Original articlesRecent constructions of low-discrepancy sequences
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 be a given dimension and let be a point set consisting of points in the -dimensional unit cube . For an arbitrary subinterval of , we introduce the local discrepancy where is the number of integers with such that and denotes the -dimensional Lebesgue measure. Then we define the star discrepancy where the supremum is extended over all subintervals of with one vertex at the origin. For an infinite sequence of points in and any integer , let be the star discrepancy of the point set consisting of the first terms of . We say that is a low-discrepancy sequence if where the implied constant is independent of . The asymptotic order of magnitude is the best that can currently be achieved for the star discrepancy of an infinite sequence of points in . 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 -sequences. The theory of -sequences was developed very recently, and we present the basics of this theory in Section 2. The principal constructions of -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
-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 -nets and -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 -sequences over a finite field , and it was realized by Tezuka [32] that they are also digital -sequences over with a choice of 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 .
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 in base via an ergodic transformation is to consider the ring of -adic integers and the transformation which is given by addition of the element
References (33)
- et al.
A variant of Atanassov’s method for -sequences and -sequences
J. Complexity
(2014) - et al.
A construction of -sequences with finite-row generating matrices using global function fields
Finite Fields Appl.
(2013) Quasi-Monte Carlo, low-discrepancy sequences, and ergodic transformations
J. Comput. Appl. Math.
(1985)Low-discrepancy and low-dispersion sequences
J. Number Theory
(1988)Factorization of polynomials and some linear-algebra problems over finite fields
Linear Algebra Appl.
(1993)- et al.
Low-discrepancy sequences and global function fields with many rational places
Finite Fields Appl.
(1996) Distribution of points in a cube and approximate evaluation of integrals
USSR Comp. Math. Math. Phys.
(1967)On the discrepancy of generalized Niederreiter sequences
J. Complexity
(2013)On the discrepancy of the Halton sequences
Math. Balkanica (N.S.)
(2004)- et al.
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration
(2010)
Discrépance de suites associées à un système de numération (en dimension )
Acta Arith.
The dynamical point of view of low-discrepancy sequences
Unif. Distrib. Theory
On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
Numer. Math.
Propagation rules for -nets and -sequences
J. Complexity
Monte Carlo and Quasi-Monte Carlo Sampling
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