Parametric mathematical models such as parameterizations of 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：参数化数学模型（如具有随机系数的偏微分方程参数化）在不确定性量化领域受到广泛关注。模型不确定性通常通过关于参数变量的级数展开来表示。在实际应用中，该级数展开需截断为有限项，从而在参数化数学模型的数值模拟中引入维度截断误差。近年来已有若干针对不同输入随机场模型的维度截断误差研究，但多数分析是在数值积分背景下开展的。本文研究了参数化模型问题的$L^2$维度截断误差。此类估计出现在高维函数逼近的维度截断误差评估中。此外，我们证明维度截断误差率对参数变量的某些变换具有不变性。文中给出的数值结果验证了理论结果的尖锐性。