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.

Vandermonde matrices with nodes in the unit disk and the large sieve

“…. = |z K | = A, we not only get a much larger range of validity for our upper bound (54) on κ(V N ×K ) than for the general upper bound (52), but we also obtain sharper bounds on σ 2 min (V N ×K ) and σ 2 max (V N ×K ) as the corresponding results are based on (36), which, as pointed out in [53], is best possible. One would hope that the constant 42/π in (35) could be improved to be closer to the corresponding constant 1/2 in (25) or that 42/π could be turned into a smaller constant which would possibly depend on min 1 k K |z k | and/or max 1 k K |z k | as in the Graham-Vaaler result (36).…”

“…A(x, y, z) = 1/(z(e 2x/z − 1)) B(x, y, z) = e 2x/z /(z(e 2x/z − 1)), for x > 0, y > 0, and z > 0, to (36) then yields (42).…”

“…Van der Corput sequences are used, e.g., in quasi-Monte Carlo simulation algorithms [36] and are known to have excellent uniform distribution properties. It is shown in [22,Cor.…”

“…. = |z K | = A, however, our bound (54) is based on the Graham-Vaaler result (36) for which the (non-optimal) constant 42/π does not appear. Specifically, the asymptote (in N , with K fixed) of our bound (54) satisfies κ a A 1/2−1/δ (w) .…”

Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -adic Halton Sequences

“…A large number of other variants of van der Corput and Halton sequences has been studied in the literature and for many of them strong discrepancy bounds can be derived (see e.g. [12,18,19,21,26,27,38]).…”