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On the optimal Halton sequence

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  • Chi, H.
  • Mascagni, M.
  • Warnock, T.

Abstract

Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo’s pseudorandom numbers. Clearly, the generation of appropriate high-quality quasirandom sequences is crucial to the success of quasi-Monte Carlo methods. The Halton sequence is one of the standard (along with (t,s)-sequences and lattice points) low-discrepancy sequences, and one of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this phenomenon is to use a randomized (scrambled) version of the Halton sequence. An alternative approach to this is to find an optimal Halton sequence within a family of scrambled sequences. This paper presents a new algorithm for finding an optimal Halton sequence within a linear scrambling space. This optimal sequence is numerically tested and shown empirically to be far superior to the original. In addition, based on analysis and insight into the correlations between dimensions of the Halton sequence, we illustrate why our algorithm is efficient for breaking these correlations. An overview of various algorithms for constructing various optimal Halton sequences is also given.

Suggested Citation

  • Chi, H. & Mascagni, M. & Warnock, T., 2005. "On the optimal Halton sequence," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 9-21.
    • Handle: RePEc:eee:matcom:v:70:y:2005:i:1:p:9-21
    DOI: 10.1016/j.matcom.2005.03.004
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    References listed on IDEAS

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    1. Spassimir H. Paskov & Joseph F. Traub, 1995. "Faster Valuation of Financial Derivatives," Working Papers 95-03-034, Santa Fe Institute.
    2. Mascagni Michael & Chi Hongmei, 2004. "On the Scrambled Halton Sequence," Monte Carlo Methods and Applications, De Gruyter, vol. 10(3-4), pages 435-442, December.
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    Cited by:

    1. Chi Hongmei, 2013. "Generation of parallel modified Kronecker sequences," Monte Carlo Methods and Applications, De Gruyter, vol. 19(4), pages 261-271, December.
    2. Lin Gui & Xinyu Li & Qingfu Zhang & Liang Gao, 2024. "A uniform sampling method for permutation space," Annals of Operations Research, Springer, vol. 338(2), pages 925-945, July.
    3. Bayousef Manal & Mascagni Michael, 2019. "A computational investigation of the optimal Halton sequence in QMC applications," Monte Carlo Methods and Applications, De Gruyter, vol. 25(3), pages 187-207, September.
    4. Vandewoestyne, Bart & Chi, Hongmei & Cools, Ronald, 2010. "Computational investigations of scrambled Faure sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 522-535.
    5. Jaromił Najman & Dominik Bongartz & Alexander Mitsos, 2021. "Linearization of McCormick relaxations and hybridization with the auxiliary variable method," Journal of Global Optimization, Springer, vol. 80(4), pages 731-756, August.
    6. Miriyala, Srinivas Soumitri & Subramanian, Venkat & Mitra, Kishalay, 2018. "TRANSFORM-ANN for online optimization of complex industrial processes: Casting process as case study," European Journal of Operational Research, Elsevier, vol. 264(1), pages 294-309.
    7. Dong, Gracia Y. & Lemieux, Christiane, 2022. "Dependence properties of scrambled Halton sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 240-262.
    8. Chi, Hongmei & Beerli, Peter, 2014. "Quasi-Monte Carlo method in population genetics parameter estimation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 103(C), pages 33-38.

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