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Simultaneous solution of large-scale linear systems and eigenvalue problems with a parallel GMRES method. (English) Zbl 1161.65027

Summary: A method for simultaneous solution of large and sparse linearized equation sets and the corresponding eigenvalue problems is presented. Such problems arise from the discretization and the solution of nonlinear problems with the finite element method and Newton iteration. The method is based on a parallel version of the preconditioned GMRES\((m)\) by deflation. The parallel code exploits the architecture of the computational clusters using the MPI (Message Passing Interface). The convergence rate, the parallel speedup and the memory requirements of the proposed method are reported and evaluated.

MSC:

65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65Y05 Parallel numerical computation

Software:

FE-BUI
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brown, P. N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comp., 11, 3, 450-481 (1990) · Zbl 0708.65049
[2] Burrage, K.; Erhel, J., On the performance of various adaptive preconditioned GMRES strategies, Numer. Linear Algebra Appl., 5, 101-121 (1998) · Zbl 0937.65036
[3] Chan, T. F.C.; Keller, H. B., Arc-length continuation and multi-grid techniques for nonlinear elliptic eigenvalue problems, SIAM J. Sci. Stat. Comp., 3, 2, 173-194 (1982) · Zbl 0497.65028
[4] Chapman, A.; Saad, Y., Deflated and augmented Krylov subspace techniques, Numer. Linear Algebra Appl., 4, 1, 43-66 (1997) · Zbl 0889.65028
[5] Dennis, J. E.; Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1996), SIAM: SIAM Philadelphia, PA · Zbl 0847.65038
[6] Dossou, K.; Pierre, R., A Newton-GMRES approach for the analysis of the postbuckling behavior of the solutions of the Von Karman equations, SIAM J. Sci. Comput., 24, 6, 1994-2012 (2003) · Zbl 1032.74051
[7] Duff, I. S.; Van der Vorst, H. A., Developments and trends in the parallel solution of linear systems, Parallel Comput., 25, 1931-1970 (1999)
[8] Ehrel, J.; Burrage, K.; Pohl, B., Restarted GMRES preconditioned by deflation, J. Comput. Appl. Math., 69, 303-318 (1996) · Zbl 0854.65025
[9] Ferng, W. R.; Kelley, C. T., Mesh independence of matrix-free methods for path following, SIAM J. Sci. Comput., 21, 5, 1835-1850 (2000) · Zbl 0957.65053
[10] Garcia-Archilla, B.; Sanchez, J.; Simo, C., Krylov methods and determinants for detecting bifurcations in one parameter dependent partial differential equations, BIT, 46, 4, 731-757 (2006) · Zbl 1111.65112
[11] Glowinski, R.; Keller, H. B.; Reinhart, L., Continuation-Conjugate Gradient methods for the least squares solution of nonlinear boundary value problems, SIAM J. Sci. Stat. Comput., 6, 793-832 (1985) · Zbl 0589.65075
[12] Golub, G. H.; Van Loan, C. F., Matrix Computations (1983), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore · Zbl 0559.65011
[13] Graves-Morris, P. R., BiCGStab, VPAStab and an adaptation to mildly nonlinear systems, J. Comput. Appl. Math., 201, 284-299 (2007) · Zbl 1109.41009
[14] Joubert, W., On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems, Numer. Linear Algebra Appl., 1, 5, 427-447 (1994) · Zbl 0838.65029
[15] Keller, H. B., Numerical solution of bifurcation and nonlinear eigenvalue problems, (Rabinowitz, P. H., Applications of Bifurcation Theory (1977), Academic Press: Academic Press New York), 359-384 · Zbl 0581.65043
[16] Moret, I., A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34, 2, 513-516 (1997) · Zbl 0873.65054
[17] Morgan, R. B., A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl., 16, 4, 1154-1171 (1995) · Zbl 0836.65050
[18] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7, 856-869 (1986) · Zbl 0599.65018
[19] Saad, Y., Analysis of augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., 18, 2, 435-449 (1997) · Zbl 0871.65026
[20] Shroff, G. M.; Keller, H. B., Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal., 30, 4, 1099-1120 (1993) · Zbl 0789.65037
[21] Simoncini, V., On the convergence of the restarted Krylov subspace methods, SIAM J. Matrix Anal. Appl., 22, 2, 430-452 (2000) · Zbl 0969.65023
[22] A.N. Spyropoulos, J.A Palyvos, A.G. Boudouvis, Finite element computations on cluster of PC’s and workstations, in: Proc. EURO-PDP 00, IEEE Computer Society, Los Alamitos, CA, USA, 1999, pp. 56-61; A.N. Spyropoulos, J.A Palyvos, A.G. Boudouvis, Finite element computations on cluster of PC’s and workstations, in: Proc. EURO-PDP 00, IEEE Computer Society, Los Alamitos, CA, USA, 1999, pp. 56-61
[23] Spyropoulos, A. N.; Palyvos, J. A.; Boudouvis, A. G., Bifurcation detection with the (un)preconditioned GMRES \((m)\), Comput. Methods Appl. Mech. Engrg., 193, 4707-4716 (2004) · Zbl 1112.76395
[24] A.N. Spyropoulos, A.G. Papathanasiou, J.A. Palyvos, A.G. Boudouvis, FE-BUI — Finite Element Beowulf User Interface: A homemade package for automated parallel finite element computations, in: Proceedings of the 5th GRACM International Congress on Computational Mechanics, Limassol, Cyprus, 2005, pp. 721-727; A.N. Spyropoulos, A.G. Papathanasiou, J.A. Palyvos, A.G. Boudouvis, FE-BUI — Finite Element Beowulf User Interface: A homemade package for automated parallel finite element computations, in: Proceedings of the 5th GRACM International Congress on Computational Mechanics, Limassol, Cyprus, 2005, pp. 721-727
[25] G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Englewood Cliffs, NJ, 1973; G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Englewood Cliffs, NJ, 1973 · Zbl 0356.65096
[26] Van der Vorst, H.; Vuik, C., The superlinear convergence behavior of GMRES, J. Comput. Appl. Math., 48, 3, 327-341 (1993) · Zbl 0797.65026
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