Abstract
The Halton sequence is one of the classical low-discrepancy sequences. It is effectively used in numerical integration when the dimension is small, however, for larger dimensions, the uniformity of the sequence quickly degrades. As a remedy, generalized (scrambled) Halton sequences have been introduced by several researchers since the 1970s. In a generalized Halton sequence, the digits of the original Halton sequence are permuted using a carefully selected permutation. Some of the permutations in the literature are designed to minimize some measure of discrepancy, and some are obtained heuristically.In this paper, we investigate how these carefully selected permutations differ from a permutation simply generated at random. We use a recent genetic algorithm, test problems from numerical integration, and a recent randomized quasi-Monte Carlo method, to compare generalized Halton sequences with randomly chosen permutations, with the traditional generalized Halton sequences. Numerical results suggest that the random permutation approach is as good as, or better than, the “best” deterministic permutations.
This material is based upon work supported by the National Science Foundation under Grant No. DMS 0703849.
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Notes
- 1.
In the numerical results of Sect. 2.1 we will give interval estimates for star discrepancy; a lower bound using the genetic algorithm, and an upper bound using Thiémard’s algorithm. In this table, we only report lower bounds since computing upper bounds with these parameters was expensive.
- 2.
The complexity of Thiémard’s algorithm grows at least as \(s/{\epsilon }^{s},\) where s is the dimension and \(\epsilon \) is the parameter that specifies the difference between the upper and lower bounds for the star-discrepancy (see [16] for a proof of the result on complexity and [15] for empirical results on complexity). We were able to go as low as \(\epsilon = 0.05\) in Table 3, and \(\epsilon = 0.2\) in Table 4. The genetic algorithm gave tighter lower bounds than Thiémard’s algorithm in computing times roughly one-fifth (Table 3) and one-fortieth (Table 4) of Thiemard’s algorithm.
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Acknowledgements
We thank Dr. Hongmei Chi for supplying us with the permuted Halton sequence code used in Chi et al. [4]. We also thank the anonymous referees for their helpful comments.
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Ökten, G., Shah, M., Goncharov, Y. (2012). Random and Deterministic Digit Permutations of the Halton Sequence. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_35
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