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.
References
E. I. Atanassov, On the discrepancy of the Halton sequences, Math. Balkanica, New Series 18 (2004) 15–32.
E. Braaten, G. Weller, An improved low-discrepancy sequence for multidimensional quasi-monte Carlo integration, Journal of Computational Physics 33 (1979) 249–258.
H. Chaix, H. Faure, Discrepance et diaphonie en dimension un, Acta Arithmetica LXIII (1993) 103–141.
H. Chi, M. Mascagni, T. Warnock, On the optimal { H}alton sequence, Mathematics and Computers in Simulation 70 (2005) 9–21.
H. Faure, Good permutations for extreme discrepancy, Journal of Number Theory 42 (1992) 47–56.
H. Faure, C. Lemieux, Generalized Halton sequences in 2008: A comparative study, ACM Transactions on Modeling and Computer Simulation 19 (2009) 15:1–31.
Y. Goncharov, G. Ökten, M. Shah, Computation of the endogenous mortgage rates with randomized quasi-Monte Carlo simulations, Mathematical and Computer Modelling 46 (2007) 459–481.
L. Kocis, W. J. Whiten, Computational investigations of low-discrepancy sequences, ACM Transactions on Mathematical Software 23 (1997) 266–294.
J. Matoušek, On the L2-discrepancy for anchored boxes, Journal of Complexity 14 (1998) 527–556.
H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.
L. Kuipers and H. Niederreiter Uniform Distribution of Sequences, Dover Publications, Mineola, NY, 2006.
G. Ökten, Generalized von Neumann-Kakutani transformation and random-start scrambled Halton sequences, Journal of Complexity 25 (2009) 318–331.
M. Shah, A genetic algorithm approach to estimate lower bounds of the star discrepancy, Monte Carlo Methods and Applications, Monte Carlo Methods Appl. 16 (2010) 379–398.
M. Shah, Quasi-Monte Carlo and Genetic Algorithms with Applications to Endogenous Mortgage Rate Computation, Ph.D. Dissertation, Department of Mathematics, Florida State University, 2008.
E. Thiémard, An algorithm to compute bounds for the star discrepancy, Journal of Complexity 17 (2001) 850–880.
M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, Journal of Complexity 24 (2008), 154–172.
B. Tuffin, A New Permutation Choice in Halton Sequences, in: H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, Vol 127, Springer Verlag, New York, 1997, pp 427–435.
B. Vandewoestyne, R. Cools, Good permutations for deterministic scrambled Halton sequences in terms of L 2-discrepancy, Journal of Computational and Applied Mathematics 189 (2006) 341–361.
T. T. Warnock, Computational Investigations of Low-discrepancy Point Sets II, in: Harald Niederreiter and Peter J.-S. Shiue, editors, Monte Carlo and quasi-Monte Carlo methods in scientific computing, Springer, New York, 1995, pp. 354–361.
X. Wang, F. J. Hickernell, Randomized { H}alton sequences, Mathematical and Computer Modelling 32 (2000) 887–899.
I. Radovic, I. M. Sobol’, R. F. Tichy, Quasi-Monte Carlo methods for numerical integration: Comparison of different low-discrepancy sequences, Monte Carlo Methods Appl. 2 (1996) 1–14.
R. E. Caflisch, W. Morokoff, A. B. Owen, Valuation of mortgage backed securities using { B}rownian bridges to reduce effective dimension, Journal of Computational Finance 1 (1997) 27–46.
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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