SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 872-892, April 2024.
Abstract. Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi–Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings.

中文翻译：

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高维数值积分的广义维截断误差分析：对数正态设置及其他

SIAM 数值分析杂志，第 62 卷，第 2 期，第 872-892 页，2024 年 4 月。
摘要。许多不确定性量化研究都考虑了具有不确定或随机输入的偏微分方程（PDE）。在正向不确定性量化中，人们感兴趣的是分析 PDE 对输入不确定性的随机响应，这通常涉及求解 PDE 输出在一系列随机变量上的高维积分。在实际计算中，通常需要以多种方式离散化问题：用有限维随机场逼近无限维输入随机场，使用有限元等对偏微分方程进行空间离散，以及使用有限元逼近高维积分诸如准蒙特卡罗方法之类的立方体。在本文中，我们重点关注输入随机场的维度截断所产生的误差。我们展示了如何使用泰勒级数来导出各种问题的理论尺寸截断率，并且我们提供了参数数学模型需要满足的简单条件清单，以便我们的尺寸截断误差成立。我们的方法的一些新颖特征包括我们的结果适用于非仿射参数算子方程、参数偏微分方程的尺寸截断一致有限元离散解，甚至是具有感兴趣的平滑非线性量的偏微分方程解的组合。作为我们方法的具体应用，我们推导了具有对数正态参数化扩散系数的椭圆偏微分方程的改进维数截断误差界。数值例子支持我们的理论发现。