×

On generalized parameterized inexact Uzawa method for a block two-by-two linear system. (English) Zbl 1291.65115

Summary: Recently, F. Chen and Y.-L. Jiang [Appl. Math. Comput. 206, No. 2, 765–771 (2008; Zbl 1159.65034)] presented a parameterized inexact Uzawa (PIU) algorithm for solving symmetric saddle point problems, where the \((1,2)\)- and the \((2,1)\)-blocks are the transpose of each other. In this paper, we extend the PIU method to the block two-by-two linear system by allowing the \((1,2)\)-block to be not equal to the transpose of the \((2,1)\)-block and the \((2,2)\)-block may not be zero. We prove that the iteration method is convergent under certain conditions. With different choices of the parameter matrices, we obtain several new algorithms for solving the block two-by-two linear system. Numerical experiments confirm our theoretical results and show that our method is feasible and effective.

MSC:

65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65F08 Preconditioners for iterative methods
65F35 Numerical computation of matrix norms, conditioning, scaling

Citations:

Zbl 1159.65034

Software:

IFISS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Elman, H. C., Preconditioners for saddle point problems arising in computational fluid dynamics, Appl. Numer. Math., 43, 75-89 (2002) · Zbl 1168.76348
[2] Elman, H. C.; Ramage, A.; Silvester, D. J., Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow, ACM Trans. Math. Software, 33, 1-18 (2007) · Zbl 1365.65326
[3] Elman, H. C.; Silvester, D. J.; Wathen, A. J., Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90, 665-688 (2002) · Zbl 1143.76531
[4] Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0575.65123
[5] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002
[6] Perugia, I.; Simoncini, V., Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl., 7, 585-616 (2000) · Zbl 1051.65038
[7] Axelesson, O.; Neytcheva, M., Preconditioning methods for linear systems arising in constrained optimization problems, Numer. Linear Algebra Appl., 10, 3-31 (2003) · Zbl 1071.65527
[9] Gill, P. E.; Murray, W.; Wright, M. H., Practical Optimization (1981), Academic Press: Academic Press New York, NY · Zbl 0503.90062
[10] Gould, N. I.M.; Hribar, M. E.; Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23, 1375-1394 (2001)
[11] Keller, C.; Gould, N. I.M.; Wathen, A. J., Constrained preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21, 1300-1317 (2000) · Zbl 0960.65052
[12] Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming (2001), SIAM: SIAM Philadelphia, PA · Zbl 0995.49017
[13] Björck, A., Numerical Methods for Least Squares Problems (1996), SIAM: SIAM Philadelphia, PA · Zbl 0847.65023
[14] Sartoris, G. E., A 3D rectangular mixed finite element method to solve the stationary semiconductor equations, SIAM J. Sci. Stat. Comput., 19, 387-403 (1998) · Zbl 0911.65131
[15] Selberherr, S., Analysis and Simulation of Semiconductor Devices (1984), Springer-Verlag: Springer-Verlag New York
[16] Haber, E.; Ascher, U. M.; Oldenburg, D., On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 16, 1263-1280 (2000) · Zbl 0974.49021
[17] Bai, Z.-Z.; Parlett, B. N.; Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102, 1-38 (2005) · Zbl 1083.65034
[18] Bai, Z.-Z.; Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428, 2900-2932 (2008) · Zbl 1144.65020
[19] Bramble, J. H.; Pasciak, J. E.; Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34, 1072-1092 (1997) · Zbl 0873.65031
[20] Chen, F.; Jiang, Y.-L., A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput., 206, 765-771 (2008) · Zbl 1159.65034
[21] Wang, Z.-Q., Optimization of the parameterized Uzawa preconditioners for saddle point problems, J. Comput. Appl. Math. (2008)
[22] Zhou, Y.-Y.; Zhang, G.-F., A generalization of parameterized inexact Uzawa method for generalized saddle point problems, Appl. Math. Comput., 215, 599-607 (2009) · Zbl 1173.65318
[23] Bai, Z.-Z., Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16, 447-479 (2009) · Zbl 1224.65081
[24] Bai, Z.-Z., Splitting iteration methods for non-Hermitian positive definite systems of linear equations, Hokkaido Math. J., 36, 801-814 (2007) · Zbl 1138.65027
[25] Bai, Z.-Z.; Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting method for saddle-point problems, IMA J. Numer. Anal., 27, 1-23 (2007) · Zbl 1134.65022
[26] Bai, Z.-Z.; Golub, G. H.; 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) · Zbl 1036.65032
[27] Duff, I. S.; Gould, N. I.M.; Reid, J. K.; Scott, J. A.; Turner, K., The factorization of sparse symmetric indefinite matrices, IMA J. Numer. Anal., 11, 181-204 (1991) · Zbl 0739.65018
[28] Duff, I. S.; Reid, J. K., Exploiting zeros on the diagonal in the direct solution of indefinite sparse symmetric linear systems, ACM Trans. Math. Software, 22, 227-257 (1996) · Zbl 0884.65020
[29] Golub, G. H.; Wu, X.; Yuan, J.-Y., SOR-like methods for augmented systems, BIT, 55, 71-85 (2001) · Zbl 0991.65036
[30] Heinrichs, W., Splitting techniques for the unsteady Stokes equations, SIAM J. Numer. Anal., 35, 1646-1662 (1998) · Zbl 0911.76058
[31] Bai, Z.-Z., Construction and analysis of structured preconditioners for block two-by-two matrices, J. Shanghai Univ., 8, 397-405 (2004)
[32] Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comp., 75, 791-815 (2006) · Zbl 1091.65041
[33] Bai, Z.-Z.; Golub, G. H.; Pan, J.-Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98, 1-32 (2004) · Zbl 1056.65025
[34] Bai, Z.-Z.; Li, G.-Q., Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23, 561-580 (2003) · Zbl 1046.65018
[35] Benzi, M.; Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26, 20-41 (2004) · Zbl 1082.65034
[36] Bai, Z.-Z.; Wang, Z.-Q., Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, J. Comput. Appl. Math., 187, 202-226 (2006) · Zbl 1083.65045
[37] Dollar, H. S., Constraint-style preconditioners for regularized saddle point problems, SIAM J. Matrix Anal. Appl., 29, 672-684 (2007) · Zbl 1144.65032
[38] Ipsen, I. C.F., A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23, 1050-1051 (2001) · Zbl 0998.65049
[39] Siefert, C.; de Sturler, E., Preconditioners for generalized saddle-point problems, SIAM J. Numer. Anal., 44, 1275-1296 (2006) · Zbl 1120.65061
[40] Simoncini, V., Block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math., 49, 63-80 (2004) · Zbl 1053.65033
[41] Sturler, E. D.; Liesen, J., Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems, I: theory, SIAM J. Sci. Comput., 26, 5, 1598-1619 (2005) · Zbl 1078.65027
[42] Yin, J.-F.; Bai, Z.-Z., The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems, J. Comput. Math., 26, 240-249 (2008) · Zbl 1174.65014
[43] Zhang, G.-F.; Ren, Z.-R.; Zhou, Y.-Y., On HSS-based constraint preconditioners for generalized saddle-point problems, Numer. Algorithms, 57, 273-287 (2011) · Zbl 1220.65038
[44] Bai, Z.-Z.; Ng, M. K., On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput., 26, 1710-1724 (2005) · Zbl 1077.65043
[45] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 1-137 (2005) · Zbl 1115.65034
[46] Bai, Z.-Z., Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math., 237, 295-306 (2013) · Zbl 1252.15022
[47] Young, D. M., Iterative Solution for Large Linear Systems (1971), Academic press: Academic press New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.