任意高阶交错加扰多项式格点规则的构造
Construction of interlaced scrambled polynomial lattice rules of arbitrary high order
AI总结：
提出使用加扰多项式格点集替代数字网，结合数字交错函数，实现高阶随机拟蒙特卡洛规则，并证明分量构造方法可达到最优收敛速率。
摘要：
高阶加扰数字网是随机拟蒙特卡洛规则，最近在 [J. Dick, Ann. Statist., 39 (2011), 1372--1398] 中引入，并证明在定义于 $s$ 维单位立方体上的光滑函数数值积分中，均方根误差达到最优收敛速率。其中的关键是将数字交错函数应用于随机加扰数字网的分量，其分量数为 $ds$，其中整数 $d$ 是所谓的交错因子。在本文中，我们将随机加扰数字网替换为随机加扰多项式格点集，这允许我们在仍达到最优收敛速率的同时，获得更好的维度依赖性。我们的结果适用于 Owen 的全加扰方案以及 Hickernell、Matoušek 和 Owen 研究的简化方案。我们考虑具有一般权重的加权函数空间，其元素具有直到 $α\ge 1$ 阶的平方可积部分混合导数，并推导了高阶加扰多项式格点规则估计量方差的上界。利用我们获得的界作为质量准则，我们证明了分量构造方法可用于获得良好多项式格点集的显式构造。通过首先构造基 $b$ 和维度 $ds$ 的经典多项式格点集，然后对其应用 $d$ 阶交错方案，我们在乘积权重情况下获得了算法构造代价为 $\mathcal{O}(dsmb^m)$ 次操作，使用 $\mathcal{O}(b^m)$ 内存，其中 $b^m$ 是多项式格点集中的点数。
英文摘要：
Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372--1398] and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the $s$-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is $ds$, where the integer $d$ is the so-called interlacing factor. In this paper, we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to $α\ge 1$, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base $b$ and dimension $ds$, to which we then apply the interlacing scheme of order $d$, we obtain a construction cost of the algorithm of order $\mathcal{O}(dsmb^m)$ operations using $\mathcal{O}(b^m)$ memory in case of product weights, where $b^m$ is the number of points in the polynomial lattice point set.
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