A partitioning algorithm for block-diagonal matrices with overlap

Abstract

We present an algorithm for partitioning a sparse matrix into a matrix that has blocks on the diagonal such that two consecutive blocks can overlap. We refer to this form of the matrix as block diagonal matrix with overlap. The partitioned matrix is suitable for applying the explicit formulation of multiplicative Schwarz (EFMS) [G.A. Atenekeng Kahou, E. Kamgnia, B. Philippe, An explicit formulation of the multiplicative Schwarz preconditioner, Journal of Applied Numerical Mathematics 57 (2007) 1197–1213] used as a preconditioner for solving a sparse unsymmetric system of linear equations Ax = b . The proposed algorithm partitions the graph of the matrix A into k parts such that every part V i has connecting edges with at most two neighbors V i - 1 and V i + 1 . First, an ordering algorithm that reduces the profile of the matrix, and an initial block-diagonal partition with overlap is obtained. Second, an iterative strategy is used to further refine the partitioning by allowing vertices to be moved between partitions. Experiments performed on real-world matrices show the usefulness of this approach.