Making sparse matrix computations scalable (invited talk abstract)

Sparse matrix computations are ubiquitous in science and engineering. In particular, solving a sparse linear system of equations is the bottleneck in many computations. Of the many available algorithms, some are easy to parallelize, because they reduce to sparse-matrix-vector multiplication, for which good algorithms based on graph partitioning exist. But others, which are widely used sequentially because of their numerical properties, remain hard to parallelize s&ably. In this talk we highlight the challenges of parallelizing two widely used methods: sparse Gaussian Elimination with pivoting, and multigrid for linear systems arising from solid mechanincs problems on irregular meshes. In both cases our current algorithms are among the fastest available (the first one getting parallel efficiencies up to 20% on 512 processors, and the latter up to 50% on 960 processors), but many algorithm design and performance analysis problems remain. In the case of multigrid: