论文标题
随机对梯形规则进行随机化，给出了高斯Sobolev空间中最佳的RMSE速率
Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces
论文作者
论文摘要
考虑到在$α\ ge 1 $的Sobolev空间中集成功能的随机四倍，其中集成性条件相对于高斯度量。在此功能空间中，建立了最坏情况均值均方方误差（RMSE）的最佳速率。在这里，最优性适用于一般类四二次，其中还允许使用具有不同函数评估数量的自适应非线性算法。通过显示匹配界限给出最佳速率。首先，证明了$ o（n^{ - α-1/2}）$最差的RMSE的下限，其中$ n $表示对预期的功能评估次数的上限。事实证明，适当的随机梯形规则达到了这一速度，最高为对数因素。还提出了该梯形规则的实用误差估计器。数值结果支持我们的理论。
Randomized quadratures for integrating functions in Sobolev spaces of order $α\ge 1$, 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 $O(n^{-α-1/2})$ 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.
