We propose two novel unbiased estimators of the integral $\int_{[0,1]^{s}}f(u) du$ for a function $f$, which depend on a smoothness parameter $r\in\mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1/2-r/s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1]^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.
翻译:我们建议使用两个新颖的公正估算元件$[0,1, ⁇ s ⁇ f(u) $(f) $(f), 函数美元, 这取决于一个顺畅的参数$r\ in\mathb{N}$。 第一个估算值完全融合了多度的美元p<r$, 并实现了最佳误差 $%-1/2-r/s}( 美元是评估次数的美元), 美元是持续差异的倍数。 第二个估算值在计算上比较便宜, 但仅限于在$[0, 1\s] 的边界上消失的函数。 两个估算值的构建取决于基于数字衍生物的立体分数和控制量值的组合。 我们提供了数字证据,表明它们表现良好, 即使中值为$, $( $) 。