发表机构
School of Mathematical Sciences, Xiamen University; Department of Mathematics, City University of Hong Kong（厦门大学数学科学学院; 香港城市大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究球面上的嵌套QMC设计，证明在$s>d/2$时等权积分率与嵌套点集兼容，并提出基于块敏感估计的补全定理，实现所有$s\ge d$的最优速率嵌套设计。
AI 中文摘要
嵌套求积规则在多层和自适应积分中很自然，其中每次细化保留所有先前使用的节点，从而重用先前的函数求值。对于每个固定的$s>d/2$，我们证明了$\nathbb S^d$上的等权QMC积分率与这样的嵌套点集兼容。在亚临界范围$d/2<s<d$内，几何增长的QMC块的累积并集产生嵌套QMC设计序列，其连续基数之比收敛到任意给定的$\rho>1$。在临界指数$s=d$及以上，其中块平均估计不再产生最优速率，我们基于低频差异抵消证明了一个等权补全定理。一个块敏感的估计改进了迭代，并为所有$s\ge d$生成了嵌套QMC设计，满足$N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$。每个给定的有限点集也允许最优速率的补全。
英文摘要
Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2<s<d$, cumulative unions of geometrically growing QMC blocks yield nested QMC design sequences whose successive cardinality ratios converge to any prescribed $ρ>1$. At and above the critical index $s=d$, where the block-averaging estimate no longer yields the optimal rate, we prove an equal-weight completion theorem based on low-frequency discrepancy cancellation. A block-sensitive estimate sharpens the iteration and yields nested QMC designs for all $s\ge d$ with $N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$. Every prescribed finite point set also admits an optimal-rate completion.
Comments12 pages
