Abstract
We present experiments with two new solvers for large sparse symmetric matrix eigenvalue problems: (1) the implicitly restarted Lanczos algorithm and (2) the Jacobi-Davidson algorithm. The eigenvalue problems originate from in the computation of a few of the lowest frequencies of standing electromagnetic waves in cavities that have been discretized by the finite element method. The experiments have been conducted on up to 12 processors of an HP Exemplar X-Class multiprocessor computer.
Chapter PDF
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
St. Adam, P. Arbenz, and R. Geus, Eigenvalue solvers for electromagnetic fields in cavities, Tech. Report 275, ETH Zurich, Computer Science Department, October 1997, (Available at URL http://www.inf.ethz.ch/publications/tr.html).
P. Arbenz and R. Geus, Eigenvalue solvers for electromagnetic fields in cavities, FORTWIHR International Conference on High Performance Scientific and Engineering Computing (F. Durst et al., ed.), Springer-Verlag, 1998, (Lecture Notes in Computational Science and Engineering).
R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, Ch. Romine, and H. van der Vorst, Templates for the solution of linear systems: Building blocks for iterative methods, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994, (Available from Netlib at URL http://www.net1ib.org/templates/index.html).
A. N. Bespalov, Finite element method for the eigenmode problem of a RF cavity resonator, Soviet Journal of Numerical Analysis and Mathematical Modelling 3 (1988), 163–178.
D. Calvetti, L. Reichel, and D. C. Sorensen, An implicitely restarted Lanczos method for large symmetric eigenvalue problems, Electronic Transmissions on Numerical Analysis 2 (1994), 1–21.
D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils, Preprint 941, revised version, Utrecht University, Department of Mathematics, Utrecht, The Netherlands, January 1997.
R. Grimes, J. G. Lewis, and H. Simon, A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems, SIAM J. Matrix Anal. Appl. 15 (1994), 228–272.
J. Jin, The finite element method in electromagnetics, Wiley, New York, 1993.
F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Computer Methods in Applied Mechanics and Engineering 64 (1987), 509–521.
R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK users’ guide: Solution of large scale eigenvalue problems by implicitely restarted Arnoldi methods, Department of Mathematical Sciences, Rice University, Houston TX, October 1997, (Available at URL http://www.caam.rice.edu/software/ARPACK/index.html).
R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen Medien, Mathematische Zeitschrift 106 (1968), 213–224.
J. C. Nédélec, Mixed finite elements in ℝ3, Numerische Mathematik 35 (1980), 315–341.
C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), 617–629.
B. N. Parlett, The symmetric eigenvalue problem, Prentice Hall, Englewood Cliffs, NJ, 1980.
G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (1996), 595–633.
D. Sorensen, Implicite application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), 357–385.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arbenz, P., Geus, R. (1998). Parallel solvers for large eigenvalue problems originating from Maxwell’s equations. In: Pritchard, D., Reeve, J. (eds) Euro-Par’98 Parallel Processing. Euro-Par 1998. Lecture Notes in Computer Science, vol 1470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0057929
Download citation
DOI: https://doi.org/10.1007/BFb0057929
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64952-6
Online ISBN: 978-3-540-49920-6
eBook Packages: Springer Book Archive