加权Walsh空间中好多项式格规则的逐分量逐位构造
Component-by-component digit-by-digit construction of good polynomial lattice rules in weighted Walsh spaces
摘要：
我们考虑多项式格规则，这是所谓的拟蒙特卡罗方法（QMC）规则的特殊情况下，有效的建设。这些对于多维积分的近似计算特别感兴趣，其中维度$d$可以是数百或数千。我们研究了一种构造方法，组装的生成向量，这是在这种情况下，向量的多项式在一个有限域，多项式格规则在一个数字的数字（或等价地，系数的系数）的方式。正如我们将展示的，相应的QMC规则的积分误差达到了很好的收敛阶，并且在适当的条件下，我们可以通过考虑配备有坐标权重的函数空间来克服维数灾难。构造算法基于独立于函数空间的潜在平滑度的质量度量，并且可以以快速方式（不使用快速傅立叶变换）实现。此外，我们用大量的数值结果说明了我们的发现。
英文摘要：
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.
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