Keywords: iterative methods, preconditioning, spectral low rank updates, Jacobi-Davidson, computational electromagnetics.

In this work we consider the solution of the linear systems arising from electromagentism applications by preconditioned Krylov subspace methods [6]. Simulation of electromagnetic wave propagation phenomena requires the numerical solution of Maxwell's equations which it is often done by means of integral equation methods. The discretization of the integral equations with the boundary element method results in dense linear systems with complex entries that are challenging to solve.

Sparse approximate inverse preconditioners based on Frobenius norm minimization [4] are quite effective on solving these linear systems. In [2] a number of preconditioners are compared and it is shown that the SPAI preconditioner performs the best. This class of preconditioners are able to cluster the spectrum of the preconditioned matrix around one but still leaves a small subset of them close to the origin which makes the fast convergence of the Krylov method difficult. Removing some of these eigenvalues can be done via low rank updates. In [1,2] the authors propose the explicit computation of the invariant subspace associated to the smallest eigenvalues solving the preconditioned system in this low dimensional space. The numerical results show that this technique is quite effective on reducing the number of iterations needed to converge and the extra computation time required can be reduced provided that the same linear system has to be solved with many different right-hand sides (which it is the case for some electromagnetism applications, for instance the computation of the cross radar section).

In [1,2] the IRA method (implicit restarted Arnoldi method) implemented in the ARPACK package [5] is used to compute the smallest eigenvalues and its corresponding eigenvectors. In this work we experiment with different variants of the Jacobi-Davidson [3,7] method. The results obtained show that the number of linear systems needed to recuce the extra eigencomputation can be significantly improved.

1

B. Carpentieri, "A class of spectral two-level preconditioners", SIAM J. Sci. Comput., 25(2):749-765, 2003. doi:10.1137/S1064827502408591

2

B. Carpentieri, "Sparse preconditioners for dense linear systems, from electromagnetics applications", PhD thesis, l'Institut National Polytechnique de Toulouse, CERFACS, 2002.

3

D.R. Fokkema, G.L. Sleijpen, H.A. Van der Vorst, "Jacobi-davidson style qr and qz algorithms for the reduction of matrix pencils", SIAM J. Sci. Comput., 20(1):94-125, 1998. doi:10.1137/S1064827596300073

4

M. Grote, T. Huckle, "Parallel preconditioning with sparse approximate inverses", SIAM Journal on Scientific Computing, 18(3):838-853, 1997. doi:10.1137/S1064827594276552

5

D.C. Sorensen, R.B. Lehoucq, C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods", SIAM, Philadelphia, 1998.

6

Y. Saad, "Iterative Methods for Sparse Linear Systems", PWS Publishing Company, Boston, 1996.

7

S.L. Sleijpen, H.A. Van der Vorst, "A Jacobi-Davidson iteration method for linear eigenvalue problems", SIAM J. Matrix Anal. Appl., 17:401-425, 1996. doi:10.1137/S0895479894270427

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