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.
翻译:在本文中,我们提出了一个以随机系数解决非本地方程式的零星兼容网状方法,描述了不同介质的传播。特别是,随机的异差系数用一个有限维随机变量或随机变量与Karhunen-Lo ⁇ ⁇ e}分解的截断组合来描述,然后用一种稀疏网格的概率合用法(PCM)来抽样抽查过程。在每一个样本中,确定性的非本地扩散问题与一个基于优化的网格自由二次曲线规则分开。我们对拟议的办法进行严格分析,并展示一些基准问题的趋同性,表明它保持了空间上的零位兼容性,并在随机系数空间中实现了升温或亚稀释趋同率,随着合用点的增加。最后,为了证实这种方法是否适用,我们认为,与给定的空间相关结构存在随机的异差非本地问题。我们证明,拟议的PCM方法与传统的蒙特卡洛模拟模型相比,实现了大幅度的加速。