基于Hilbert空间填充曲线分层抽样的星偏差界及其应用
Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications
AI总结：
本文研究基于Hilbert空间填充曲线抽样的概率星偏差上界，将收敛阶从$O(N^{-1/2})$提升至$O(N^{-1/2-1/(2d)})$，并给出加权空间积分误差界及Koksma-Hlawka不等式改进等应用。
摘要：
本文考虑基于Hilbert空间填充曲线抽样的概率星偏差上界。该问题源于多元积分近似，但主要结果去除了经典基于网格的抖动采样对采样数的严格条件。主要内容分为三个部分。首先，我们继承了这种新采样的优点，获得了比使用Monte Carlo采样更好的随机星偏差上界。此外，上界的收敛阶从$O(N^{-\ rac{1}{2}})$提高到$O(N^{-\ rac{1}{2}-\ rac{1}{2d}})$。其次，得到了加权空间中函数更好的均匀积分近似误差界。第三，将给出其他应用，例如Hilbert空间中的采样定理以及经典Koksma-Hlawka不等式的改进。最后，该思想也可应用于星偏差版本的强划分原理的证明。
英文摘要：
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict conditions on the sampling number of the classical grid-based jittered sampling. The main content has three parts. First, we inherit the advantages of this new sampling and achieve a better upper bound of the random star discrepancy than the use of Monte Carlo sampling. In addition, the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$ to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integral approximation error bound of the function in the weighted space is obtained. Third, other applications will be given. Such as the sampling theorem in Hilbert spaces and the improvement of the classical Koksma-Hlawka inequality. Finally, the idea can also be applied to the proof of the strong partition principle of the star discrepancy version.
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