SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024.
Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional 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.

中文翻译：

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通用中值拟蒙特卡罗积分

SIAM 数值分析杂志，第 62 卷，第 1 期，第 533-566 页，2024 年 2 月。
摘要。我们研究了具有不同平滑度类别的多个加权函数空间中多维单位立方体上的拟蒙特卡罗（QMC）积分。我们考虑在独立和随机选择基础 QMC 点集（线性置乱数字网络或无限精度多项式格点集）的情况下，通过几个积分估计的中值来近似函数的积分。尽管我们的方法不需要关于目标函数空间的平滑度和权重的任何信息作为输入，但我们可以证明相应加权函数空间的最坏情况误差的概率上限，其中故障概率收敛到 0随着估计数量的增加呈指数增长。对于具有有限光滑度的函数空间，我们获得的收敛率几乎是最优的，并且对于一类无限可微函数，我们可以获得与维数无关的超多项式收敛。这意味着我们的基于中值的 QMC 规则是通用的，因为它不需要根据函数空间的平滑度和权重进行调整，但仍表现出接近最优的收敛速度。数值实验支持我们的理论结果。