Hybrid Sparse Linear Solutions with Substituted Factorization

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 increased 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 × k 5-point finite-difference problem by 10% and empirical test 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.