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.

中文翻译：

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加权方差分析空间中的拟蒙特卡罗方法

在本文中，我们研究了准蒙特卡罗规则，用于逼近正则为 1 的加权 Sobolev 空间中函数的 $d$ 维单位立方体上的积分。虽然锚定 Sobolev 空间可以很好地理解这些规则的性质，但 ANOVA 空间并非如此，ANOVA 空间是准蒙特卡罗规则的另一种非常重要的参考空间类型。我们使用直接方法为加权方差分析空间中的函数提供了拟蒙特卡罗规则的最坏情况误差的公式。因此，我们根据采用的集成节点的加权差异限制了上面的最坏情况误差。另一方面，我们还根据使用的集成节点的数量 $n$ 获得了一般下限。