Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each uniformly distributed on an interval, with these random variables entering linearly in the input random field (the so-called affine model). In this paper we consider an alternative model of the random field, in which the random variables have the same uniform distribution on an interval, but the random variables enter the input field as periodic functions. The field is constructed in such a way as to have the same mean and covariance function as the affine random field. Higher moments differ from the affine case, but in general the periodic model seems no less desirable. The new model of the random field is used to compute expected values of a quantity of interest arising from an elliptic PDE with random coefficients. The periodicity is shown to yield a higher order cubature convergence rate of $\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction with rank-1 lattice cubature rules constructed using suitably chosen smoothness-driven product and order dependent weights, where $n$ is the number of lattice points and $p$ is the summability exponent of the fluctuations in the series expansion of the random coefficient. We present numerical examples that assess the performance of our method.

中文翻译：

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使用周期性随机变量的不确定性量化

许多不确定性量化的研究都是在输入随机场的假设下进行的，其中可数个独立随机变量均匀分布在一个区间上，这些随机变量线性进入输入随机场（所谓的仿射模型）。在本文中，我们考虑随机场的另一种模型，其中随机变量在一个区间上具有相同的均匀分布，但随机变量作为周期函数进入输入场。该场的构造方式与仿射随机场具有相同的均值和协方差函数。较高的矩不同于仿射情况，但一般而言，周期模型似乎同样可取。随机场的新模型用于计算由具有随机系数的椭圆偏微分方程引起的感兴趣量的期望值。The periodicity is shown to yield a higher order cubature convergence rate of $\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction with rank-1 lattice cubature rules constructed using suitably chosen smoothness -驱动的产品和顺序相关的权重，其中 $n$ 是格点的数量，$p$ 是随机系数级数展开中波动的可和指数。我们提供了评估我们方法性能的数值例子。The periodicity is shown to yield a higher order cubature convergence rate of $\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction with rank-1 lattice cubature rules constructed using suitably chosen smoothness -驱动的产品和顺序相关的权重，其中 $n$ 是格点的数量，$p$ 是随机系数级数展开中波动的可和指数。我们提供了评估我们方法性能的数值例子。The periodicity is shown to yield a higher order cubature convergence rate of $\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction with rank-1 lattice cubature rules constructed using suitably chosen smoothness -驱动的产品和顺序相关的权重，其中 $n$ 是格点的数量，$p$ 是随机系数级数展开中波动的可和指数。我们提供了评估我们方法性能的数值例子。