论文标题
高阶随机整合通过立方分层
Higher-order stochastic integration through cubic stratification
论文作者
论文摘要
我们建议对功能$ f $的积分$ \ int _ {[0,1]^{s}} f（u）du $的两个新颖的无偏估计量，这取决于平滑度参数$ r \ in \ mathbb {n} $。第一个估计器完全集成了度的多项式$ p <r $，并达到最佳错误$ n^{ - 1/2-r/s} $（其中$ n $是$ f $的评估数）是$ f $ $ f $ $ r $ $ r $ ru $ a $连续可相互不同的次数。第二个估计器在计算上更便宜，但仅限于在$ [0,1]^s $的边界上消失的功能。两个估计量的构建依赖于基于数值衍生物的立方分层和控制膜的组合。我们提供数值证据表明，即使对于中等值$ n $，它们也表现出良好的性能。
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 $n$.
