Adaptive Quasi-Monte Carlo Methods for Cubature

  • Fred J. HickernellEmail author
  • Lluís Antoni Jiménez Rugama
  • Da Li


High dimensional integrals can be approximated well by quasi-Monte Carlo methods. However, determining the number of function values needed to obtain the desired accuracy is difficult without some upper bound on an appropriate semi-norm of the integrand. This challenge has motivated our recent development of theoretically justified, adaptive cubatures based on digital sequences and lattice nodeset sequences. Our adaptive cubatures are based on error bounds that depend on the discrete Fourier transforms of the integrands. These cubatures are guaranteed for integrands belonging to cones of functions whose true Fourier coefficients decay steadily, a notion that is made mathematically precise. Here we describe these new cubature rules and extend them in two directions. First, we generalize the error criterion to allow both absolute and relative error tolerances. We also demonstrate how to estimate a function of several integrals to a given tolerance. This situation arises in the computation of Sobol’ indices. Second, we describe how to use control variates in adaptive quasi-Monte cubature while appropriately estimating the control variate coefficient.


  1. 1.
    Bratley, P., Fox, B.L., Niederreiter, H.: Implementation and tests of low-discrepancy sequences. ACM Trans. Model. Comput. Simul. 2, 195–213 (1992)CrossRefGoogle Scholar
  2. 2.
    Choi, S.C.T., Ding, Y., Hickernell, F.J., Jiang, L., Jiménez Rugama, Ll.A., Tong, X., Zhang, Y., Zhou, X.: GAIL: Guaranteed Automatic Integration Library (versions 1.0–2.2). MATLAB software (2013–2017).
  3. 3.
    Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)Google Scholar
  4. 4.
    Dick, J., Kuo, F., Sloan, I.H.: High dimensional integration—the Quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Genz, A.: Comparison of methods for the computation of multivariate normal probabilities. Comput. Sci. Stat. 25, 400–405 (1993)Google Scholar
  6. 6.
    Giles, M.: Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Halton, J.H.: Quasi-probability: why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically. Monte Carlo Methods Appl. 11, 203–350 (2005)Google Scholar
  8. 8.
    Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67, 299–322 (1998).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hickernell, F.J., Hong, H.S.: Computing multivariate normal probabilities using rank-1 lattice sequences. In: Golub, G.H., Lui, S.H., Luk, F.T., Plemmons, R.J. (eds.) Proceedings of the Workshop on Scientific Computing, pp. 209–215. Springer, Singapore (1997)Google Scholar
  10. 10.
    Hickernell, F.J., Jiménez Rugama, Ll.A.: Reliable adaptive cubature using digital sequences. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, April 2014. Springer Proceedings in Mathematics and Statistics, vol. 163, pp. 367–383. Springer, Berlin (2016)Google Scholar
  11. 11.
    Hickernell, F.J., Hong, H.S., L’Écuyer, P., Lemieux, C.: Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22, 1117–1138 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hickernell, F.J., Lemieux, C., Owen, A.B.: Control variates for quasi-Monte Carlo. Stat. Sci. 20, 1–31 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jiménez Rugama, Ll.A.: Adaptive quasi-Monte Carlo cubature. Ph.D. thesis, Illinois Institute of Technology (2016)Google Scholar
  14. 14.
    Jiménez Rugama, Ll.A., Gilquin, L.: Reliable error estimation for Sobol’ indices. Stat. Comput. (2017+, in press)Google Scholar
  15. 15.
    Jiménez Rugama, Ll.A., Hickernell, F.J.: Adaptive multidimensional integration based on rank-1 lattices. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, April 2014. Springer Proceedings in Mathematics and Statistics, vol. 163, pp. 407–422. Springer, Berlin (2016)Google Scholar
  16. 16.
    Li, D.: Reliable quasi-Monte Carlo with control variates. Master’s thesis, Illinois Institute of Technology (2016)Google Scholar
  17. 17.
    Maize, E.: Contributions to the theory of error reduction in quasi-Monte Carlo methods. Ph.D. thesis, The Claremont Graduate School (1981)Google Scholar
  18. 18.
    Maize, E., Sepikas, J., Spanier, J.: Accelerating the convergence of lattice methods by importance sampling-based transformations. In: Plaskota, L., Woźniakowski, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics and Statistics, vol. 23, pp. 557–572. Springer, Berlin (2012)CrossRefGoogle Scholar
  19. 19.
    Matoušek, J.: On the L 2-discrepancy for anchored boxes. J. Complexity 14, 527–556 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992)Google Scholar
  21. 21.
  22. 22.
    Owen, A.B.: Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo. In: Niederreiter H., Spanier J. (eds.) Monte Carlo, Quasi-Monte Carlo, and Randomized Quasi-Monte Carlo, pp. 86–97. Springer, Berlin (2000)zbMATHGoogle Scholar
  23. 23.
    Owen, A.B.: On the Warnock-Halton quasi-standard error. Monte Carlo Methods Appl. 12, 47–54 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  25. 25.
    Sobol’, I.M.: On sensitivity estimation for nonlinear mathematical models. Matem. Mod. 2(1), 112–118 (1990)Google Scholar
  26. 26.
    Sobol’, I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55(1–3), 271–280 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    The MathWorks, Inc.: MATLAB 9.2. The MathWorks, Inc., Natick, MA (2017)Google Scholar
  28. 28.
    Wasilkowski, G.W.: On tractability of linear tensor product problems for -variate classes of functions. J. Complex. 29, 351–369 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Fred J. Hickernell
    • 1
    Email author
  • Lluís Antoni Jiménez Rugama
    • 1
  • Da Li
    • 1
  1. 1.Illinois Institute of TechnologyChicagoUSA

Personalised recommendations