论文标题
在加权方差分析空间中的准蒙特卡洛方法上
On Quasi-Monte Carlo Methods in Weighted ANOVA Spaces
论文作者
论文摘要
在本文中，我们研究了准蒙特卡罗规则，以近似于$ d $尺寸的单位立方体近似积分，以适用于加权Sobolev sobolev空间的功能。尽管这些规则的属性对于锚定的Sobolev空间已经很好地理解了，但对于ANOVA空间而言并非如此，这是Quasi-Monte Carlo规则的另一种非常重要的参考空间类型。 使用直接方法，我们为准方差分析空间的功能的准蒙特卡洛规则最坏情况下提供了公式。结果，我们根据所采用的集成节点的加权差异来限制上述最坏的错误。另一方面，我们还可以根据使用的集成节点的数量$ n $获得一般的下限。 对于一维情况，我们的结果导致最佳整合规则，在二维情况下，我们提供了产生最佳收敛率的规则。
In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules. Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number $n$ of used integration nodes. For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.
