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

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 ∈ ℝN × N is possibly nonsymmetric and moderately large; i.e., 10000 ≤ N ≤ 500000. The it split and parallelize (tt SaP) approach seeks to partition the matrix bf A into diagonal sub-blocks bf Ai, 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 Ai. 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.