Multiple Explicitly Restarted Arnoldi Method for Solving Large Eigenproblems

Abstract

In this paper we propose a new approach for calculating some eigenpairs of large sparse non-Hermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is particularly well suited for environments that combine different parallel programming paradigms. This technique is based on a multiple use of the Explicitly Restarted Arnoldi method (ERAM) and improves its convergence.This technique is implemented and tested on a distributed environment consisting of two interconnected parallel machines. The MERAM technique is compared with ERAM, and one can notice that the convergence is improved. In some cases, more than a twofold improvement can be seen in MERAM results.We also implemented MERAM on a cluster of workstations. According to our experiments, MERAM converges better than the Explicitly Restarted Block Arnoldi method and, for some matrices, more quickly than the PARPACK package, which implements the Implicitly Restarted Arnoldi method.

A key to choose subspace size in implicitly restarted Arnoldi method

“…The matrix, initial vector and tolerance value are bf w782a, x = t n / t n 2 and 10 −10 respectively. MIRAMns (8,18,20) (a) MIRAMns (8,17,20) with bf w782a matrix 6. Conclusion.…”

“…This approach, multiple IRAM, has the advantage to update (implicitly) the initial vector of an IRAM by taking into account the eigen-information obtained by several different (non nested) subspaces. The increase cost engendered by these different subspaces could be compensated by the implementation of the method in a large-scale distributed environment as for the multiple explicitly Arnoldi method in [8].…”

“…Moreover, the problem of the choice of the size of the subspace remains. An approach based on the Arnoldi projection onto several Krylov subspaces is proposed by Emad et al [8]. The latter are formed with different initial vectors and have different sizes.…”

“…It updates the restarting vector of each ERAM process by taking into account the eigen-information of interest obtained by all ERAM processes. MERAM improves often the convergence of ERAM but the restarting issues intrinsic to ERAM remain [8].…”

“…This approach called multiple implicitly restarted Arnoldi method with nested subspaces (MIRAMns), is based upon the projection of the problem on several Krylov subspaces instead of a single one. These subspaces differ by their size while the subspaces in multiple restarted methods such as MERAM can differ by their size and their initial vectors [8]. MIRAMns makes use of Arnoldi method to compute the Ritz elements of a large matrix A in a set of ℓ nested Krylov subspaces K mi,v (for i = 1, .…”