# high performance computing on graphics processing units: hgpu.org

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

Ang Li, Radu Serban, Dan Negrut
arXiv:1509.07919 [cs.DC], (25 Sep 2015)

@article{li2015analysis,

title={Analysis of A Splitting Approach for the Parallel Solution of Linear Systems on GPU Cards},

author={Li, Ang and Serban, Radu and Negrut, Dan},

year={2015},

month={sep},

archivePrefix={"arXiv"},

primaryClass={cs.DC}

}

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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.
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