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Convergence of Uniformity Criteria and the Application in Numerical Integration

Author

Listed:
  • Yang Huang

    (School of Statistics and Data Science, Nankai University, Tianjin 300071, China)

  • Yongdao Zhou

    (School of Statistics and Data Science, Nankai University, Tianjin 300071, China)

Abstract

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of order of some improved discrepancies, such as centered L 2 -discrepancy, wrap-around L 2 -discrepancy, and mixture discrepancy, and propose a randomized QMC method based on a uniform design constructed by the mixture discrepancy and Baker’s transformation. Moreover, the numerical results show that the proposed method has better approximation than the Monte Carlo method and many other QMC methods, especially when the number of dimensions is less than 10.

Suggested Citation

  • Yang Huang & Yongdao Zhou, 2022. "Convergence of Uniformity Criteria and the Application in Numerical Integration," Mathematics, MDPI, vol. 10(19), pages 1-20, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3717-:d:938366
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    References listed on IDEAS

    as
    1. Fred J. Hickernell, 2002. "Uniform designs limit aliasing," Biometrika, Biometrika Trust, vol. 89(4), pages 893-904, December.
    2. Pierre L’Ecuyer & Christiane Lemieux, 2002. "Recent Advances in Randomized Quasi-Monte Carlo Methods," International Series in Operations Research & Management Science, in: Moshe Dror & Pierre L’Ecuyer & Ferenc Szidarovszky (ed.), Modeling Uncertainty, chapter 0, pages 419-474, Springer.
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