Sparse Monte Carlo Method for Nonlocal Diffusion Problems

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