Nonlocal models feature a finite length scale, referred to as the horizon, such that points separated by a distance smaller than the horizon interact with each other. Such models have proven to be useful in a variety of settings. However, due to the reduced sparsity of discretizations, they are also generally computationally more expensive compared to their local differential equation counterparts. We introduce a multifidelity Monte Carlo method that combines the high-fidelity nonlocal model of interest with surrogate models that use coarser grids and/or smaller horizons and thus have lower fidelities and lower costs. Using the multifidelity method, the overall computational cost of uncertainty quantification is reduced without compromising accuracy. It is shown for a one-dimensional nonlocal diffusion example that speedups of up to two orders of magnitude can be achieved using the multifidelity method to estimate the expectation of an output of interest.

非局部模型的特征是有限的长度标度，称为水平线，这样，距离小于水平线的点之间便会相互影响。事实证明，这样的模型在各种设置中都是有用的。但是，由于离散稀疏性的降低，与局部微分方程的对应物相比，它们的计算量通常也更高。我们引入了多保真度蒙特卡罗方法，该方法将关注的高保真度非局部模型与使用较粗网格和/或较小视野的替代模型相结合，从而具有较低的保真度和较低的成本。使用多重保真度方法，可以在不影响准确性的情况下减少不确定性量化的总体计算成本。