An Analysis of Spectral Envelope-reduction via Quadratic Assignment Problems

A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work involved in an envelope Cholesky factorization scheme. We formulate the latter two problems as quadratic assignment problems, and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a projected quadratic assignment problem, and then show that finding a permutation matrix closest to an orthogonal matrix attaining one of the lower bounds justifies the spectral envelope reduction algorithm. The lower bound on the 2-sum is seen to be tight for reasonably "uniform" finite element meshes. We also obtain asymptotically tight lower bounds for the envelope size for certain classes of meshes. * Written Sep. 1994. t Department of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1 Canada (jageorge_sparsel.u_aterloo.ca). This author was supported by the Canadian Natural Sciences and Engineering Research Council under grant OGP0008111. Department of Computer Science, Old Dominion University, Norfolk, VA 23529-0162 and ICASE, NASA Langley Research Center, Hampton, VA 23681-0001 ([email protected], pothen_icase, edu). This author was supported by National Science Foundation grant CCR-9024954, by U. S. Department of Energy grants DE-FG02-91ER25095 and DE-FG05-94ER25216, by the National Aeronautics and Space Administration under NASA Contract NAS1-19480, and by the Canadian Natural Sciences and Engineering Research Council under grant OGP0008111. i