Numerically Aware Orderings for Sparse Symmetric Indefinite Linear Systems

Abstract

Sparse symmetric indefinite problems arise in a large number of important application areas; they are often solved through the use of an LDL T factorization via a sparse direct solver. While for many problems prescaling the system matrix A is sufficient to maintain stability of the factorization, for a small but important fraction of problems numerical pivoting is required. Pivoting often incurs a significant overhead, and consequently, a number of techniques have been proposed to try and limit the need for pivoting. In particular, numerically aware ordering algorithms may be used, that is, orderings that depend not only on the sparsity pattern of A but also on the values of its (scaled) entries. Current approaches identify large entries of A and symmetrically permute them onto the subdiagonal, where they can be used as part of a 2 × 2 pivot. This is numerically effective, but the fill in the factor L and hence the runtime of the factorization and subsequent triangular solves may be significantly increased over a standard ordering if no pivoting is required. We present a new algorithm that combines a matching-based approach with a numerically aware nested dissection ordering. Numerical comparisons with current approaches for some tough symmetric indefinite problems are given.