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:我们描述了一种使用在格点集上的内核插值来快速求解具有不确定系数的椭圆偏微分方程的方法。通过使用Kaarnioja,Kuo和Sloan提出的模型(其中无限个独立随机变量作为周期函数进入随机场),表示系统的输入随机场,Kaarnioja,Kazashi,Kuo,Nobile和Sloan(Numer. Math. 2022)表明,可以使用快速傅里叶变换(FFT)以非常有效的方式构造基于格点法的内核插值法解决PDE解的随机变量的函数。在这项工作中,我们讨论了我们的模型与在不确定系数的PDE的不确定性量化文献中广泛研究的流行的“正交和均匀模型”的联系。我们还提出了一类新的权重,其进入内核插值的构造 - “意外权重”。这些权重极大地改善了具有不确定系数的PDE问题的内核插值的计算性能,并使我们能够处理非常高维度的函数逼近问题。通过数值实验展示了意外权重的性能。