Advertisement

Latin Hypercube Sampling and Fibonacci Based Lattice Method Comparison for Computation of Multidimensional Integrals

  • Stoyan Dimitrov
  • Ivan Dimov
  • Venelin TodorovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)

Abstract

We perform computational investigations to compare the performance of Latin hypercube sampling (LHS method) and a particular QMC lattice rule based on generalized Fibonacci numbers (FIBO method) for integration of smooth functions of various dimensions. The two methods have not been compared before and both are generally recommended in case of smooth integrands. The numerical results suggests that the FIBO method is better than LHS method for low-dimensional integrals, while LHS outperforms FIBO when the integrand dimension is higher. The Sobol nets, which performence is given as a reference, are outperformed by at least one of the two discussed methods, in any of the considered examples.

Keywords

Importance Sampling Latin Hypercube Sampling Lattice Rule Latin Hypercube Sampling Method Sobol Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported by the Program for career development of young scientists, BAS, Grant No. DFNP-91/04.05.2016, by the Bulgarian National Science Fund under grant DFNI I02-20/2014, and the financial funds allocated to the Sofia University St. Kl. Ohridski, grant No. 197/2016.

References

  1. 1.
    Dimov, I.: Monte Carlo Methods for Applied Scientists, New Jersey, London, Singapore, World Scientific, 291 p. (2008). ISBN-10 981–02-2329-3Google Scholar
  2. 2.
    Dimov, I., Karaivanova, A.: Error analysis of an adaptive Monte Carlo method for numerical integration. Math. Comput. Simul. 47, 201–213 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ermakov, S.M.: Monte Carlo Methods and Mixed Problems. Nauka, Moscow (1985)Google Scholar
  4. 4.
    Jarosz, W.: Efficient Monte Carlo Methods for Light Transport in Scattering Media, Ph.D. dissertation, UCSD (2008)Google Scholar
  5. 5.
    Kroese, D.P., Taimre, T., Botev, Z.: Handbook of Monte Carlo Methods. Wiley Series in Probability and Statistics. Wiley, Hoboken (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kucherenko, S., Albrecht, D., Saltelli, A.: Exploring multi-dimensional spaces: a Comparison of Latin Hypercube and Quasi Monte Carlo Sampling Techniques, arXiv preprint arXiv:1505.02350 (2015)
  7. 7.
    McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44(247), 335–341 (1949)CrossRefzbMATHGoogle Scholar
  9. 9.
    Niederreiter, H., Talay, D.: Monte Carlo and Quasi-Monte Carlo Methods, 2010. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  10. 10.
    Owen, A.: Monte Carlo theory, methods and examples (2013)Google Scholar
  11. 11.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  12. 12.
    Sloan, I.H., Kachoyan, P.J.: Lattice methods for multiple integration: theory, error analysis and examples. J. Numer. Anal. 24, 116–128 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vose, D.: The pros and cons of Latin Hypercube sampling (2014)Google Scholar
  14. 14.
    Wang, X., Sloan, I.H.: Low discrepancy sequences in high dimensions: how well are their projections distributed? J. Comput. Appl. Math. 213(2), 366–386 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sofia UniversityPalo AltoUSA
  2. 2.IICTBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations