Partitioning sparse rectangular matrices for parallel computing of AAtX

Abstract

Many scientific applications involve repeated sparse matrix-vector and matrixtranspose-vector product computations. Graph and hypergraph partitioning based approaches used in the literature aim at minimizing the total communication volume while maintaining computational load balance through one dimensional partitioning of sparse matrices. In this thesis, we consider two approaches which consider minimizing both the total message count and communication volume while maintaining balance on the communication loads of the processors. Two communication schemes are investigated for the fold and expand operations needed in the parallel algorithm. For the global communication scheme, we show that the problem of minimizing concurrent communication volume can be formulated as the problem of permuting the sparse matrix into a singly-bordered block-diagonal form, where the total and concurrent message count is determined by the interconnection topology. For the personalized communication scheme, a two stage approach is proposed. In the first stage, the total communication volume is minimized while maintaining balance on the computational loads of the processors. In the second stage, a novel communication hypergraph model is proposed which enables the minimization of the total message count while maintaining balance on the communication loads of the processors through hypergraph-partitioning-like methods. The solution methods are tested on various matrices and results, which are quite attractive in terms of solution quality and running times, are obtained.