Highly Parallel Sparse Cholesky Factorization

We develop and compare several fine-grained parallel algorithms to compute the Cholesky factorisation of a sparse matrix. Our experimental implementations are on the Connection Machine, a distributedmemory SIMD machine whose programming model conceptually supplies one processor per data element. In contrast to special-purpose algorithms in which the matrix structure conforms to the connection structure of the machine, our focus is on matrices with arbitrary sparsity structure. Our most promising algorithm is one whose inner loop performs several dense factorisations simultaneously on a twodimensional grid of processors. Virtually any massively parallel dense factorisation algorithm can be used as the key subroutine. The sparse code attains execution rates comparable to those of the dense subroutine. Although at present architectural limitations prevent the dense factorisation from realising its potential efficiency, we conclude that s regular data parallel architecture can be used efficiently to solve arbitrarily structured sparse problems. We also present a performance model and use it to analyse our algorithms. We find that asymptotic analysis combined with experimental measurement of parameters is accurate enough to be useful in choosing among alternative algorithms for a complicated problem. *Xerox Palo Alto Research Center, 3333 Coyote Hi]] Road, Palo Alto, California 94304. Copyright _) 1990 Xerox Corporation. All rights reserved. tResearch Institute for Advanced Computer Science, MS 230-5, NASA Ames Research Center, Moffett Field, CA 94035. This author's work was supported by the NAS Systems Division and DARPA via Cooperative Agreement NCC 2-387 between NASA and the University Space Research Association (USRA).