On the Use of Incomplete LU Decomposition as a Preconditioning Technique for Density Fitting in Electronic Structure Computations

Abstract

Incomplete factorization preconditioners combined with Krylov subspace accelerators are currently among the most effective methods for iteratively solving large systems of linear equations. In this paper we consider the use of a dual threshold incomplete LU factorization (ILUT) preconditioner for the iterative solution of the linear equation systems encountered when performing electronic structure calculations that involve density fitting. Two questions are addressed, how the overall performance of the ILUT method varies as a function of the accuracy of the preconditioning matrix, and whether it is possible to make approximations to the original matrix on which the LU decomposition is based and still obtain a good preconditioner. With respect to overall performance both computational and memory storage requirements are considered, while in terms of approximations both those based on numerical and physical arguments are considered. The results indicate that under the right circumstances the ILUT method is superior to fully direct approaches such as singular value decomposition.