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Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan pp 597–619Cite as

Adaptive Quasi-Monte Carlo Methods for Cubature

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Abstract

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.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday with many thanks for his friendship and leadership.

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Authors and Affiliations

  1. Illinois Institute of Technology, Chicago, IL, USA

    Fred J. Hickernell, Lluís Antoni Jiménez Rugama & Da Li

Authors
  1. Fred J. Hickernell
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  2. Lluís Antoni Jiménez Rugama
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  3. Da Li
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Corresponding author

Correspondence to Fred J. Hickernell .

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Editors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Josef Dick

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Frances Y. Kuo

  3. Department of Computer Science, Columbia University, New York, USA

    Henryk Woźniakowski

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Hickernell, F.J., Jiménez Rugama, L.A., Li, D. (2018). Adaptive Quasi-Monte Carlo Methods for Cubature. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_27

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