xu, Z.Y., Neccessary and sufficient conditions for a polynomial of degree 3 or 4 with real coefficients to be a Von Neumann polynomial, Numer. Math. J. Chinese Universities. 3, 274–280 (1984).Google Scholar
[29]Notay, Y., A new analysis of block preconditioners for saddle point problems, SIAM J. Matrix Anal. Appl. 35, 143–173 (2014).CrossRefGoogle Scholar
[30]Pan, J.Y., Ng, M.K. and Bai, Z.Z., New preconditioners for saddle point problems, Appl. Math. Comput. 172, 762–771 (2006).Google Scholar
[31]Perugia, I. and Simoncini, V., New Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl. 7, 585–616 (2000).3.0.CO;2-F>CrossRefGoogle Scholar
[32]Saad, Y. and Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).CrossRefGoogle Scholar
[33]Shao, X.H., Li, Z. and Li, C.J., Modified SOR-like method for the augmented system, Int. J. Comput. Math. 84, 1653–1662 (2007).CrossRefGoogle Scholar
[34]Wright, S., Stability of augmented system factorizations in interior-point methods, SIAM J. Matrix Anal. Appl. 18, 191–222 (1997).CrossRefGoogle Scholar
[35]Wang, S.S. and Zhang, G.F., Preconditioned AHSS iteration method for singular saddle point problems, Numer. Algor. 63, 521–535 (2013).CrossRefGoogle Scholar
[36]Young, D.M., Iterative Solution of Large Linear Systems, Academic Press (1971).Google Scholar
[37]Zhu, M.Z., Zhang, G.F., Zheng, Z. and Liang, Z.Z.On HSS-based sequential two-stage method for non-Hermitian saddle point problems, Appl. Math. Comput. 242, 907–916 (2014).Google Scholar