Skip to main content
SpringerLink
Log in

Tuning the Generation of Sobol Sequence with Owen Scrambling

  • Conference paper
Large-Scale Scientific Computing (LSSC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5910))

Included in the following conference series:

Abstract

Sobol sequence is the most widely used low discrepancy sequence for numerical solving of multiple integrals and other quasi-Monte Carlo computations. Owen first proposed scrambling of this sequence through permutation in a manner that maintained its low discrepancy. Scrambling is necessary not only for error analysis but for parallel implementations. Good scrambling is especially important for GRID applications. However, scrambling is often difficult to implement and time consuming. In this paper we propose fast generation of Sobol sequence with Owen scrambling, tuned to specific hardware. Numerical and timing results, demonstrating the advantages of our approach are presented and discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atanassov, E.: A New Efficient Algorithm for Generating the Scrambled Sobol Sequence. In: Dimov, I.T., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) NMA 2002. LNCS, vol. 2542, pp. 83–90. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  2. Bromley, B.C.: Quasirandom Number Generation for Parallel Monte Carlo Algorithms. Journal of Parallel Distributed Computing 38(1), 101–104 (1996)

    Article  Google Scholar 

  3. Caflisch, R.: Monte Carlo and quasi-Monte Carlo methods. Acta Numerica 7, 1–49 (1998)

    Article  MathSciNet  Google Scholar 

  4. Chi, H.: Scrambled Quasirandom Sequences and Their Applications, PhD dissertation, FSU (2004)

    Google Scholar 

  5. Chi, H., Jones, E.: Generating Parallel Quasirandom Sequences by using Randomization. Journal of Distributed and Parallel Computing 67(7), 876 (2007)

    Article  MATH  Google Scholar 

  6. Chi, H., Mascagni, M.: Efficient Generation of Parallel Quasirandom Sequences via Scrambling. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2007. LNCS, vol. 4487, pp. 723–730. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Cranley, R., Patterson, T.: Randomization of number theoretic methods for multiple integration. SIAM Journal of Numerical Analysis 13(6), 904–914 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Davis, P., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1984)

    MATH  Google Scholar 

  9. Genz, A.: The numerical evaluation of multiple integrals on parallel computers. In: Keast, P., Fairweather, G. (eds.) Numerical Integration, pp. 219–230. D. Reidel, Dordrecht (1987)

    Google Scholar 

  10. Goeddeke, D., Strzodka, R., Turek, S.: Performance and accuracy of hardware-oriented native-, emulated- and mixed-precision solvers in FEM simulations. International Journal of Parallel, Emergent and Distributed Systems 22(4), 221–256 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hong, H., Hickernell, F.: Algorithm 823: Implementing scrambled digital sequences. ACM Transactions on Mathematical Software 29(2), 95–109 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Joe, S., Kuo, F.: Remark on Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software 29(1), 49–57 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Niederreiter, H.: Low-discrepancy and low-dispersion sequences. Journal of Number Theory 30(1), 51–70 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Niederreiter, H.: Random Number Generations and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    Google Scholar 

  15. Owen, A.: Randomly permuted (t, m, s)-nets and (t, s)-sequences. In: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statistics, vol. 106, pp. 299–317 (1995)

    Google Scholar 

  16. Owen, A.: Scrambled net variance for integrals of smooth functions. Annals of Statistics 25, 1541–1562 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Owen, A.: Scrambling Sobo’l and Niederreiter-Xing points. Journal of Complexity 14, 466–489 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Owen, A.: Variance with alternative scramblings of digital nets. ACM Transactions on Modeling and Computer Simulation 13(4), 363–378 (2003)

    Article  Google Scholar 

  19. Sobol, I.M.: Uniformly distributed sequences with additional uniformity properties. USSR Comput. Math. and Math. Phy. 16, 236–242 (1976)

    Article  MathSciNet  Google Scholar 

  20. Radovic, I., Sobol, I.M., Tichy, R.: Quasi-Monte Carlo methods for numerical integration: Comparison of different low discrepancy sequences. Monte Carlo Methods and Appl. 2, 1–14 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tezuka, S.: Quasi-Monte Carlo discrepancy between theory and practice. In: Fang, K.T., Hickernell, F., Niederreiter, H. (eds.) MC&QMC Methods, pp. 124–140. Springer, Berlin (2002)

    Google Scholar 

  22. Wang, X., Fang, K.: The effective dimension and quasi-Monte Carlo. Journal of Complexity 19(2), 101–124 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. http://www.nvidia.com/cuda

  24. http://en.wikipedia.org/wiki/Salsa20

Download references

Author information

Authors and Affiliations

  1. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 25A, 1113, Sofia, Bulgaria

    Emanouil Atanassov, Aneta Karaivanova & Sofiya Ivanovska

Authors
  1. Emanouil Atanassov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aneta Karaivanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sofiya Ivanovska
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 25A, 1113, Sofia, Bulgaria

    Ivan Lirkov

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 25-A, 1113, Sofia, Bulgaria

    Svetozar Margenov

  3. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads - Building 321, 2800, Kongens Lyngby, Denmark

    Jerzy Waśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Atanassov, E., Karaivanova, A., Ivanovska, S. (2010). Tuning the Generation of Sobol Sequence with Owen Scrambling. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2009. Lecture Notes in Computer Science, vol 5910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12535-5_54

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-12535-5_54

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-12534-8

  • Online ISBN: 978-3-642-12535-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation