Performance-Portable Symbolic Factorization Through Common Graph Operations.

In lower-upper (LU) factorization of a sparse matrix in the form of A = LU, symbolic factorization is a pre-processing stage usually performed to discover the sparsity structure of the factors. A is usually not equal to L + U due to the fill-ins (the nonzeros that do not appear in A but element of L or U) introduced in the factorization. The symbolic factorization can be performed using only the pattern of A and the existing algorithms for that purpose usually utilize graphs by relying on the duality of graphs and matrices. In this work, we assess the viability of utilizing GraphBLAS for symbolic factorization. GraphBLAS defines a standard way to express operations on graphs in the language of linear algebra. We express edge-based and path-based symbolic factorization of the coefficient matrix using common graph operations and investigate utilization of masks and elimination trees for optimization. Our goal is to obtain a performant symbolic factorization, which can then be used on any hardware in a portable manner for which the GraphBLAS standard is realized. We demonstrate our approach on various non-symmetric sparse matrices by running symbolic factorization on multi-core and many-core architectures efficiently.