Numerical Methods for a Class of Nonlocal Diffusion Problems with the Use of Backward SDEs

TL;DR

This paper introduces a new stochastic numerical method for nonlocal diffusion equations using backward SDEs driven by Lévy processes, offering high accuracy, parallelizability, and adaptability for complex nonlinear problems.

Contribution

The paper develops a novel stochastic numerical scheme for nonlocal diffusion problems via backward SDEs, improving accuracy and computational efficiency over traditional methods.

Findings

The proposed method achieves higher accuracy than classic stochastic approaches.

It allows for embarrassingly parallel implementations and adaptive techniques.

Numerical examples demonstrate the method's effectiveness and efficiency.

Abstract

We propose a novel numerical approach for nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by L\`{e}vy processes with jumps. The nonlocal diffusion problem under consideration is converted to a BSDE,for which numerical schemes are developed and applied directly. As a stochastic approach, the proposed method does not require the solution of linear systems, which allows for embarrassingly parallel implementations and also enables adaptive approximation techniques to be incorporated in a straightforward fashion. Moreover, our method is more accurate than classic stochastic approaches due to the use of high-order temporal and spatial discretization schemes. In addition, our approach can handle a broad class of problems with general nonlinear forcing terms as long as they are globally Lipchitz continuous. Rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach.