Parallelization strategies for subspace methods to solve large eigenproblems

A. Basermann
C & C Research Laboratories, NEC Europe Ltd.,
Rathausallee 10, 53757 Sankt Augustin, Germany

B. Steffen
Research Centre Juelich GmbH,
Central Institute for Applied Mathematics (ZAM),
52425 Juelich, Germany

To find a few eigenvalues and -vectors of large sparse real symmetric or
complex Hermitian matrices, we present parallel preconditioned methods based
on the Jacobi-Davidson (JD) method by G.L.G. Sleijpen and H.A. van der
Vorst.  For peconditioning, parallel QMR (Quasi-Minimal Residual) algorithms
are applied to solve both real symmetric and complex Hermitian problems.  To
parallelize the solvers, we investigate matrix and vector partitioning as
well as dividing the spectrum of the matrix into independent parts.  If the
eigenvalues required are not extreme - this is always the case when parts of
the spectrum are calculated in parallel - the subspace part of the JD
methods may produce disastrous results and is replaced by a minimal residue
search.  The efficiency of these strategies is demonstrated on the massively
parallel systems NEC Cenju-3 and Cray T3E.