A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields. (c) 2020 The Author(s). Published by Elsevier Inc.

Automated summary

This article discusses fast direct solvers for nonlocal operators by combining wavelet representation with nested dissection ordering scheme to reduce fill-in during matrix factorization and achieve the exact inverse of the compressed system matrix with only a moderate increase in nonzero entries. Numerical experiments are conducted for various nonlocal operator applications to illustrate the efficacy of the approach.