Efficient implicit-explicit sparse stochastic method for high dimensional semi-linear nonlocal diffusion equations

TL;DR

This paper introduces a scalable sparse grid Monte Carlo method for high-dimensional semi-linear nonlocal diffusion equations, combining implicit-explicit schemes with probabilistic sampling to handle complexity and singular kernels effectively.

Contribution

The paper develops a novel implicit-explicit sparse grid Monte Carlo approach that efficiently solves high-dimensional nonlocal PDEs with singular kernels, overcoming curse of dimensionality.

Findings

Successfully applied to problems in up to 100 dimensions.

Achieved unconditional stability without discretization constraints.

Validated accuracy and robustness through extensive numerical experiments.

Abstract

In this paper, we present a sparse grid-based Monte Carlo method for solving high-dimensional semi-linear nonlocal diffusion equations with volume constraints. The nonlocal model is governed by a class of semi-linear partial integro-differential equations (PIDEs), in which the operator captures both local convection-diffusion and nonlocal diffusion effects, as revealed by its limiting behavior with respect to the interaction radius. To overcome the bottleneck of computational complexity caused by the curse of dimensionality and the dense linear systems arising from nonlocal operators, we propose a novel implicit-explicit scheme based on a direct approximation of the nonlinear Feynman-Kac representation. The incorporation of sparse grid interpolation significantly enhances the algorithm's scalability and enables its application to problems in high dimensions. To further address the challenges posed by hypersingular kernels, we design a sampling strategy tailored to their singular structure, which ensures accurate and stable treatment of the nonlocal operators within the probabilistic framework. Notably, the proposed method inherits unconditional stability from the underlying stochastic representation, without imposing constraints on the temporal and spatial discretization scales. A rigorous error analysis is provided to establish the convergence of the proposed scheme. Extensive numerical experiments, including some non-radial solutions in up to 100 dimensions, are presented to validate the robustness and accuracy of the proposed method.