Permuting Sparse Square Matrices into Block Diagonal Form with Overlap
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چکیده
In this whitepaper, we describe the problem of permuting sparse square matrices into block diagonal form with overlap (BDO) and propose a graph partitioning algorithm for solving this problem. A block diagonal matrix with overlap is a block diagonal matrix whose consecutive diagonal blocks may overlap. The objective in this permutation problem is to minimize the total overlap size, whereas the permutation constraint is to maintain balance on the number of nonzeros in the diagonal blocks. This permutation problem arises in the parallelization of an explicit formulation of multiplicative Schwarz preconditioner. We define ordered Graph Partitioning by Vertex Separator (oGPVS) problem as an equivalent problem to this permutation problem. oGPVS problem is a restricted version of Graph Partitioning by Vertex Separator (GPVS) problem and the aim is to find a partition of the vertices into K ordered vertex parts and K-1 ordered separators where each two consecutive parts can be connected through only a separator, a separator can only connect two consecutive parts, and each two consecutive separators can be adjacent. The objective in the oGPVS problem is to minimize the total number of vertices in the separators, whereas the partitioning objective is to maintain balance on the part weights where part weight is defined as the sum of the weights of vertices in that part. To solve oGPVS problem, we utilized recursive bipartitioning paradigm and fixed vertices in our proposed oGPVS algorithm. We tested the performance of our algorithm in a wide range of matrices in comparison to another graph partitioning algorithm that solves the same problem. Results showed that the oGPVS algorithm performs better than the other algorithm in terms of overlap size.
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تاریخ انتشار 2012