We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights $\gamma_1,\gamma_2,\ldots\in [0,1]$. Building upon the deterministic lattice algorithm by Kuo, Sloan, and Wo\'{z}niakowski (2006), we incorporate a randomized quadrature rule by Dick, Goda, and Suzuki (2022) to accelerate the convergence rate. This randomization involves drawing the number of points for function evaluations randomly, and selecting a good generating vector for rank-1 lattice points using the randomized component-by-component algorithm. We prove that our randomized algorithm achieves a worst-case root mean squared $L_2$-approximation error of order $M^{-\alpha/2 - 1/8 + \varepsilon}$ for an arbitrarily small $\varepsilon > 0$, where $M$ denotes the maximum number of function evaluations, and that the error bound is independent of the dimension $d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha} < \infty$. Our upper bound converges faster than a lower bound on the worst-case $L_2$-approximation error for deterministic rank-1 lattice-based approximation proved by Byrenheid, K\"{a}mmerer, Ullrich, and Volkmer (2017). We also show a lower error bound of order $M^{-\alpha/2-1/2}$ for our randomized algorithm, leaving a slight gap between the upper and lower bounds open for future research.

翻译：我们提出了一种随机化格点算法，用于逼近定义在 $d$ 维单位立方体上、来自混合光滑度 $\alpha > 1/2$ 且具有乘积权 $\gamma_1,\gamma_2,\ldots\in [0,1]$ 的加权 Korobov 空间中的多元周期函数。在 Kuo、Sloan 和 Wo\'{z}niakowski (2006) 提出的确定性格点算法基础上，我们引入了 Dick、Goda 和 Suzuki (2022) 的随机化求积法则以加速收敛速率。该随机化过程包括：随机抽取用于函数求值的点数，并利用随机化分量逐分量算法选取一个用于生成秩-1格点的优良生成向量。我们证明了，对于任意小的 $\varepsilon > 0$，我们的随机化算法实现了阶为 $M^{-\alpha/2 - 1/8 + \varepsilon}$ 的最坏情况均方根 $L_2$ 逼近误差，其中 $M$ 表示函数求值的最大次数；并且，如果权重满足 $\sum_{j=1}^\infty \gamma_j^{1/\alpha} < \infty$，该误差界与维度 $d$ 无关。我们的上界收敛速度，快于 Byrenheid、K\"{a}mmerer、Ullrich 和 Volkmer (2017) 所证明的、基于确定性秩-1格点逼近的最坏情况 $L_2$ 逼近误差的下界。我们还展示了我们的随机化算法具有阶为 $M^{-\alpha/2-1/2}$ 的下误差界，这在上界与下界之间留下了一个微小的间隙，有待未来研究。