Adaptive Parameter Selection and Deflation for Computing Clustered Eigenvalues using a Shift-Invert Block Arnoldi Method
by
David Harrar
CMA/SMS, Australian National University

We discuss a block Arnoldi method incorporating techniques of restarting, shift-invert transformation, and an implicit deflation scheme.The block variant of Arnoldi's method is particularly useful in the case that the desired eigenvalues are multiple or clustered; an additional advantage is that block methods enable the use of level-3 BLAS and hence may result in increased performance on high performance computer architectures due to increased computational density.

One of the novel features of this implementation is that heuristics are employed to adaptively select new values for parameters such as the block size, number of block Arnoldi steps in a reduction, and maximum subspace dimension as eigenpairs converge and are subsequently removed from the computation via the technique of deflation. Parameter values selected according to these heuristics are often more appropriate than the static choices made at the outset of a calculation. This is particularly true in the case that the desired eigenvalues are clustered and/or multiple since this information is not generally known a priori, and poor parameter choices for the block Arnoldi method an have a very deleterious effect on convergence for these eigenvalues. We also elucidate computational simplifications which can be taken advantage of when deflation is implemented in the manner described here.

The resulting algorithm has been implemented on the Fujitsu VPP300, a parallel array of high performance vector processors. Performance results are given for an application arising in the study of chemical reactions and serve to illustrate the convergence behavior for clustered eigenvalues. Performance profiling results show that the vast majority of the computational time is spent either in highly vectorizable subroutines or in Fujitsu library routines which have a high degree of vectorization.

Date received: October 12, 1999