Optimization of the Sparse Multi-Threaded Cholesky Factorization for A64FX

TL;DR

This paper introduces OPT-D-COST, a dynamic algorithm that optimizes task granularity in parallel sparse Cholesky factorization, significantly improving performance on A64FX processors.

Contribution

The paper presents a novel method called selective nesting and an automatic algorithm OPT-D-COST for optimizing parallel sparse Cholesky factorization based on matrix structure.

Findings

Achieves an average speedup of 1.46× over state-of-the-art methods.

Effectively leverages matrix sparsity for workload optimization.

Demonstrates improved performance on a diverse set of 60 matrices.

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 definite positive 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-COST algorithm, which automatically and dynamically applies selective nesting. OPT-D-COST 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 60 sparse matrices and a parallel machine featuring the A64FX processor. OPT-D-COST delivers an average performance speedup of 1.46$\times$ with respect to the best state-of-the-art parallel method to run direct solvers.