Some new techniques to use in serial sparse Cholesky factorization algorithms

TL;DR

This paper introduces two new variants of serial supernodal sparse Cholesky factorization, demonstrating that the second variant, RLB, is faster and more efficient, especially with multithreaded BLAS, due to optimized reordering strategies.

Contribution

The paper presents a novel RLB variant that exploits larger off-diagonal blocks and requires no working storage, improving speed over existing methods.

Findings

RLB is faster than existing competitors.

RLB performs significantly better with multithreaded BLAS.

Reordering within supernodes reduces off-diagonal blocks, enhancing efficiency.

Abstract

We present a new variant of serial right-looking supernodal sparse Cholesky factorization (RL). Our comparison of RL with the multifrontal method confirms that RL is simpler, slightly faster, and requires slightly less storage. The key to the rest of the work in this paper is recent work on reordering columns within supernodes so that the dense off-diagonal blocks in the factor matrix joining pairs of supernodes are fewer and larger. We present a second new variant of serial right-looking supernodal sparse Cholesky factorization (RLB), where this one is specifically designed to exploit fewer and larger off-diagonal blocks in the factor matrix obtained by reordering within supernodes. A key distinction found in RLB is that it uses no floating-point working storage and performs no assembly operations. Our key finding is that RLB is unequivocally faster than its competitors. Indeed, RLB is consistently, but modestly, faster than its competitors whenever Intel's MKL sequential BLAS are used. More importantly, RLB is substantially faster than its competitors whenever Intel's MKL multithreaded BLAS are used. Finally, RLB using the multithreaded BLAS achieves impressive speedups over RLB using the sequential BLAS.