随机网局部对偶抽样
Local antithetic sampling with scrambled nets
摘要：
我们考虑的问题，计算近似的积分$I=\int_{[0,1]^d}f(x) dx$。蒙特卡罗（MC）抽样通常会从$n$独立随机函数评估中获得$O(n^{-1/2})$的均方误差（RMSE）。相比之下，使用仔细等间距的评估点的准蒙特卡罗（QMC）采样可以针对任何$\varepsilon>0$获得速率$O(n^{-1+\varepsilon})$，并且随机化QMC（RQMC）可以获得RMSE $O(n^{-3/2+\varepsilon})$，两者都在$f$上的温和条件下。经典的MC方差缩减方法可以适用于QMC。发表的结果结合QMC与重要性抽样和控制变量发现值得改进，但没有改变的错误率。本文对经典的对偶抽样方差缩减方法进行了推广，并将其与RQMC相结合。一个这样的方法被证明带来了适度的改善RMSE率，达到$O(n^{-3/2-1/d+\varepsilon})$的任何$\varepsilon>0$，足够平滑的$f$。
英文摘要：
We consider the problem of computing an approximation to the integral $I=\int_{[0,1]^d}f(x) dx$. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of $O(n^{-1/2})$ from $n$ independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate $O(n^{-1+\varepsilon})$ for any $\varepsilon>0$ and randomized QMC (RQMC) can attain the RMSE $O(n^{-3/2+\varepsilon})$, both under mild conditions on $f$. Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining $O(n^{-3/2-1/d+\varepsilon})$ for any $\varepsilon>0$, for smooth enough $f$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
