Parallel Solve Phase of a Direct Sparse Solver

We propose a parallel solve phase algorithm for solving large sparse linear systems arising in wide range of science-intensive and computational applications. In many scenarios, serial solutions of systems with fixed matrix and varied right-hand sides are required. In this case matrix is factorized only once and solve phase is performed many times, so that solve phase becomes the most performance critical part of computations. In this paper, we consider parallel block algorithms of direct and backward substitutions, sequential analogues corresponding to them are also described. We propose a parallelization strategy of the solve phase based on a dynamic two-level scheduling scheme using the nested dissection algorithm. It makes possible independent tasks processing in the subtrees. This reduces synchronization events count and makes the algorithm more efficient. Our parallel solve phase algorithm uses block storage of factorized matrix using a supernodal approach, which allows exploiting the efficiency of the LAPACK and BLAS Level-3 routines.