Sparse matrix factorization on massively parallel computers

Direct methods for solving sparse systems of linear equations have a high asymptotic computational and memory requirements relative to iterative methods. However, systems arising in some applications, such as structural analysis, can often be too ill-conditioned for iterative solvers to be effective. We cite real applications where this is indeed the case, and using matrices extracted from these applications to conduct experiments on three different massively parallel architectures, show that a well designed sparse factorization algorithm can attain very high levels of performance and scalability. We present strong scalability results for test data from real applications on up to 8,192 cores, along with both analytical and experimental weak scalability results for a model problem on up to 16,384 cores---an unprecedented number for sparse factorization. For the model problem, we also compare experimental results with multiple analytical scaling metrics and distinguish between some commonly used weak scaling methods.