用于加权瓦尔什空间中数值积分的优良交错多项式格点规则
Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces
AI总结：
本文研究了使用多项式格点集进行交错组件的数值积分规则，推导了两种交错多项式格点规则的最坏情况误差上界，并探讨了组件-by-组件构造和科罗波夫构造以获得最优误差率的方法。
摘要：
使用高阶数字网和序列的求积规则已被证明可以利用函数的光滑性进行数值积分，并在光滑函数上实现比经典数字网和序列更优的收敛速度。在[J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176]中引入了一种基于数字交错函数的高阶数字网和序列的构造原理，该方法交错经典数字网或序列，其组件数量是维度的倍数。本文研究了使用多项式格点集进行交错组件的应用。我们称使用此类点集的求积规则为“交错多项式格点规则”。我们考虑包含光滑函数的加权瓦尔什空间，并推导出两种交错多项式格点规则的最坏情况误差上界，这两种上界均可作为构造交错多项式格点规则的质量标准。我们探讨了组件-by-组件构造和科罗波夫构造作为显式构造优良交错多项式格点规则的方法，以达到最坏情况误差的最优速率。通过这种方法，我们能够在某些权重条件下获得最坏情况误差界对维度的良好依赖性，同时与高阶多项式格点规则相比显著降低了构造成本。
英文摘要：
Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176], which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension. In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets {\em interlaced polynomial lattice rules}. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worst-case error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.
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