Sparse Monte Carlo Method for Nonlocal Diffusion Problems

Abstract

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media, peridynamics, and models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. Our method requires minimal assumptions on the regularity of the data. In particular, the kernel of the nonlocal diffusivity is assumed to be a square integrable function and may be singular or discontinuous. An important feature of our method is the use of sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows us to use fewer discretization points without compromising the accuracy. For kernels with singularities, more points are selected automatically in the regions near the singularities. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations, respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.