可扩展网格：在空间填充曲线上的均匀采样
Extensible grids: uniform sampling on a space-filling curve
AI总结：
研究空间填充曲线生成的点在[0,1]^d中的性质，发现确定性采样具有O(n^{-1/d})的偏差，随机分层采样和 scrambling van der Corput 点在积分Lipshitz连续函数时有O(n^{-1-2/d})的均方误差，且优于 IID 采样。
摘要：
我们研究了在[0,1]^d中由Hilbert空间填充曲线生成的点的性质。对于确定性采样，我们得到d≥2时偏差为O(n^{-1/d})。对于随机分层采样和scrambled van der Corput点，当d≥3时，积分Lipshitz连续积分函数的均方误差为O(n^{-1-2/d})。这些速率与在d维网格上的采样相同，但随着d的增加而恶化。对于Lipshitz函数，该速率在该平滑度下是最佳的，优于普通IID采样。不同于网格，空间填充曲线采样可提供任意所需的样本量，且van der Corput版本可扩展n。此外，我们证明某些具有无限变分（按Hardy和Krause意义）的不连续函数可被积分，其均方误差为O(n^{-1-1/d})。此前仅知该速率为o(n^{-1})。其他空间填充曲线，如Sierpinski和Peano曲线，也达到这些速率，而Lebesgue曲线的上界稍差，如同维度为log₂(3)倍。
英文摘要：
We study the properties of points in $[0,1]^d$ generated by applying Hilbert's space-filling curve to uniformly distributed points in $[0,1]$. For deterministic sampling we obtain a discrepancy of $O(n^{-1/d})$ for $d\ge2$. For random stratified sampling, and scrambled van der Corput points, we get a mean squared error of $O(n^{-1-2/d})$ for integration of Lipshitz continuous integrands, when $d\ge3$. These rates are the same as one gets by sampling on $d$ dimensional grids and they show a deterioration with increasing $d$. The rate for Lipshitz functions is however best possible at that level of smoothness and is better than plain IID sampling. Unlike grids, space-filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in $n$. Additionally we show that certain discontinuous functions with infinite variation in the sense of Hardy and Krause can be integrated with a mean squared error of $O(n^{-1-1/d})$. It was previously known only that the rate was $o(n^{-1})$. Other space-filling curves, such as those due to Sierpinski and Peano, also attain these rates, while upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were $\log_2(3)$ times as high.
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