Construction of interlaced polynomial lattice rules for infinitely differentiable functions
Construction of interlaced polynomial lattice rules for infinitely differentiable functions
复制标题
无限可微函数的交错多项式格规则的构造
DOI：
10.1007/s00211-017-0882-x
复制
发表时间：
2017
影响因子：
2.1
通讯作者：
Yoshiki Takehito
中科院分区：
文献类型：
--
作者：
Dick Josef;Goda Takashi;Suzuki Kosuke;Yoshiki Takehito
研究了无限可微函数加权空间中多维单位立方上的多元积分问题。由Suzuki最近的结果可知，存在一个很好的准蒙特卡罗（QMC）规则，在该函数空间中实现了最坏情况误差的超多项式收敛，并且在一定的权值条件下，这种收敛行为与维数无关。在本文中，我们提供了一种建设性的方法来寻找一个好的QMC规则，以实现这种与最坏情况误差无关的超多项式收敛。具体地说，我们证明了交错多项式格规则可以用一种快速的逐分量算法在最多的算术运算中构造出与维无关的超多项式收敛性，交错多项式格规则可以根据点的个数和权值选择合适的交错因子。证明最坏情况误差界的关键思想是使用詹森不等式的一种变体，并带有专门设计的凹函数。
We study multivariate integration over thes-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of pointsNand the weights, can be constructed using a fast component-by-component algorithm in at mostarithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen’s inequality with a purposely-designed concave function.