Skip to main content

A Pragmatic View on Numerical Integration of Unbounded Functions

  • Conference paper
Monte Carlo and Quasi-Monte Carlo Methods 2006

Summary

We take a pragmatic approach to numerical integration of unbounded functions. In this context we discuss and evaluate the practical application of a method suited also for non-specialists and application developers. We will show that this method can be applied to a rich body of functions, and evaluate it’s merits in comparison to other methods for integration of unbounded integrals. Furthermore, we will give experimental results to illustrate certain issues in the actual application and to confirm theoretic results.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
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. E. deDoncker and Y. Guan. Error bounds for integration of singular functions using equidistributed sequences. Journal of Complexity, 19(3):259–271,2003.

    Article  MathSciNet  Google Scholar 

  2. J. Hartinger and R. F. Kainhofer. Non-uniform low-discrepancy sequence generation and integration of singular integrands. In H. Niederreiter and D. Talay, editors, Proceedings of MC2QMC2004, Juan-Les-Pins France, June 2004. Springer Verlag, June 2005.

    Google Scholar 

  3. J. Hartinger, R. F. Kainhofer, and R. F. Tichy. Quasi-monte carlo algorithms for unbound, weighted integration problems. Journal of Complexity, 20 (5):654–668, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  4. JĂ¼rgen Hartinger, Reinhold Kainhofer, and Volker Ziegler. On the corner avoidance properties of various low- discrepancy sequences. INTEGERS: Electronic Journal of Combinatorial Number Theory, 5(3), 2005. paper A10.

    MathSciNet  Google Scholar 

  5. E. Hlawka and R. MĂ¼ck. Ăœ ber eine Transformation von gleichverteilten Folgen. Computing, 9:127–138, 1972.

    Article  MATH  Google Scholar 

  6. B. Klinger. Numerical Integration of Singular Integrands Using Low-Discrepancy Sequences. Computing, 59:223–236, March 1997.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. L. Keng and W. Yuan. Applications of Number Theory to Numerical Analysis. Springer Verlag, Science Press, 1981.

    MATH  Google Scholar 

  8. H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 62 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), 1992.

    Google Scholar 

  9. Art B. Owen. Quasi-Monte Carlo for integrands with point singularities at unknown locations. Technical Report 26, Stanford University, 2004.

    Google Scholar 

  10. A. B. Owen. Multidimensional variation for quasi-Monte Carlo. In Jianqing Fan and Gang Li, editors, International Conference on Statistics in honour of Professor Kai-Tai Fang’s 65th birthday, pages 49–74, 2005.

    Google Scholar 

  11. Art B. Owen. Halton sequences avoid the origin. SIAM Rev., 48(3):487–503, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  12. I.M. Sobol’. Calculation of improper integrals using uniformly distributed sequences. Soviet Math Dokl., 14(3):734–738, July 1973.

    Google Scholar 

  13. P. Zinterhof. Einige zahlentheoretische Methoden zur numerischen Quadratur und Interpolation. Sitzungsberichte der Osterreichischen Akademie der Wissenschaften, math.-nat.wiss. Klasse Abt. II, 177:51–77, 1969.

    MATH  MathSciNet  Google Scholar 

  14. P. Zinterhof. High dimensional improper integration procedures. In Civil Engineering Faculty Technical University of Košice, editor, Proceedings of Contributions of the 7th International Scientific Conference, pages 109–115, Hroncova 5, 04001 Košice, SLOVAKIA, May 2002. TULIP.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hofbauer, H., Uhl, A., Zinterhof, P. (2008). A Pragmatic View on Numerical Integration of Unbounded Functions. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_29

Download citation

Publish with us

Policies and ethics