SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2840-A2868, January 2021.
When solving partial differential equations (PDEs) with random fields as coefficients, the efficient sampling of random field realizations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the stochastic PDE (SPDE) approach of Lindgren et al. combined with a new fast algorithm for white noise sampling which is tailored to (ML)QMC. We express white noise as a wavelet series expansion that we divide into two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good quasi Monte Carlo (QMC) convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

中文翻译：

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基于快速白噪声采样的具有随机场系数的椭圆偏微分方程的多级准蒙特卡罗方法

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2840-A2868 页，2021 年 1 月。
在以随机场作为系数求解偏微分方程 (PDE) 时，随机场实现的有效采样可能具有挑战性。在本文中，我们专注于在有限元和多级准蒙特卡罗 (MLQMC) 设置中使用准随机点对高斯场进行快速采样。我们的方法使用 Lindgren 等人的随机 PDE (SPDE) 方法。结合了针对 (ML)QMC 量身定制的新的白噪声采样快速算法。我们将白噪声表示为一个小波级数展开式，我们将其分为两部分。第一部分使用准随机点进行采样，并包含有限数量的项，按重要性递减的顺序排列，以确保良好的准蒙特卡罗 (QMC) 收敛。第二部分是使用标准伪随机数采样的校正项。我们展示了如何通过超网格结构在网格单元数量的线性时间和内存复杂度中执行这两个项的采样，从而产生总体线性成本。此外，即使在非嵌套网格层次结构的情况下，我们的技术也可用于强制执行 MLQMC 耦合。我们通过数值实验证明了我们方法的有效性。