We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.

翻译：本文研究在多个具有不同光滑度类别的加权函数空间中，多维单位立方体上的拟蒙特卡洛（QMC）积分。我们考虑通过多个积分估计值的中位数来逼近函数积分，这些估计值基于独立随机选择的QMC点集（包括线性扰动的数字网络或无限精度多项式格点集）。尽管我们的方法无需输入目标函数空间的光滑度和权重信息，但我们仍能证明相应加权函数空间最坏情况误差的概率上界，且随着估计次数增加，失败概率以指数速度收敛至零。对于有限光滑度的函数空间，我们获得的收敛速率接近最优；对于一类无穷可微函数，可实现与维度无关的超多项式收敛。这表明基于中位数的QMC规则具有通用性——无需根据函数空间的光滑度和权重进行调整，却能展现近乎最优的收敛速率。数值实验支持了我们的理论结果。