Parallel sparse matrix-vector multiplies and iterative solvers

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Abstract. Sparse matrix-vector multiply (SpMxV) operations are in the kernel of many scientific computing applications. Therefore, efficient parallelization of SpMxV operations is of prime importance to scientific computing community. Previous works on parallelizing SpMxV operations consider maintaining the load balance among processors and minimizing the total message volume. We show that the total message latency (start-up time) may be more important than the total message volume. We also stress that the maximum message volume and latency handled by a single processor are important communication cost metrics that should be minimized. We propose hypergraph models and hypergraph partitioning methods to minimize these four communication cost metrics in one dimensional and two dimensional partitioning of sparse matrices.

Iterative methods used for solving linear systems appear to be the most common context in which SpMxV operations arise. Usually, these iterative methods apply a technique called preconditioning. Approximate inverse preconditioning--which can be applied to a large class of unsymmetric and symmetric matrices--replaces an SpMxV operation by a series of SpMxV operations. That is, a single SpMxV operation is only a piece of a larger computation in the iterative methods that use approximate inverse preconditioning. In these methods, there are interactions in the form of dependencies between the successive SpMxV operations. These interactions necessitate partitioning the matrices simultaneously in order to parallelize a full step of the subject class of iterative methods efficiently. We show that the simultaneous partitioning requirement gives rise to various matrix partitioning models depending on the iterative method used. We list the partitioning models for a number of widely used iterative methods. We propose operations to build a composite hypergraph by combining the previously proposed hypergraph models and show that partitioning the composite hypergraph models addresses the simultaneous matrix partitioning problem. We strove to demonstrate how the proposed partitioning methods--both the one that addresses multiple communication cost metrics and the other that addresses the simultaneous partitioning problem--help in practice.

We implemented a library and investigated the performances of the partitioning methods. These practical investigations revealed a problem that we call message ordering problem. The problem asks how to organize the send operations to minimize the completion time of a certain class of parallel programs. We show how to solve the message ordering problem optimally under reasonable assumptions.

Key words. Sparse matrices, parallel matrix-vector multiplication, iterative methods, preconditioning, approximate inverse preconditioner, hypergraph partitioning.