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：在不确定性量化研究中，常考虑具有不确定性或随机输入的偏微分方程。正向不确定性量化旨在分析输入不确定性下偏微分方程的随机响应，这通常涉及对偏微分方程输出在随机变量序列上的高维积分求解。实际计算中，通常需要从多个维度对问题进行离散化：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、以及采用准蒙特卡洛方法等数值积分格式逼近高维积分。本文聚焦于输入随机场维度截断产生的误差。我们展示如何利用泰勒级数为广泛问题类别推导理论维度截断速率，并提供参数数学模型需满足的简单条件清单，以确保维度截断误差界成立。该方法的新颖之处在于：我们的结果适用于非仿射参数算子方程、参数偏微分方程的维度截断协调有限元离散解，甚至包含偏微分方程解与光滑非线性关注量的复合。作为该方法的具体应用，我们推导了对数正态参数化扩散系数椭圆偏微分方程的改进维度截断误差界。数值实验验证了理论结果。