Analysis of A Splitting Approach for the Parallel Solution of Linear Systems on GPU Cards

TL;DR

This paper introduces SaP::GPU, a GPU-based parallel solver for large banded linear systems that partitions matrices for independent factorization, demonstrating competitive efficiency against established solvers.

Contribution

The paper presents a novel split and parallelize approach for solving large linear systems on GPUs, with an open-source implementation that performs well against standard solvers.

Findings

SaP::GPU is robust and efficient for large systems.

It compares favorably with PARDISO, SuperLU, and MUMPS.

Performs well on dense banded systems near diagonal dominance.

Abstract

We discuss an approach for solving sparse or dense banded linear systems ${\bf A} {\bf x} = {\bf b}$ on a Graphics Processing Unit (GPU) card. The matrix ${\bf A} \in {\mathbb{R}}^{N \times N}$ is possibly nonsymmetric and moderately large; i.e., $10000 \leq N \leq 500000$. The ${\it split\ and\ parallelize}$ (${\tt SaP}$) approach seeks to partition the matrix ${\bf A}$ into diagonal sub-blocks ${\bf A}_i$, $i=1,\ldots,P$, which are independently factored in parallel. The solution may choose to consider or to ignore the matrices that couple the diagonal sub-blocks ${\bf A}_i$. This approach, along with the Krylov subspace-based iterative method that it preconditions, are implemented in a solver called ${\tt SaP::GPU}$, which is compared in terms of efficiency with three commonly used sparse direct solvers: ${\tt PARDISO}$, ${\tt SuperLU}$, and ${\tt MUMPS}$. ${\tt SaP::GPU}$, which runs entirely on the GPU except several stages involved in preliminary row-column permutations, is robust and compares well in terms of efficiency with the aforementioned direct solvers. In a comparison against Intel's ${\tt MKL}$, ${\tt SaP::GPU}$ also fares well when used to solve dense banded systems that are close to being diagonally dominant. ${\tt SaP::GPU}$ is publicly available and distributed as open source under a permissive BSD3 license.