16,186 research outputs found
From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
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 -sequences
The inverse of the star-discrepancy problem and the generation of pseudo-random numbers
The inverse of the star-discrepancy problem asks for point sets of
size in the -dimensional unit cube whose star-discrepancy
satisfies where
is a constant independent of and . The first existence results in this
direction were shown by Heinrich, Novak, Wasilkowski, and Wo\'{z}niakowski in
2001, and a number of improvements have been shown since then. Until now only
proofs that such point sets exist are known. Since such point sets would be
useful in applications, the big open problem is to find explicit constructions
of suitable point sets .
We review the current state of the art on this problem and point out some
connections to pseudo-random number generators
- …