发表机构
Catholic University of Eichstätt–Ingolstadt; UNSW Sydney; University of Bonn; Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（天主教艾希施泰特-因戈尔施塔特大学; 新南威尔士大学悉尼分校; 波恩大学; 弗劳恩霍夫算法与科学计算研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于周期化和格点核插值的高阶数值积分方法，用于高维非周期函数，达到接近最优的收敛速率，并可通过插值权衡收敛速度与维度增长。
AI 中文摘要
我们提出了一种新颖的数值积分方法，用于高维单位立方体 $[0,1]^s$ 上的非周期函数，该方法通过使用变换后的格点构造的（非等权）加权求积规则，并可选地采用子采样。该方法针对混合导数直到 $\alpha$ 阶关于乘积 Chebyshev 密度平方可积的被积函数设计。其组成部分包括：(i) 对被积函数进行周期化，(ii) 在格点处用核插值逼近光滑周期函数，(iii) 精确积分核插值与不光滑密度的乘积，并为 (ii) 和 (iii) 提供子采样替代方案。以 $|J|$ 表示函数求值次数，我们实现了接近 $|J|^{-(\alpha-1/2)}$ 和 $|J|^{-\alpha}\\,($1.110721$)^s$ 的收敛速率，其中隐含常数在有利条件下与 $s$ 无关。我们可以对这两个结果进行插值，以在收敛速率和随 $s$ 的增长之间进行权衡。我们提供了数值实验来验证我们的理论。
英文摘要
We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube $[0,1]^s$ by a specially crafted (unequally-)weighted quadrature rule using transformed lattice points, with the additional option of subsampling. The method is designed for integrands whose mixed derivatives up to order $α$ are square-integrable with respect to a product Chebyshev density. The ingredients are (i) periodizing the integrand, (ii) approximating the smooth periodic function by a kernel interpolant at lattice points, (iii) integrating exactly the product of the kernel interpolant and the nonsmooth density, together with subsampling alternatives for (ii) and~(iii). With $|J|$ corresponding to the number of function evaluations, we achieve the convergence rate close to the order $|J|^{-(α-1/2)}$ and $|J|^{-α}\,($1.110721$)^s$, where the implied constants are independent of $s$ under favorable conditions. We can interpolate these two results to trade between the convergence rate and the growth in $s$. We provide numerical experiments to demonstrate our theory.
