Generalized Halton sequences in 2008

Abstract

Halton sequences have always been quite popular with practitioners, in part because of their intuitive definition and ease of implementation. However, in their original form, these sequences have also been known for their inadequacy to integrate functions in moderate to large dimensions, in which case (t, s)-sequences such as the Sobol' sequence are usually preferred. To overcome this problem, one possible approach is to include permutations in the definition of Halton sequencesthereby obtaining generalized Halton sequences-an idea that goes back to almost thirty years ago, and that has been studied by many researchers in the last few years. In parallel to these efforts, an important improvement in the upper bounds for the discrepancy of Halton sequences has been made by Atanassov in 2004. Together, these two lines of research have revived the interest in Halton sequences. In this article, we review different generalized Halton sequences that have been proposed recently, and compare them by means of numerical experiments. We also propose a new generalized Halton sequence which, we believe, offers a practical advantage over the surveyed constructions, and that should be of interest to practitioners.

Random and Deterministic Digit Permutations of the Halton Sequence

“…To answer this, we pick two permuted sequences, one by Faure [5], which was obtained by minimizing the discrepancy of one-dimensional projections, and the other by Kocis & Whiten [9], which was obtained heuristically. Both sequences were reported to be among the best in recent numerical work (see [6] Figure 2 suggests that the digit permuted Halton sequences are also prone to the same de…ciency of the Halton sequence. The bases used in the above plots were obtained by a computer search, and there are several other projections that have a similar behavior.…”

“…Numerical results are sometimes surprising. For example, even though the digit-permuted Halton sequence by Atanassov [1] has the best known bounds for its star-discrepancy, after extensive numerical results, Faure & Lemieux [6] conclude that several other digit-permuted sequences (Chi, Mascagni & Warnock [4], Faure & Lemieux [6], Kocis & Whiten [9]) are as good as the one by Atanassov [1].…”

“…Since it is not very practical to compare all digit permuted sequences, we will proceed as follows: Following the conclusion of Faure & Lemieux [6], we will pick the Kocis-Whiten permutation as a representative of the permutations that performed equally well in their numerical results. We will also consider the permutation by Faure [5], which was used successfully in previous numerical studies of the authors (Goncharov, Ökten, Shah [8]), and the permutation by Braaten-Weller [2].…”

“…Since the publication of Braaten and Weller [2], several authors introduced di¤erent permutations to scramble the digits of the Halton sequence; see, for example, [1], [3], [4], [5], [6], [9], [17], [18], [19]. Some of these permutations were obtained using heuristics, such as [9] & [18], and some others were obtained by searching for the optimal permutations that minimize the discrepancy of the one-dimensional or two-dimensional projections of the Halton sequence, such as [2], [3], [4], [5], and [6].…”

“…Some of these permutations were obtained using heuristics, such as [9] & [18], and some others were obtained by searching for the optimal permutations that minimize the discrepancy of the one-dimensional or two-dimensional projections of the Halton sequence, such as [2], [3], [4], [5], and [6].…”