Original articles
Recent constructions of low-discrepancy sequences

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Abstract

We present a survey of the recently developed theory of (u,e,s)-sequences which has led to new constructions of low-discrepancy sequences. We also review recent constructions of low-discrepancy sequences by means of ergodic theory.

Introduction

Low-discrepancy sequences are basic ingredients of quasi-Monte Carlo methods for numerical integration. The standard error bounds for quasi-Monte Carlo integration, such as the Koksma–Hlawka inequality (see  [9, Section 2.5]), guarantee a rate of convergence faster than the Monte Carlo rate of convergence if the integration nodes are taken from a low-discrepancy sequence. The widespread use of quasi-Monte Carlo methods in areas like scientific computing, computational finance, and computer graphics implies a heavy demand for concrete sequences with as low a discrepancy as possible. The explicit construction of low-discrepancy sequences has a long history, with the first construction of a low-discrepancy sequence for any dimension dating back to 1960 (see Halton  [6] and Section  3). Many exciting developments in this area have taken place since then, and in the present paper we survey construction principles that were found in the last few years. For a general background on low-discrepancy sequences and quasi-Monte Carlo methods, we refer to the monographs  [2] and  [11].

Let s1 be a given dimension and let P be a point set consisting of N points x1,,xN in the s-dimensional unit cube [0,1]s. For an arbitrary subinterval J of [0,1]s, we introduce the local discrepancyR(J;P)=A(J;P)Nλs(J), where A(J;P) is the number of integers n with 1nN such that xnJ and λs denotes the s-dimensional Lebesgue measure. Then we define the star discrepancyDN(P)=supJ|R(J;P)|, where the supremum is extended over all subintervals J of [0,1]s with one vertex at the origin. For an infinite sequence S of points in [0,1]s and any integer N1, let DN(S) be the star discrepancy of the point set consisting of the first N terms of S. We say that S is a low-discrepancy sequence if DN(S)=O(N1(logN)s)for all  N2, where the implied constant is independent of N. The asymptotic order of magnitude N1(logN)s is the best that can currently be achieved for the star discrepancy of an infinite sequence of points in [0,1]s. We refer to  [2, Section 3.2] for a survey of known lower bounds on the star discrepancy.

The emphasis in this paper is on types of low-discrepancy sequences that are called (u,e,s)-sequences. The theory of (u,e,s)-sequences was developed very recently, and we present the basics of this theory in Section  2. The principal constructions of (u,e,s)-sequences, which use global function fields, are described in Section  3. Some other recent constructions of low-discrepancy sequences are reviewed in Section  4.

Section snippets

(u,e,s)-sequences

Many of the classical constructions of low-discrepancy sequences, such as the constructions of Sobol’ sequences  [29], Faure sequences  [3], Niederreiter sequences  [17], generalized Niederreiter sequences  [31], and Niederreiter–Xing sequences  [23], [33], are based on the theory of (t,m,s)-nets and (t,s)-sequences introduced in  [16]. Expositions of this theory can be found in the monographs  [2] and  [18] as well as in the recent handbook article  [20]. Therefore we will just recall the

Constructions using global function fields

We already mentioned Niederreiter sequences and generalized Niederreiter sequences in Section  2. These sequences were originally constructed as digital (t,s)-sequences over a finite field Fq, and it was realized by Tezuka  [32] that they are also digital (0,e,s)-sequences over Fq with a choice of eNs inherent in the construction (compare also with the paragraph following Remark 2). The construction of (generalized) Niederreiter sequences makes heavy use of the rational function field over Fq.

Miscellaneous constructions

In recent years there has been an increased interest in the construction of low-discrepancy sequences by means of ergodic theory. It is known for a long time (see e.g. Lambert  [10]) that the classical van der Corput sequences can be generated by ergodic transformations. The easiest way to produce the van der Corput sequence (ϕb(n))n=0 in base b via an ergodic transformation is to consider the ring Zb of b-adic integers and the transformation Tb:ZbZb which is given by addition of the element 1

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