Parametric mathematical models such as partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.
翻译:参数数学模型,如带有随机系数的局部差分方程式,在不确定性量化领域受到了很多关注。模型不确定性通常通过参数变量的系列扩展来体现。在实践中,这一序列扩展需要缩短到一定数量,对参数数学模型的数值模拟引入一个尺寸脱线错误。近年来,对与输入随机字段不同模型相对应的尺寸脱轨错误进行了若干研究,但其中许多分析是在数字整合的背景下进行的。在本文件中,我们研究了参数模型问题的数值脱轨误差。在评估高维值函数近似的尺寸脱轨误差时,会得出这种估计。此外,我们表明,相对于参数随机变量的某些变异,尺寸脱轨误差率是变化不定的。提出了数值结果,显示了理论结果的精确性。