Structure-Adaptive Parallel Solution of Sparse Triangular Linear Systems
PPL Technical Report 2012
Publication Type: Paper
Solution of sparse triangular systems of linear equations is a performance bottleneck in many methods for solving more general sparse systems. In both direct methods and iterative preconditioners, it is used to solve the system or refine the solution, often across many iterations. Triangular solution is notoriously resistant to parallelism, however, and existing parallel linear algebra packages appear to be ineffective in exploiting much parallelism for this problem. We develop a novel parallel algorithm based on various heuristics that adapts to the structure of the matrix and extracts parallelism that is unexploited by conventional methods. By analysis and reordering operations, our algorithm can even extract parallelism in some cases where most of the nonzero matrix entries are near the diagonal. We describe the implementation of our algorithm in Charm++ and MPI and present promising results on up to 512 cores of BlueGene/P, using numerous sparse matrices from real applications.