A Selective Nesting Approach for the Sparse Multi-threaded Cholesky Factorization

Abstract

Sparse linear algebra routines are fundamental building blocks of a large variety of scientific applications. Direct solvers, which are methods for solving linear systems via the factorization of matrices into products of triangular matrices, are commonly used in many contexts. The Cholesky factorization is the fastest direct method for symmetric and positive definite matrices. This paper presents selective nesting, a method to determine the optimal task granularity for the parallel Cholesky factorization based on the structure of sparse matrices. We propose the Opt-D algorithm, which automatically and dynamically applies selective nesting. Opt-D leverages matrix sparsity to drive complex task-based parallel workloads in the context of direct solvers. We run an extensive evaluation campaign considering a heterogeneous set of 35 sparse matrices and a parallel machine featuring the A64FX processor. Opt-D delivers an average performance speedup of 1.75× with respect to the best state-of-the-art parallel methods to run direct solvers.