发表机构
Tsinghua University; Zhejiang University（清华大学; 浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究结合粗置乱与中位数技巧，提出一种鲁棒的高维积分RQMC方法，实现了维度无关的误差界及近最优收敛速率，并通过数值实验验证了其有效性。
AI 中文摘要
我们研究使用随机化拟蒙特卡洛（RQMC）方法对高维积分的数值逼近，重点关注置乱Sobol'序列。虽然渐近速度比蒙特卡洛快，但经典RQMC的误差界随维度$s$呈指数增长，其均方根误差（RMSE）收敛速率通常不优于$O(N^{-3/2})$。为克服这些限制，我们结合了两项近期进展：Suzuki的粗置乱和中位数技巧。粗置乱根据Sobol'序列的生成基多项式对其进行随机化，并显著降低最大增益系数。通过适当选择基多项式，我们证明该系数可以在$s$中一致有界，从而对于$L^2$被积函数，RMSE达到$O(N^{-1/2})$，且常数与维度无关。中位数技巧则使得对于超出$L^2$的函数类实现近乎最优收敛：对于$p\in(1,2)$的$L^p$被积函数，独立粗置乱估计的中位数以高概率达到$O(N^{-1+1/p})$的误差；对于Haar小波空间$\mathcal H_{\mathrm{wav},\alpha,s,p,q}$中的被积函数，其中$\alpha_p:=\alpha-(1/p-1/2)_+>0$，我们证明对任意$\varepsilon>0$，高概率误差界为$O(N^{-\alpha_p-1/2+\varepsilon})$；当被积函数具有属于$\mathcal H_{\mathrm{wav},\alpha,s,p,q}$的$r\ge1$阶混合主导导数时，该界改进为$O(N^{-r-\alpha_p-1/2+\varepsilon})$。后两个界在关于被积函数ANOVA分量的适当条件下对$s$一致。数值实验证实了预测的收敛速率，并展示了所提方法的鲁棒性。
英文摘要
We study the numerical approximation of high-dimensional integrals using randomized quasi-Monte Carlo (RQMC) methods, with a focus on scrambled Sobol' sequences. While asymptotically faster than Monte Carlo, classical RQMC suffers from error bounds that grow exponentially in the dimension $s$, and its root mean squared error (RMSE) convergence rate is generally no better than $O(N^{-3/2})$. To overcome these limitations, we combine two recent developments: Suzuki's coarse scrambling and the median trick. Coarse scrambling randomizes Sobol' sequences according to their generating base polynomials and significantly reduces the maximal gain coefficient. By appropriately choosing the base polynomials, we show that the coefficient can be made uniformly bounded in $s$, yielding an $O(N^{-1/2})$ RMSE for $L^2$ integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond $L^2$: for $L^p$ integrands with $p\in(1,2)$, the median of independent coarsely scrambled estimates achieves an error of $O(N^{-1+1/p})$ with high probability; for integrands in the Haar wavelet space $\mathcal H_{\mathrm{wav},α,s,p,q}$ with $α_p:=α-(1/p-1/2)_+>0$, we prove a high-probability error bound of $O(N^{-α_p-1/2+\varepsilon})$ for any $\varepsilon>0$; the bound improves to $O(N^{-r-α_p-1/2+\varepsilon})$ when the integrand has dominating mixed derivatives of order $r\ge1$ that belong to $\mathcal H_{\mathrm{wav},α,s,p,q}$. The latter two bounds are uniform in $s$ under suitable conditions on the ANOVA components of the integrands. Numerical experiments confirm the predicted convergence rates and demonstrate the robustness of the proposed approach.
