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 non-affine 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.

翻译：具有不确定或随机输入的偏微分方程在不确定性量化的众多研究中备受关注。在前向不确定性量化中，人们致力于分析受输入不确定性影响的偏微分方程的随机响应，这通常涉及基于一系列随机变量对偏微分方程输出进行高维积分求解。在实际计算中，通常需要通过多种方式对问题进行离散化：用有限维随机场逼近无限维输入随机场、利用有限元等方法对偏微分方程进行空间离散化，以及借助准蒙特卡洛方法等求积法逼近高维积分。本文重点研究输入随机场维度截断所产生的误差。我们展示了如何利用泰勒级数为广泛的问题类别推导理论上的维度截断速率，并给出了参数化数学模型所需满足的简单条件清单，以确保我们的维度截断误差界成立。本方法的新颖之处包括：我们的结果适用于非仿射参数算子方程、参数化偏微分方程的维度截断协调有限元离散解，甚至适用于偏微分方程解与光滑非线性目标量的复合情形。作为本方法的具体应用，我们为具有对数正态参数化扩散系数的椭圆型偏微分方程推导了改进的维度截断误差界。数值算例验证了我们的理论发现。