Abstract
For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSS) preconditioner as well as the MGSS iteration method is derived in this paper, which generalize the modified shift-splitting (MSS) preconditioner and the MSS iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 317:535–546, 2017), respectively. The convergent and semi-convergent analyses of the MGSS iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. Meanwhile, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSS preconditioner and the MGSS iteration method and also illustrate that the MGSS iteration method outperforms the generalized shift-splitting (GSS) and the generalized modified shift-splitting (GMSS) iteration methods, and the MGSS preconditioner is superior to the shift-splitting (SS), GSS, modified SS (M-SS), GMSS and MSS preconditioners for the generalized minimal residual (GMRES) method for solving the nonsymmetric saddle point problems.
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Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273–285 (2000)
Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comp. 75, 791–815 (2006)
Bai, Z.-Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)
Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)
Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)
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 (2005)
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)
Bai, Z.-Z., Golub, G.H., 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)
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)
Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1–38 (2005)
Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)
Bai, Z.-Z., Yin, J.-F., Su, Y.-F.: A shift-splitting preconditioner for non-Hermitian positive definite matrices. J Comput. Math. 24, 539–552 (2006)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia (1994)
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)
Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer, New York (1991)
Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 272, 239–250 (2014)
Cao, Y., Li, S., Yao, L.-Q.: A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Lett. 49, 20–27 (2015)
Cao, Y., Miao, S.-X.: On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems. Comput. Math. Appl. 71, 1503–1511 (2016)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for saddle point problems. Appl. Math. Lett. 43, 49–55 (2015)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput. 269, 947–955 (2015)
Chen, F.: On choices of iteration parameter in HSS method. Appl. Math. Comput. 271, 832–837 (2015)
Dai, L.-F., Liang, M.-L., Fan, H.-T.: A new generalized parameterized inexact uzawa method for solving saddle point problems. Appl. Math. Comput. 265, 414–430 (2015)
Dou, Y., Yang, A.-L., Wu, Y.-J.: Modified parameterized inexact Uzawa method for singular saddle-point problems. Numer. Algorithms 72, 325–339 (2016)
Dyn, N., Ferguson, W.E. Jr: The numerical solution of equality constrained quadratic programming problems. Math. Comput. 41, 165–170 (1983)
Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)
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)
Fischer, B., Ramage, R., Silvester, D.J., Wathen, A.J.: Minimum residual methods for augmented systems. BIT Numer. Math. 38, 527–543 (1998)
Gill, P.E., Murray, W., Wright, M.H.: Practical optimization. Academic Press, New York (1981)
Golub, G.H., Wu, X., Yuan, J.-Y.: SOR-like methods for augmented systems. BIT Numer. Math. 41, 71–85 (2001)
Guo, P., Li, C.-X., Wu, S.-L.: A modified SOR-like method for the augmented systems. J. Comput. Appl Math. 274, 58–69 (2015)
Huang, Y.-M.: A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math. 255, 142–149 (2014)
Huang, Z.-G., Wang, L.-G., Xu, Z., Cui, J.-J.: The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Comput. 299, 95–118 (2017)
Huang, Z.-H., Su, H.: A modified shift-splitting method for nonsymmetric saddle point problems. J. Comput. Appl Math. 317, 535–546 (2017)
Li, J., Zhang, N.-M.: A triple-parameter modified SSOR method for solving singular saddle point problems. BIT Numer. Math. 56, 501–521 (2016)
Li, X., Wu, Y.-J., Yang, A.-L., Yuan, J.-Y.: Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems. Appl. Math. Comput. 244, 552–560 (2014)
Liang, Z.-Z., Zhang, G.-F.: PU-STS method for non-Hermitian saddle-point problems. Appl. Math Lett. 46, 1–6 (2015)
Ma, C.-F., Zheng, Q.-Q.: The corrected Uzawa method for solving saddle point problems. Numer. Linear Algebra Appl. 22, 717–730 (2015)
Meng, G.-Y.: A practical asymptotical optimal SOR method. Appl. Math. Comput. 242, 707–715 (2014)
Njeru, P.N., Guo, X.-P.: Accelerated SOR-like method for augmented linear systems. BIT Numer. Math. 56, 557–571 (2016)
Wu, X., Silva, B.P.B., Yuan, J.-Y.: Conjugate gradient method for rank deficient saddle point problems. Numer. Algorithms 35, 139–154 (2004)
Yang, A.-L., Li, X., Wu, Y.-J.: On semi-convergence of the uzawa-HSS method for singular saddle-point problems. Appl. Math. Comput. 252, 88–98 (2015)
Zhang, G.-F., Wang, S.-S.: A generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 219, 4225–4231 (2013)
Zhang, N.-M., Lu, T.-T., Wei, Y.-M.: Semi-convergence analysis of Uzawa methods for singular saddle point problems. J. Comput. Appl. Math. 255, 334–345 (2014)
Zheng, B., Bai, Z.-Z., Yang, X.: On semi-convergence of parameterized Uzawa methods for singular saddle point problems. Linear Algebra Appl. 431, 808–817 (2009)
Zheng, Q.-Q., Lu, L.-Z.: On semi-convergence of ULT iteration method for the singular saddle point problems. Comput. Math. Appl. 72, 1549–1555 (2016)
Zhou, S.-W., Yang, A.-L., Dou, Y., Wu, Y.-J.: The modified shift-splitting preconditioners for nonsymmetric saddle-point problems. Appl. Math. Lett. 59, 109–114 (2016)
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We would like to express our sincere thanks to the anonymous reviewers for their valuable suggestions and construct comments which greatly improved the presentation of this paper.
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This research was supported by the National Natural Science Foundation of China (No. 11171273) and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX201628).
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Huang, ZG., Wang, LG., Xu, Z. et al. A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems. Numer Algor 78, 297–331 (2018). https://doi.org/10.1007/s11075-017-0377-y
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DOI: https://doi.org/10.1007/s11075-017-0377-y
Keywords
- Nonsymmetric saddle point problem
- Modified generalized shift-splitting
- Convergence
- Semi-convergence
- Spectral properties