A Shift-Invert Block Arnoldi Method with Deflation and Adaptive Parameter Selection
David L. Harrar II
Abstract:
We discuss here 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.
The implementation developed here attempts to select adaptively
new values for parameters such as the block size, number of
block Arnoldi steps in a reduction, and maximum Hessenberg
dimension (equivalently, subspace dimension) as eigenpairs
converge and are subsequently removed from the computation
via the technique of deflation.
This enables one to work with parameter values which may be
more appropriate than the static choices made at the outset
of a calculation, particularly in the case that the desired
eigenvalues are clustered and/or multiple.
Computational simplifications which can be taken advantage
of when deflation is implemented in the manner described
here are also elucidated.
The resulting algorithm has been implemented on the Fujitsu
VPP300, a parallel array of high performance vector processors.
Performance results on an application arising in the study of
chemical reactions serve to illustrate the convergence behavior
for clustered eigenvalues, and 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.
The paper closes with a brief review/survey of some of
the contemporary literature on Arnoldi methods.
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