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:许多不确定性量化研究考虑了具有不确定或随机投入的局部差异方程式(PDEs) 。在前期不确定性量化研究中,人们有兴趣分析PDE在投入不确定性条件下的随机反应,这通常涉及解决PDE输出的高度构件,在一个随机输入变量序列中解决PDE输出的高度分解。在实际计算中,人们通常需要以多种方式将问题分解:以一个有限维度随机字段来接近一个无线输入随机字段,PDE在空间上离散,使用例如有限元素和以准蒙特卡洛方法等缩放法对高维构件进行对等分析。在本文中,我们注重的是因输入随机字段的尺寸突变而出现的错误。我们展示了泰勒系列如何用多种方法来为一系列广泛的问题得出理论维度分解率,我们提供了一个简单的条件核对清单,为了维持我们维度调的参数错误,例如,使用有限的元素元素分解,我们的方法中的一些新特点包括:我们关于分解法的分解法的分解性分析结果,我们用于不精确的分解的分解法的分解方法的分解的分解的分解方法。