We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.

翻译：我们研究了加权Korobov空间中周期函数在随机化设定下的多元积分问题。本文提出了一种具有随机选取点数的新型随机化秩-1格点规则，该规则在寻找优质生成向量时避免了逐分量构造的需求，同时仍能实现接近最优的随机化误差收敛速率。我们的核心思路是利用以下事实：在确定性设定下，至少半数的可能生成向量能产生接近最优的最坏情况误差收敛速率。通过随机选取$r$次生成向量并比较其对应的最坏情况误差，我们能够以极高概率找到满足期望最坏情况误差界的生成向量，且（较小的）失败概率可通过将$r$按点数对数函数增长的方式加以控制。数值实验验证了我们的理论结果。