An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

TL;DR

This paper introduces an asymptotically compatible meshfree probabilistic collocation method for efficiently solving nonlocal diffusion problems with random heterogeneous coefficients, demonstrating convergence and significant computational speed-up.

Contribution

The paper develops a novel meshfree PCM that is asymptotically compatible and effective for nonlocal problems with random coefficients, with rigorous analysis and improved computational efficiency.

Findings

Achieves convergence in both spatial and stochastic domains.

Demonstrates algebraic or sub-exponential convergence rates.

Provides substantial speed-up over Monte Carlo methods.

Abstract

In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Lo\`{e}ve decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.