gSoFa: Scalable Sparse Symbolic LU Factorization on GPUs

Decomposing a matrix $\mathbf {A}$ into lower {L}$ and an upper {U}$ , which is also known as LU decomposition, essential operation in numerical linear algebra. For sparse matrix, decomposition often introduces more nonzero entries the factors than original matrix. A symbolic factorization step needed to identify structures of matrices. Attracted by enormous potentials Graphics Processing Units (GPUs), array efforts have surged deploy various factorization steps except for symbolic factorization, best our knowledge, on GPUs. This article gSoFa first G PU-based s ymb o lic fa ctorization design with following three optimizations enable scalable nonsymmetric pattern matrices First, we introduce novel fine-grained parallel algorithm that well suited Single Instruction Multiple Thread (SIMT) architecture Second, tailor supernode detection SIMT friendly process strive balance workload, minimize communication saturate GPU computing resources during detection. Third, three-pronged optimization reduce excessive space consumption problem faced multi-source concurrent factorization. Taken together, achieves up 31× speedup from 1 44 Summit nodes (6 264 GPUs) outperforms state-of-the-art CPU project, average, 5×. Notably, 47 percent peak memory throughput V100 Supercomputer.