论文标题
无施工的中值准蒙特卡罗标准Carlo规则，用于具有未指定平稳性和一般重量的功能空间
Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights
论文作者
论文摘要
我们研究了在多维单元立方体上定义的平滑功能的准蒙特卡洛（QMC）集成。受Pan和Owen最近的工作的启发，我们研究了一项新的无施工中位数QMC规则，该规则可以自适应地利用功能空间的平稳性和权重。对于加权的Korobov空间，我们绘制了$ r $独立生成级别晶格规则的vectors的样本，计算每个级别的积分估计值，并近似这些$ r $估计值的中位数的真实积分。对于加权的Sobolev空间，我们使用相同的方法，但是用高阶多项式晶格规则代替了等级-1晶格规则。比现有方法的主要优势是，我们不需要通过计算机搜索算法来构建良好的生成向量，而我们的中位数QMC规则几乎实现了各个功能空间的最佳最差案例错误率，并具有任何平滑度和权重，并且概率呈额定范围，即$ r $增加。数值实验说明并支持我们的理论发现。
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.
