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Cited by 13Mathematical Approximation and IntegrationMathematical functions and polynomialsMarkov Chains and Monte Carlo Methods

The DOI matches its Crossref record; 40 of the 42 references it cites were found in a registry or an index.

The Trio Identity for Quasi-Monte Carlo Error

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Hickernell, Fred J.·2018·Springer Proceedings in Mathematics & Statistics
DOI 10.1007/978-3-319-91436-7_1
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Abstract

Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Although the variation and the discrepancy are common in the literature, the confounding is relatively unknown and under-appreciated. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might decay at a rate different from that of the discrepancy.

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Research works printed but not linked: 5

  1. Giles, M.: Multilevel Monte Carlo methods. In: Dick, J., Kuo, F.Y., Peters, G.W., Sloan, I.H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics and Statistics, vol. 65. Springer, Berlin (2013)
  2. Jiang, L.: Guaranteed adaptive Monte Carlo methods for estimating means of random variables. Ph.D. thesis, Illinois Institute of Technology (2016)
  3. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems Volume II: Standard Information for Functionals. EMS Tracts in Mathematics, vol. 12. European Mathematical Society, Zürich (2010) · DOI
  4. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems: Volume III: Standard Information for Operators. EMS Tracts in Mathematics, vol. 18. European Mathematical Society, Zürich (2012) · DOI
  5. Rasmussen, C.E., Ghahramani, Z.: Bayesian Monte Carlo. In: Thrun, S., Saul, L.K., Obermayer, K. (eds.) Advances in Neural Information Processing Systems, vol. 15, pp. 489–496. MIT Press, Cambridge (2003)

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Metadata

Authors
Hickernell, F. J.
Affiliations
Illinois Institute of Technology (US)
Corresponding author
Hickernell, Fred J.
ORCID
Hickernell, Fred J.
Venue
Springer Proceedings in Mathematics & Statistics
Publisher
Springer International Publishing
Published
2018
Pages
3-27
ISSN
  • 2194-1009 (ISSN-L)
  • 2194-1017
DOI
10.1007/978-3-319-91436-7_1
OpenAlex
W2749558669
Cited by
9
References
42
Citations by year
2026: 1 · 2024: 1 · 2023: 3 · 2021: 1 · 2020: 1 · 2019: 1 · 2018: 1
FWCI
2.00
Type
conference paper
Language
en
Open access
Licence
www.springer.com/tdm (unspecified)
Locations
  • Springer proceedings in mathematics & statistics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
National Science Foundation (1522687)
Topics
  • Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
  • Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
Keywords
quasi-Monte Carlo, discrepancy, confounding, low-discrepancy sampling, Bayesian integration

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