From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

详细信息    查看全文

• 作者：Henri Faure^a ; ^{henri.faure@univ-amu.fr" class="auth_mail" title="E-mail the corresponding author} ; Peter Kritzer^b ; ^{peter.kritzer@jku.at" class="auth_mail" title="E-mail the corresponding author} ; Friedrich Pillichshammer^b ; ^{friedrich.pillichshammer@jku.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Van der Corput sequence ; Uniform distribution ; Discrepancy ; Bounded remainder sets ; von Neumann&ndash ; Kakutani transform ; Halton sequence ; Hammersley point set ; (t ; s)(t ; s)-sequences
• 刊名：Indagationes Mathematicae
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：26
• 期：5
• 页码：760-822
• 全文大小：679 K

文摘

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results.

On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput’s ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter’s (t,s)$\left(t\text{,}s\right)$-sequences.