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An Efficient Variant of the GMRES(m) Method Based on the Error Equations

Published online by Cambridge University Press:  28 May 2015

Akira Imakura ,
Tomohiro Sogabe  and
Shao-Liang Zhang
Akira Imakura*
Affiliation:
Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
Tomohiro Sogabe
Affiliation:
Aichi Prefectural University, 1522-3 Ibaragabasama, Kumabari, Nagakute-cho, Aichi-gun, Aichi, 480-1198, Japan
Shao-Liang Zhang
Affiliation:
Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
*
Corresponding author. Email: a-imakura@na.cse.nagoya-u.ac.jp
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Abstract

The GMRES(m) method proposed by Saad and Schultz is one of the most successful Krylov subspace methods for solving nonsymmetric linear systems. In this paper, we investigate how to update the initial guess to make it converge faster, and in particular propose an efficient variant of the method that exploits an unfixed update. The mathematical background of the unfixed update variant is based on the error equations, and its potential for efficient convergence is explored in some numerical experiments.

Keywords

65F10Nonsymmetric linear systemsGMRES(m) methodrestarterror equations

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Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 1 , February 2012 , pp. 19 - 32
Copyright
Copyright © Global-Science Press 2012

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References

[1]Baglama, J., Calvetti, D., Golub, G. H., and Reichel, L., Adaptively preconditioned GMRES algorithms, SIAM J. Sci. Comput., 20 (1999), pp. 17–29.Google Scholar
[2]Baglama, J. and Reichel, L., Augmented GMRES-type methods, Numer. Linear Alg. Appl., 14 (2007), pp. 337–350.Google Scholar
[3]Baker, A. H., Jessup, E. R., and Kolev, T.V., A simple strategy for varying the parameter in GMRES(m), J. Comput. Appl. Math., 230 (2009), pp. 751–761.CrossRefGoogle Scholar
[4]Baker, A. H., Jessup, E. R., and Manteuffel, T., A technique for accelerating the convergence of restarted GMRES, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 962–984.Google Scholar
[5]Burrage, K. and Erhel, J., On the performance of various adaptive preconditioned GMRES strategies, Numer. Linear Alg. Appl., 5 (1998), pp. 101–121.Google Scholar
[6]Davis, T. A., The University of Florida Sparse Matrix Collection, Available from http://www.cise.ufl.edu/research/sparse/matrices/.Google Scholar
[7]Erhel, J., Burrage, K., and Pohl, B., Restarted GMRES preconditioned by deflation, J. Comput. Appl. Math., 69 (1996), pp. 303–318.Google Scholar
[8]Golub, G. H. and van der Vorst, H. A., Closer to the solution: Iterative linear solvers, in The State of the Art in Numerical Analysis, Duff, I. S. and Watros, G. A. eds. Clarendon Press, Oxford, 1997, pp. 63–92.Google Scholar
[9]Gutknecht, M. H., Variants of BiCGStab for matrices with complex spectrum, SIAM J. Sci. Stat. Comput., 14 (1993), pp. 1020–1033.CrossRefGoogle Scholar
[10]Hestenes, M. R. and Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49 (1952), pp. 409–436.Google Scholar
[11] Matrix Market, Available from http://math.nist.gov/MatrixMarket/.Google Scholar
[12]Morgan, R. B., A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1154–1171.Google Scholar
[13]Morgan, R. B., Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1112–1135.CrossRefGoogle Scholar
[14]Morgan, R. B., GMRES with deflated restarting, SIAM J. Sci. Comput., 24 (2002), pp. 20–37.CrossRefGoogle Scholar
[15]Moriya, K. and Nodera, T., The DEFLATED-GMRES(m, k) method with switching the restart frequency dynamically, Numer. Linear Alg. Appl., 7 (2000), pp. 569–584.Google Scholar
[16]Paige, C. C. and Saunders, M. A., Solution of Sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617–729.CrossRefGoogle Scholar
[17]Saad, Y. and Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869.Google Scholar
[18]Saad, Y. and van der Vorst, H. A., Iterative solution of linear systems in the 20th century, J. Comput. Appl. Math., 123 (2000), pp. 1–33.Google Scholar
[19]Simoncini, V. and Szyld, D. B., Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Alg. Appl., 14 (2007), pp. 1–59.Google Scholar
[20]Sleijpen, G. L. G. and Fokkema, D. R., BiCGStab(l) for linear equations involving unsymmetric matrices with complex spectrum, Electronic Trans. Numer. Anal., 1 (1993), pp. 11–32.Google Scholar
[21]Sleijpen, G. L. G., Sonneveld, P., and van Gijzen, M. B., Bi-CGSTAB as an induced dimension reduction method, Appl. Numer. Math. 60 (2010), pp. 1100–1114.Google Scholar
[22]Sonneveld, P. and Van Gijzen, M. B., IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062.Google Scholar
[23]Sosonkina, M., Watson, L. T., Kapania, R. K., and Walker, H. F., A new adaptive GMRES algorithm for achieving high accuracy, Numer. Linear Alg. Appl., 5 (1998), pp. 275–297.3.0.CO;2-B>CrossRefGoogle Scholar
[24]Tanio, M. and Sugihara, M., GBi-CGSTAB(s, L): IDR(s) with higher-order stabilization polynomials, J. Comput. Appl. Math., 235 (2010), pp. 765–784.CrossRefGoogle Scholar
[25]Van Der Vorst, H. A., Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 631–644.CrossRefGoogle Scholar
[26]Wilkinson, J. H., Rounding Errors in Algebraic Processes, Prentice Hall, Englewood Cliffs, NJ, 1963.Google Scholar
[27]Zhang, L. and Nodera, T., A new adaptive restart for GMRES(m) method, ANZIAM J., 46 (2005), pp. 409–426.Google Scholar
[28]Zhang, S.-L., GPBi-CG: Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput., 18 (1997), pp. 537–551.CrossRefGoogle Scholar