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

Abstract

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

arXiv comment38 pages