Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

TL;DR

This paper introduces a finite element approach for high-dimensional nonlocal diffusion problems, leveraging multilevel Toeplitz structures to efficiently handle the curse of dimensionality, with numerical validation in 1D, 2D, and 3D.

Contribution

It develops a novel finite element method that exploits Toeplitz structures in the stiffness matrix for high-dimensional nonlocal diffusion problems.

Findings

Efficient handling of high-dimensional nonlocal problems using Toeplitz structures.

Numerical results demonstrating the method in 1D, 2D, and 3D.

Reduction of computational complexity in nonlocal diffusion simulations.

Abstract

We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel function. We assume that the domain is an arbitrary $d$-dimensional hyperrectangle and the kernel is translation invariant. Under these assumptions, we carefully analyze the structure of the stiffness matrix resulting from a continuous Galerkin method with multilinear elements and exploit this structure in order to cope with the curse of dimensionality associated to nonlocal problems. For the purpose of illustration we choose a particular kernel, which is related to space-fractional diffusion and present numerical results in 1d, 2d and for the first time also in 3d.