A fast direct solver for nonlocal operators in wavelet coordinates

TL;DR

This paper introduces a fast direct solver for nonlocal operators that combines wavelet-based matrix compression with nested dissection ordering, enabling efficient inversion for complex applications like boundary integral equations and fractional Laplacians.

Contribution

The paper presents a novel approach that integrates wavelet representations with nested dissection to efficiently invert nonlocal operator matrices, reducing computational complexity.

Findings

Effective in solving boundary integral equations in 3D

Accelerates simulation of Gaussian random fields

Handles fractional Laplacian problems efficiently

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.