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Numerical Algorithms
Published in: Numerical Algorithms 4/2020

18-11-2019 | Original Paper

Preconditioned Krylov subspace and GMRHSS iteration methods for solving the nonsymmetric saddle point problems

Authors: A. Badahmane, A. H. Bentbib, H. Sadok

Published in: Numerical Algorithms | Issue 4/2020

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Abstract

In the present paper, we propose a separate approach as a new strategy to solve the saddle point problem arising from the stochastic Galerkin finite element discretization of Stokes problems. The preconditioner is obtained by replacing the (1,1) and (1,2) blocks in the RHSS preconditioner by others well chosen and the parameter α in (2,2) −block of the RHSS preconditioner by another parameter β. The proposed preconditioner can be used as a preconditioner corresponding to the stationary itearative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergence properties of the GMRHSS iteration method are derived. Meanwhile, we analyzed the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Finally, numerical results show the effectiveness of the proposed preconditioner as compared with other preconditioners.
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Literature
1.
Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951)MathSciNetCrossRef
2.
Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2004)MathSciNetCrossRef
3.
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)MathSciNetCrossRef
4.
Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)MathSciNetCrossRef
5.
6.
Benzi, M., Wathen, J.A.: Some preconditioning techniques for saddle point problems. Model Order Reduction: Theory. Res. Aspects and Appl. 13, 195–211 (2004)MATH
7.
8.
Benzi, M., Simoncini, V.: On the eigenvalues of a class of saddle point matrices. Numer. Math. 103, 173–196 (2006)MathSciNetCrossRef
9.
Bellalij, M., Jbilou, K., Sadok, H.: New convergence results on the global GMRES method for diagonalizable matrices. J. Comput. Appl. Math. 219, 350–358 (2008)MathSciNetCrossRef
10.
Benner, P., Saak, J., Stoll, M., Weichelt, H.K.: Efficient solution of large-scale saddle point systems arising in Riccati-based boundary feedback stabilization of incompressible stokes flow. SIAM J. Sci. Comput. 35, S150–S170 (2013)MathSciNetCrossRef
11.
Bissuel, A., Allaire, G., Daumas, L., Chalot, F., Mallet, M.: Linear systems with multiple right-hand sides with GMRES, an application to aircraft design. ECCOMAS Congress (2016)
12.
Bouyouli, R., Jbilou, K., Sadaka, R., Sadok, H.: Convergence proprieties of some block Krylov subspace methods for multiple linear systems. J. Comput. Appl. Math. 196, 498–511 (2006)MathSciNetCrossRef
13.
Cao, Z.-H.: Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks. Linear Algebra Appl. 15, 515–533 (2008)MathSciNetCrossRef
14.
Dollar, H.S., Wathen, A.J.: Approximate factorization constraint preconditioners for saddle point matrices. SIAM J. Sci. Comput. 27, 1555–1572 (2006)MathSciNetCrossRef
15.
Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)MATH
16.
Elman, H.C., Ramage, A., Silvester, D.J.: Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2–14 (2007)CrossRef
17.
Elbouyahyaoui, L., Messaoudi, A., Sadok, H.: Algebraic properties of the block GMRES and block Arnoldi methods. Elect Trans Numer Analysis. 33, 207–220 (2009)MathSciNetMATH
18.
Ernst, O.G., Powell, C.E., Silvester, D.J., Ullmann, E.: Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31, 1424–1447 (2009)MathSciNetCrossRef
19.
Gould, N., Orban, D., Rees, T.: Projected Krylov methods for saddle-point system. SIAM J. Matrix Anal. Appl. 35, 1329–1343 (2014)MathSciNetCrossRef
20.
Huang, Z.-G., Wang, G.-L., LG, Xu, Z. , Cui, J.-J.: A generalized variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems. Numer. Algor. 75, 1161–1191 (2017)MathSciNetCrossRef
21.
Jbilou, K., Messaoudi, A., Sadok, H.: Global FOM and GMRES algorithms for matrix equation. Appl. Numer. Math. 31, 49–43 (1999)MathSciNetCrossRef
22.
Jiang, M.-Q., Cao, Y., Yao, L.-Q.: On parametrized block triangular preconditioners for generalized saddle point problems. Appl. Math. Comput. 216, 1777–1789 (2010)MathSciNetMATH
23.
Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21, 1969–1972 (2000)MathSciNetCrossRef
24.
Pestana, J., Wathen, A.J.: Combination preconditioning of saddle point systems for positive definiteness. Linear Algebra Appl. 20, 785–808 (2012)MathSciNetCrossRef
25.
Pestana, J., Wathen, A.J.: On the choice of preconditioner for minimum residual methods for non-Hermitian matrices. J. Comput. Appl. Math. 249, 57–68 (2013)MathSciNetCrossRef
26.
Pestana, J., Wathen, A.J.: Natural preconditioning and iterative methods for saddle point systems. SIAM Rev. 57, 71–91 (2015)MathSciNetCrossRef
27.
Pestana, J., Wathen, A.J.: A preconditioned MINRES method for nonsymmetric Toeplitz matrices. SIAM J. Matrix Anal. Appl. 36, 273–288 (2015)MathSciNetCrossRef
28.
Powell, C.E., Silvester, D.J: Preconditioning steady-state Navier-Stokes equations with random data. SIAM J. Sci. Comput. 34, A2482–A2506 (2012)MathSciNetCrossRef
29.
Rozloznik, M., Simoncini, V.: Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J. Matrix Anal. Appl. 24, 368–391 (2002)MathSciNetCrossRef
30.
Saad, Y., Schultz, M.: GMRES: A generalised minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)CrossRef
31.
Sadok, H.: Analysis of the convergence of the minimal and the orthogonal residual methods. Numer. Algor. 40, 101–115 (2005)MathSciNetCrossRef
32.
Salkuyeh, D.K., Masoudi, M.: A new relaxed HSS preconditioner for saddle point problems. Numer. Algor. 74, 781–795 (2017)MathSciNetCrossRef
33.
Schöberl, J., Zulehner, W.: Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29, 752–773 (2007)MathSciNetCrossRef
34.
Stoll, M., Wathen, A.: Combination preconditioning and the Bramble-Pasciak preconditioner. SIAM J. Matrix Anal. Appl. 30, 582–608 (2008)MathSciNetCrossRef
35.
Zhang, J.-L., Gu, C.-Q.: A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems. BIT Numer. Math. 56, 587–604 (2016)MathSciNetCrossRef
36.
Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71, 479–50 (2001)MathSciNetCrossRef
Metadata
Title
Preconditioned Krylov subspace and GMRHSS iteration methods for solving the nonsymmetric saddle point problems
Authors
A. Badahmane
A. H. Bentbib
H. Sadok
Publication date
18-11-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00833-4

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