High Performance Parallel Block Supernode Direct Solver for Stiffness Equation

Abstract

The process of direct method to solve large sparse linear equation mainly includes reordering, symbolic factorization, numerical factorization and triangular solving. Traditional symbolic factorization predicts the pattern of $L$ based on single column and single row index. We propose to directly partition supernodes based on characteristics of matrix reordered by METIS, and then perform parallel symbolic factorization based on supernodes and row index fragments. A parallel block supernode numerical factorization strategy is proposed based on the concept of task pool here. In triangular solving stage, unlike traditional algorithms based on DAXPY and DDOT operations, we propose a new parallel triangular solving algorithm based on DGEMM and DTRSM operations. We name the parallel solver as finite element analysis direct solver (FEADS) and compare it with the advanced MKL PARDISO and MUMPS. The stiffness equations of 394770 and 719871 dimensions are solved using the solvers on two different computers. On the first computer, the solving efficiency of FEADS and MKL PARDISO is comparable, while MUMPS is relatively backward. On the second computer, FEADS performs especially well. For solving the case with 394770 dimensions, FEADS leads MKL PARDISO and MUMPS by 21.92% and 42.35%, respectively. For solving the case with 719871 dimensions, FEADS leads 34.75% and 38.38% respectively.