A fast direct solver for nonlocal operators in wavelet coordinates
Abstract
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.
Cite
@article{arxiv.2007.01541,
title = {A fast direct solver for nonlocal operators in wavelet coordinates},
author = {Helmut Harbrecht and Michael Multerer},
journal= {arXiv preprint arXiv:2007.01541},
year = {2021}
}