面向未指定光滑度及一般权重函数空间的免构造中位数拟蒙特卡洛规则
Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights
AI总结：
提出一种免构造的中位数 QMC 规则，通过随机生成向量或高阶多项式格点并取积分估计中位数，在任意光滑度和一般权重下近乎最优且概率随样本数指数趋于1。
摘要：
我们研究定义在多维单位立方体上的光滑函数的拟蒙特卡洛（QMC）积分。受 Pan 和 Owen 近期工作启发，我们研究一种新的免构造中位数 QMC 规则，它能够自适应地利用函数空间的光滑度和权重。对于加权 Korobov 空间，我们抽取 r 个独立的秩-1格点规则生成向量样本，对每个向量计算积分估计，并用这 r 个估计的中位数逼近真实积分。对于加权 Sobolev 空间，我们采用相同方法，但将秩-1格点规则替换为高阶多项式格点规则。与现有方法相比，一个主要优势在于我们无需通过计算机搜索算法构造优良的生成向量；同时，对于具有任意光滑度和权重的相应函数空间，我们的中位数 QMC 规则几乎能达到最优的最坏情形误差率，并且随着 r 增大，达到该误差率的概率以指数速度收敛至 1。数值实验阐明并支持了我们的理论发现。
英文摘要：
We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of $r$ independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these $r$ estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as $r$ increases. Numerical experiments illustrate and support our theoretical findings.
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。
