Multi-Level Preconditioners for the Parallel PCG Package.

                         R. T. McLay and G. F. Carey

In this study we consider parallel solution of sparse linear systems arising
from discretized PDE's.  The main problem of in terest in the present work is
the construction of preconditioners in the context of the parallel PCG solver
we are developing [1, 2].  Here the problem is partitioned over a set of
processors subdomains and the matrix-vector product for PCG is carried out in
parallel for overlapping grid subblocks.  For problems of scaled speedup, the
actual rate of convergence of the unpreconditioned system deteriorates as the
mesh is refined.  Multigrid and subdomain strategies provide a logical
approach to resolving the problem.  We consider the parallel trade-offs
between communication and computation and provide a complexity analysis of a
representative algorithm.  Some preliminary calculations using the parallel
package and comparisons with other preconditioners are provided together with
parallel performance results.

Acknowledgement:  This research is supported by ARPA Grant DABT63-92-C-0024.

                                  References

[1] Joubert, W. et al, PCG Reference Manual, CNA-274, Center for Numerical
Analysis, University of Texas at Austin, January 1995.

[2] McLay , R. T., Swift, S., and G. F. Carey , "Maximizing Sparse
Matrix-Vector Product Performance in MIMD Computers", Colorado Conference of
Iterative Methods, Breckenridge CO, April 1994.