Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings

Abstract

In the last two decades, many heuristics have been developed for finding good elimination orderings for sparse Cholesky factorization. These heuristics aim to find elimination orderings with either low fill, low operatian count, or low elimination height. Though many heuristics seem to perform weil in practice, there has been a marked absence of much theoretieal analysis to back these heuristics. Indeed, few heuristics are known to provide any guarantee on the quality of the elimination ordering produced for arbitrary matrices. In this work, we present the first polynomial-time ordering algorithm that guarantees approximately optimal fill. Our algorithm is a variant of the well-known nested dissectian algorithm. Our ordering performs particularly weil when the number of elements in each row (and hence each column) of the coeflicient matrix is small. Fortunately, many problems in practice, especially those arising from finite-element methods, have such a property due to the physical constraints of the problems being modeled. Our ordering heuristie guarantees not only low fill, but also approximately optimal operatian count, and approximately optimal elimination height. Elimination orderings with small height and low fill are of much interest when performing factorization on parallei machines. No previous orderi ng heuristic guaranteed even small elimina.tion height. We wiJJ describe our ordering algorithm and prove its performance bounds. We shall also present some experimental resuIts comparing the quality of the orderings produced by our heuristic to those produced by two other well-known heuristics.