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：参数化数学模型（如具有随机系数的偏微分方程）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参数化变量的级数展开来表示。实际应用中，这种级数展开需要截断为有限项，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，针对输入随机场不同模型对应的维度截断误差已有若干研究，但多数分析均在数值积分框架下开展。本文研究了参数化模型问题的$L^2$维度截断误差。这类估计出现在高维函数逼近中维度截断误差的评估过程中。此外，我们证明了维度截断误差速率对于参数化变量的特定变换具有不变性。数值结果展示了理论结果的精确性。