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：我们研究了多维单位立方体上不同光滑类加权函数空间中的拟蒙特卡洛（QMC）积分。考虑在独立随机选择的QMC点集（线性扰动的数字网或无限精度多项式格点集）下，通过多个积分估计的中位数来近似函数的积分。尽管该方法无需输入目标函数空间的光滑性和权重信息，我们仍能证明相应加权函数空间最坏情形误差的概率上界，其中失败概率随估计数量增加以指数速度收敛至零。对于有限光滑性的函数空间，我们获得的收敛速率接近最优；对于一类无穷可微函数，可实现与维数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——它无需根据函数空间的光滑性与权重进行调整，却依然能展现近乎最优的收敛速率。数值实验验证了我们的理论结果。