A recursive graph bipartitioning algorithm by vertex separators with fixed vertices for permuting sparse matrices into block diagonal form with overlap

Abstract

Solving sparse system of linear equations Ax=b using preconditioners can be effi- ciently parallelized using graph partitioning tools. In this thesis, we investigate the problem of permuting a sparse matrix into a block diagonal form with overlap which is to be used in the parallelization of the multiplicative schwarz preconditioner. A matrix is said to be in block diagonal form with overlap if the diagonal blocks may overlap. In order to formulate this permutation problem as a graph-theoretical problem, we introduce a restricted version of the graph partitioning by vertex separator problem (GPVS), where the objective is to find a vertex partition whose parts are only connected by a vertex separator. The modified problem, we refer as ordered GPVS problem (oGPVS), is restricted such that the parts should exhibit an ordered form where the consecutive parts can only be connected by a separator. The existing graph partitioning tools are unable to solve the oGPVS problem. Thus, we present a recursive graph bipartitioning algorithm by vertex separators together with a novel vertex fixation scheme so that a GPVS tool supporting fixed vertices can effectively and efficiently be utilized. We also theoretically verified the correctness of the proposed approach devising a necessary and sufficient condition to the feasibility of a oGPVS solution. Experimental results on a wide range of matrices confirm the validity of the proposed approach.