论文标题
良好多项式晶格规则在加权沃尔什空间
Component-by-component digit-by-digit construction of good polynomial lattice rules in weighted Walsh spaces
论文作者
论文摘要
我们考虑了多项式晶格规则的有效构建，这些规则是所谓的准蒙特卡洛（QMC）规则的特殊情况。这些对于多元积分的近似计算而言是特别感兴趣的，其中尺寸$ d $可能是数百或数千个。我们研究了一种组装生成矢量的构造方法，在这种情况下，这是在有限磁场上，多项式晶格规则的多项式矢量，以数字数字（或等效地，系数为单位）的方式。正如我们将要显示的那样，相应的QMC规则的集成误差达到了出色的收敛顺序，在适当的条件下，我们可以通过考虑配备有坐标重量的功能空间来消除维度的诅咒。构造算法基于一种质量度量，该质量度量与功能空间的潜在平滑度无关，并且可以快速实现（而无需使用快速傅立叶变换）。此外，我们以广泛的数值结果来说明我们的发现。
We consider the efficient construction of polynomial lattice rules, which are special cases of so-called quasi-Monte Carlo (QMC) rules. These are of particular interest for the approximate computation of multivariate integrals where the dimension $d$ may be in the hundreds or thousands. We study a construction method that assembles the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As we will show, the integration error of the corresponding QMC rules achieves excellent convergence order, and, under suitable conditions, we can vanquish the curse of dimensionality by considering function spaces equipped with coordinate weights. The construction algorithm is based on a quality measure that is independent of the underlying smoothness of the function space and can be implemented in a fast manner (without the use of fast Fourier transformations). Furthermore, we illustrate our findings with extensive numerical results.
