一种快速概率逐分量构造秩1格的方法及其应用
A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications
摘要：
在过去的几十年里，一些越来越有效的组件-由-组件（CBC）的建设，适当的秩-1格的发展。一方面，存在基于最小化某些误差泛函的构造。另一方面，有可能构建秩-1格，其相应的体积规则正好整合了多元三角多项式空间内的所有元素。在本文中，我们重点介绍第二种方法，即，秩1格规则的精确性。的主要贡献是一个已知的算法，实现了CBC建设这样的秩-1格的概率版本的分析。事实证明，已知的确定性算法的计算工作量可以通过简单的随机化在平均上得到相当大的改善。此外，我们给出了一个详细的分析的计算成本相对于一定的故障概率，然后导致概率CBC算法的发展。特别是，所提出的方法将是非常有益的所谓的重建秩-1格，这是实际相关的函数逼近的建设。在对CBC算法进行严格分析的基础上，我们提出了一种算法，该算法以高概率确定重构合理格尺寸的秩1格。我们提供的估计所产生的晶格尺寸和边界上发生故障的概率。此外，我们还讨论了该算法的计算复杂性。各种数值试验说明了所提出的算法的效率。除其他外，我们演示了如何利用我们的算法的效率，甚至完全集成秩-1格的建设，提供了一定的属性的三角多项式的处理空间是已知的。
英文摘要：
Several more and more efficient component--by--component (CBC) constructions for suitable rank-1 lattices were developed during the last decades. On the one hand, there exist constructions that are based on minimizing some error functional. On the other hand, there is the possibility to construct rank-1 lattices whose corresponding cubature rule exactly integrates all elements within a space of multivariate trigonometric polynomials. In this paper, we focus on the second approach, i.e., the exactness of rank-1 lattice rules. The main contribution is the analysis of a probabilistic version of an already known algorithm that realizes a CBC construction of such rank-1 lattices. It turns out that the computational effort of the known deterministic algorithm can be considerably improved in average by means of a simple randomization. Moreover, we give a detailed analysis of the computational costs with respect to a certain failure probability, which then leads to the development of a probabilistic CBC algorithm. In particular, the presented approach will be highly beneficial for the construction of so-called reconstructing rank-1 lattices, that are practically relevant for function approximation. Subsequent to the rigorous analysis of the presented CBC algorithms, we present an algorithm that determines reconstructing rank-1 lattices of reasonable lattice sizes with high probability. We provide estimates on the resulting lattice sizes and bounds on the occurring failure probability. Furthermore, we discuss the computational complexity of the presented algorithm. Various numerical tests illustrate the efficiency of the presented algorithms. Among others, we demonstrate how to exploit the efficiency of our algorithm even for the construction of exactly integrating rank-1 lattices, provided that a certain property of the treated space of trigonometric polynomials is known.
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