Construction algorithms for higher order polynomial lattice rules
Construction algorithms for higher order polynomial lattice rules
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高阶多项式格规则的构造算法
DOI：
10.1016/j.jco.2010.06.002
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
F. Pillichshammer
中科院分区：
文献类型：
--
作者：
Jan Baldeaux;J. Dick;Julia Greslehner;F. Pillichshammer
高阶多项式格点集是一种特殊类型的数字高阶网络，当在拟蒙特卡罗算法中用于近似单位立方体上的高维积分时，它可以达到几乎最优的收敛速度。高阶多项式格点集的“好”质量的存在性已被证实，但其构造问题尚未得到解决。我们使用一种逐分量的方法构造高阶多项式格规则，对任意高平滑的函数实现最优收敛速率，同时在一定的权重条件下-（强）多项式可跟踪性。将这种方法与筛型算法相结合，可以得到高阶多项式点阵规则，该规则可以根据被积函数的平滑程度调整到一定程度。高阶Korobov多项式格规则得到了类似的结果。
Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of “good” quality has recently been established, but their construction was not addressed. We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time–under certain conditions on the weights–(strong) polynomial tractability. Combining this approach with a sieve-type algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.