发表机构
Illinois Institute of Technology（伊利诺伊理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对任意样本量的可扩展格序列，建立了品质因子 $P_\alpha$ 的上界，发现限定 $n$ 为 $\lambda b^p$ 时可恢复最优衰减，还得到 $P_{\alpha,2}$ 的上界并证实最优样本权重可降低求积误差。
AI 中文摘要
可扩展格是通过样本均值近似多维积分时常用的节点选择，通常使用形如 $n = b^m$ 的样本量，其中 $b$ 为素数，$m$ 为非负整数。然而，计算时间预算或硬件故障导致的提前终止可能使我们无法选择最优样本量。本文中，我们针对**任意 $n$ 的可扩展格序列**，在巴拿赫空间框架下以最坏情况误差为依据，建立了品质因子 $P_\alpha$ 的上界。我们证明，对于一般 $n$，$P_\alpha$ 的上界衰减速度不会快于 $\mathcal{O}(n^{-1})$；但当 $n$ 限定为 $n = \lambda b^p$（$\lambda$ 为固定整数）时，随着 $p$ 趋于无穷，我们可恢复接近 $\mathcal{O}(n^{-\alpha})$ 的最优衰减。我们还研究了与积分算子希尔伯特空间最坏情况误差相关的另一品质因子 $P_{\alpha,2}$，在样本权重相等的条件下，我们得到了以 $P_\alpha$ 表示的 $P_{\alpha,2}$ 的理论上界，并通过数值计算表明，最优样本权重可进一步降低求积误差界。
英文摘要
Extensible lattices are a common choice of nodes for approximating multidimensional integrals by a sample mean, typically using sample sizes of the form $n = b^m$ for a prime base $b$ and non-negative integer $m$. However, a computational time budget or early termination due to hardware failure may prevent us from choosing the preferred number of samples. In this paper, we establish an upper bound on the figure of merit $P_α$ for extensible lattice sequences \emph{with arbitrary $n$}, derived as a worst-case error in a Banach space setting. We show that although the upper bound on $P_α$ can decay no faster than $\mathcal O(n^{-1})$ for general $n$, restricting $n$ to the form $n =λb^p$, where $λ$ is a fixed integer, we can recover the optimal decay of nearly $\mathcal O(n^{-α})$ as $p \to \infty$. We also investigate a related figure of merit, $P_{α,2}$, arising as a worst-case error for Hilbert spaces of integrands. We obtain a theoretical upper bound on $P_{α,2}$ in terms of $P_α$ for equal sample weights and show numerically that optimal sample weights can improve the cubature error bound even further.
