SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 229-247, February 2024.
Abstract. We propose two novel unbiased estimators of the integral [math] for a function [math], which depend on a smoothness parameter [math]. The first estimator integrates exactly the polynomials of degrees [math] and achieves the optimal error [math] (where [math] is the number of evaluations of [math]) when [math] is [math] times continuously differentiable. The second estimator is also optimal in terms of convergence rate and has the advantage of being computationally cheaper, but it is restricted to functions that vanish on the boundary of [math]. The construction of the two estimators relies on a combination of cubic stratification and control variates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of [math].

中文翻译：

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通过三次分层的高阶蒙特卡罗

SIAM 数值分析杂志，第 62 卷，第 1 期，第 229-247 页，2024 年 2 月。
摘要。我们提出了函数 [math] 的积分 [math] 的两个新颖的无偏估计器，它依赖于平滑参数 [math]。第一个估计器精确积分次数 [math] 的多项式，并在 [math] 为 [math] 倍连续可导时实现最佳误差 [math]（其中 [math] 是 [math] 的评估次数）。第二个估计器在收敛速度方面也是最优的，并且具有计算成本更低的优点，但它仅限于在[数学]边界上消失的函数。这两个估计量的构造依赖于三次分层和基于数值导数的控制变量的组合。我们提供的数字证据表明，即使对于中等的[数学]值，它们也表现出良好的性能。