A GMRES(m) method with two stage deflated preconditioners

Jungo Shiroishi, Takashi Nodera


The GMRES(m) method is often used to compute Krylov subspace solutions of large sparse linear systems of equations. Morgan developed a new procedure that deflates the smallest eigenvalues and improves the eigenvalue distribution. Several preconditioning techniques have been explored in numerous research papers. In particular, the deflated GMRES proposed by Erhel and others replaces the smallest eigenvalues of the original coefficient matrix of the linear system with the largest modulus of the eigenvalues. We explore a new deflated GMRES which uses a two stage deflation technique. Further, the results of the numerical experiments for test matrices are tabulated to illustrate that our approach is effective in solving a wide range of problems.

  • J. Baglama, D. Calvetti, G. H. Golub and L. Reichel, Adaptively pre-conditioned GMRES algorithms, SIAM J. Sci. Comput., 20 (1998), pp. 243--269. doi:10.1137/S1064827596305258
  • K. Burrage and J. Erhel, On the performance of various adaptive preconditioned GMRES strategies, Numer. Linear Algebra Appl., 5(1998), pp. 101--121. doi:10.1002/(SICI)1099-1506(199803/04)5:2<101::AID-NLA127>3.0.CO;2-1
  • J. Erhel, K. Burrage and B. Pohl, Restarted GMRES preconditioned by deflation, J. Comput. Appl. Math., 69 (1996), pp. 303--318. doi:10.1016/0377-0427(95)00047-X
  • W. Joubert. Lanczos methods for the solution of nonsymmetric systems of linear equations, SIAM J. Matrix. Anal. Appl., 13 (1992), pp. 926--943. doi:10.1137/0613056
  • R. B. Morgan, GMRES with deflated restarting, SIAM J. Sci. Comput., 24 (2002), pp. 20--37. doi:10.1137/S1064827599364659
  • R. B. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1154--1171. doi:10.1137/S0895479893253975
  • R. B. Morgan, Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1112--1135. doi:10.1137/S0895479897321362
  • R. B. Morgan, Harmonic projection methods for large non-symmetric eigenvalue problems, Numer. Linear Algebra Appl., 5(1998), pp. 33--55. doi:10.1002/(SICI)1099-1506(199801/02)5:1<33::AID-NLA125>3.0.CO;2-1
  • S. Rollin and W. Fichtner, Improving the accuracy of GMRES with deflated restarting, SIAM J. Sci. Comput., 30(2007), pp. 232--245. doi:10.1137/060656127
  • Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869. doi:10.1137/0907058
  • H. A. van der Vorst and C. Vuik, The superlinear convergence behavior of GMRES, J. Comput. Appl. Math., 48 (1993), pp. 327--341. doi:10.1016/0377-0427(93)90028-A


GMRES deflation preconditioner

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DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3984

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