We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights.

翻译：本文描述了一种利用格点集上的核插值快速求解含不确定系数的椭圆型偏微分方程(PDE)的方法。通过采用Kaarnioja、Kuo和Sloan (SIAM J.~Numer.~Anal.~2020)提出的模型表示系统的输入随机场（该模型中可数个独立随机变量以周期函数形式进入随机场），Kaarnioja、Kazashi、Kuo、Nobile和Sloan (Numer.~Math.~2022)证明：利用快速傅里叶变换(FFT)，可高效构造作为随机变量函数的PDE解的基于格点的核插值。本文探讨了该模型与不确定系数PDE不确定性量化文献中广泛研究的"仿射统一模型"之间的联系。我们提出了一类新的权重——偶然权重——用于核插值的构造，该权重显著提升了含不确定系数PDE问题的核插值计算性能，使得我们能够处理极高维度的函数逼近问题。通过数值实验展示了偶然权重的性能。