Low-discrepancy point sets for non-uniform measures
Low-discrepancy point sets for non-uniform measures
复制标题
非均匀测量的低差异点集
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. Dick
中科院分区：
文献类型：
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作者：
C. Aistleitner;J. Dick
本文证明了d维单位立方体上关于任意非一致测度mu的低偏差点集的存在性的几个结果。本文改进了Beck的一个定理，证明了对任意$d geq 1$，$N geq 1，$和[0，1]^d$上的任意非负的标准化Borel测度$mu$，在[0，1]^d $上存在一个点集$x_1，dots，x_N，其关于$mu$的星差为$$ D_N^*（x_1，点，x_N; mu）ll压裂{（log N）^{（3d+1）/2}}{N}。为了证明，我们使用了Banaszczyk关于向量平衡的定理，这意味着超图的线性差异的上界。此外，理论的大偏差范围内的经验过程索引集进行了讨论，我们证明了一个数值显式的上限为Vapnik-v{C}ervonenkis类的差异的逆。最后，使用最近版本的Koksma-Hlawka不等式由于Brandolini，Colzani，Gigante和Travaglini，我们表明，我们的研究结果意味着存在的体积规则产生快速收敛速度的数值积分的功能具有一定形式的不连续性。
In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for any $d geq 1$, $N geq 1,$ and any non-negative, normalized Borel measure $mu$ on $[0,1]^d$ there exists a point set $x_1, dots, x_N in [0,1]^d$ whose star-discrepancy with respect to $mu$ is of order $$ D_N^*(x_1, dots, x_N; mu) ll frac{(log N)^{(3d+1)/2}}{N}. $$ For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik--v{C}ervonenkis classes. Finally, using a recent version of the Koksma--Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.