Randomized quadratures for integrating functions in Sobolev spaces of order $\alpha \ge 1$, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of $O(n^{-\alpha-1/2})$ is proven, where $n$ denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.

翻译：针对Sobolev空间（阶次$\alpha \ge 1$，且可积性条件关于高斯测度）中的函数积分，本文研究了随机化求积方法。在此函数空间中，我们建立了最坏情形均方根误差（RMSE）的最优速率。该最优性适用于广义求积法则类，其中允许采用自适应非线性算法且函数求值次数可动态变化。通过匹配上下界证明给出最优速率：首先证明最坏情形RMSE的下界为$O(n^{-\alpha-1/2})$（$n$表示期望函数求值次数的上界），随后发现适当随机化的梯形法则在忽略对数因子后可达到该速率。文中还给出了该梯形法则的实用误差估计器，数值实验结果与理论分析相符。