×

zbMATH — the first resource for mathematics

Preconditioners for nonsymmetric indefinite linear systems. (English) Zbl 1425.65043
Summary: In this paper, we develop algorithms for solving nonsymmetric indefinite linear systems by considering the augmented linear systems resulting from a weighted linear least squares problem. Even though the augmented system is more ill-conditioned than the original linear system, one can construct preconditioned GMRES methods for solving these augmented systems capable of obtaining reasonable approximation of the solution in fewer iterations than the classical ILU preconditioned GMRES method for solving the original linear system. More specifically, we present two different preconditioners for these augmented systems, examine the spectral properties of these preconditioned augmented systems, and report numerical results to illustrate the effectiveness of these preconditioners.
MSC:
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Stynes, M.; Stynes, D., (Convection Diffusion Problems: An Introduction to their Analysis and Numerical Solution. Convection Diffusion Problems: An Introduction to their Analysis and Numerical Solution, Graduate Studies in Mathematics, vol. 196 (2018), American Mathematical Society) · Zbl 1422.65005
[2] Douglas, J.; Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 5, 871-885 (1982) · Zbl 0492.65051
[3] Roos, H.-G.; Stynes, M.; Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, Vol. 24 (2008), Springer Science & Business Media
[4] Varga, R. S., Matrix Iterative Analysis, Vol. 27 (2009), Springer Science & Business Media
[5] Bai, Z.-Z., Quasi-hss iteration methods for non-hermitian positive definite linear systems of strong skew-Hermitian parts, Numer. Linear Algebra Appl., 25, 4, Article e2116 pp. (2018) · Zbl 06945795
[6] Axelsson, O.; Bai, Z.-Z.; Qiu, S.-X., A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms, 35, 2-4, 351-372 (2004) · Zbl 1054.65028
[7] Benzi, M., Preconditioning techniques for large linear systems: a survey, J. Comput. Phys., 182, 2, 418-477 (2002) · Zbl 1015.65018
[8] Benzi, M.; Haws, J. C.; Tuma, M., Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput., 22, 4, 1333-1353 (2000) · Zbl 0985.65036
[9] Saad, Y., Preconditioning techniques for nonsymmetric and indefinite linear systems, J. Comput. Appl. Math., 24, 1-2, 89-105 (1988) · Zbl 0662.65028
[10] Chow, E.; Saad, Y., Experimental study of ilu preconditioners for indefinite matrices, J. Comput. Appl. Math., 86, 2, 387-414 (1997) · Zbl 0891.65028
[11] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034
[12] Benzi, M.; Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26, 1, 20-41 (2004) · Zbl 1082.65034
[13] Fischer, B.; Ramage, A.; Silvester, D. J.; Wathen, A. J., Minimum residual methods for augmented systems, BIT Numer. Math., 38, 3, 527-543 (1998) · Zbl 0914.65026
[14] Saad, Y.; Schultz, M. H., Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018
[15] Davis, T. A.; Hu, Y., The university of Florida sparse matrix collection, ACM Trans. Math. Softw. (TOMS), 38, 1, 1 (2011) · Zbl 1365.65123
[16] 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, 3, 603-626 (2003) · Zbl 1036.65032
[17] Zhang, Z.; Sameh, A. H., A parallel sparse linear system solver based on hermitian/skew-Hermitian splitting, Comput. Math. Appl., 72, 8, 2000-2007 (2016) · Zbl 1359.65068
[18] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic Press · Zbl 0231.65034
[19] Liu, X.; Wen, Z.; Zhang, Y., Limited memory block Krylov subspace optimization for computing dominant singular value decompositions, SIAM J. Sci. Comput., 35, 3, A1641-A1668 (2013) · Zbl 1278.65045
[20] Axelsson, O., Preconditioning of indefinite problems by regularization, SIAM J. Numer. Anal., 16, 1, 58-69 (1979) · Zbl 0416.65071
[21] Axelsson, O.; Neytcheva, M., Preconditioning methods for linear systems arising in constrained optimization problems, Numer. Linear Algebra Appl., 10, 1-2, 3-31 (2003) · Zbl 1071.65527
[22] Hundsdorfer, W.; Verwer, J. G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Vol. 33 (2013), Springer Science & Business Media
[23] Maffeo, C.; Bhattacharya, S.; Yoo, J.; Wells, D.; Aksimentiev, A., Modeling and simulation of ion channels, Chem. Rev., 112, 12, 6250-6284 (2012)
[24] Brenner, S.; Scott, L., The Mathematical Theory of Finite Element Methods (2008), Springer-Verlag: Springer-Verlag New York · Zbl 1135.65042
[25] Logg, A.; Mardal, K.-A.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Vol. 84 (2012), Springer Science & Business Media · Zbl 1247.65105
[26] Simoncini, V.; Benzi, M., Spectral properties of the hermitian and skew-hermitian splitting preconditioner for saddle point problems, SIAM J. Matrix Anal. Appl., 26, 2, 377-389 (2004) · Zbl 1083.65047
[27] Benzi, M.; Gander, M. J.; Golub, G. H., Optimization of the hermitian and skew-hermitian splitting iteration for saddle-point problems, BIT Numer. Math., 43, 5, 881-900 (2003) · Zbl 1052.65015
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.