用中位数均值法提升高维积分精度，尤其在大样本时表现更优。
Median of Means Sampling for the Keister Function
- 采用中位数均值替代传统均值，增强采样稳定性。
- 样本量超1000时，中位数均值误差显著低于传统方法。
- 适合高维数值积分，尤其适用于大样本场景的科研计算。
本研究对比了中位数均值采样与传统均值均值采样在随机准蒙特卡洛（RQMC）方法中计算凯斯特函数积分的表现。实验使用格点与数字网作为点分布，在2、3、5和8维下测试了从2^8到2^19个样本点的规模。结果表明，当样本量大于10^3时，中位数均值采样始终优于传统均值；而小样本时，尤其是数字网情形下，传统均值更具精度。研究验证了此前理论预测——中位数均值在大样本下优势明显，并反映了高维积分中保持精度的固有挑战。这些发现支持将中位数均值作为数值积分中一种有前景的替代方法，但其在样本量与维度上的局限性仍需通过更多测试函数和更大参数空间进一步探索。
原文摘要 · Abstract (English)
This study investigates the performance of median-of-means sampling compared to traditional mean-of-means sampling for computing the Keister function integral using Randomized Quasi-Monte Carlo (RQMC) methods. The research tests both lattice points and digital nets as point distributions across dimensions 2, 3, 5, and 8, with sample sizes ranging from 2^8 to 2^19 points. Results demonstrate that median-of-means sampling consistently outperforms mean-of-means for sample sizes larger than 10^3 points, while mean-of-means shows better accuracy with smaller sample sizes, particularly for digital nets. The study also confirms previous theoretical predictions about median-of-means' superior performance with larger sample sizes and reflects the known challenges of maintaining accuracy in higher-dimensional integration. These findings support recent research suggesting median-of-means as a promising alternative to traditional sampling methods in numerical integration, though limitations in sample size and dimensionality warrant further investigation with different test functions and larger parameter spaces.
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