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

Abstract

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 $(t,s)$-sequences.

References in corpus (7)

• Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order
• Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions
• Discrepancy of generalized Hammersley type point sets in Besov spaces of dominating mixed smoothness
• Ergodic properties of β-adic Halton sequences
• The Supremum Norm of the Discrepancy Function: Recent Results and Connections
• Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2
• On the lower bound of the discrepancy of $(t,s)$ sequences: II

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• Optimal $L_p$-discrepancy bounds for second order digital sequences
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