非均匀分区设计显著降低采样误差，提升高维积分精度
The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy
- 用非均匀分块替代传统随机抖动采样，优化点分布
- 理论证明新方法期望星偏差低于经典抖动采样
- 为高维数值积分提供更优的采样框架，适合算法研究者
我们研究了一类新型非均匀体积分块下的期望星偏差。主要贡献有两点：第一，建立了星偏差的强分块原理，证明新设计的非均匀体积分块生成的分层采样点集，其期望星偏差低于经典随机抖动采样，即 $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$，其中 $Y$ 和 $Z$ 分别代表抖动采样与我们的非均匀体积分块采样；第二，推导出该模型下期望星偏差的显式上界，优于现有抖动采样的上界。结果为高维数值积分中使用非均匀体积分块提供了理论支持。
原文摘要 · Abstract (English)
We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.
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