Hybrid Sparse Linear Solutions with Substituted Factorization

Abstract

We develop a computationally less expensive alternative to the direct solution of a large sparse symmetric positive definite system arising from the numerical solution of elliptic partial differential equation models. Our method, substituted factorization , replaces the computationally expensive factorization of certain dense submatrices that arise in the course of direct solution with sparse Cholesky factorization with one or more solutions of triangular systems using substitution. These substitutions fit into the tree-structure commonly used by parallel sparse Cholesky, and reduce the initial factorization cost at the expense of a slight increase cost in solving for a right-hand side vector. Our analysis shows that substituted factorization reduces the number of floating-point operations for the model $k \times k$ 5-point finite-difference problem by $10\,\%$ and empirical tests show execution time reduction on average of $24.4\,\%$. On a test suite of three-dimensional problems we observe execution time reduction as high as $51.7\,\%$ and $43.1\,\%$ on average.

J.D. Booth—This work was completed while attending The Pennsylvania State University.