A backward stochastic differential equations approach for the numerical solution of a class of nonlocal diffusion problems

We propose a novel numerical approach for linear nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and a class of backward stochastic differential equations (BSDEs) driven by Lèvy processes with jumps. The nonlocal diffusion problem under consideration is converted into a BSDE, for which numerical schemes are developed and applied directly. The most significant advantage of our approach is that the BSDE can be solved independently along each trajectory of the underlying Lèvy processes, and therefore, completely avoiding the difficulties associated with solving sequences of linear systems required by traditional numerical approximations. In addition, the inherent independence of our procedure allows for embarrassingly parallel implementations and also enables adaptive approximation techniques to be incorporated in a straightforward fashion. Rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach.