Skip to main content

Complexity of Banach Space Valued and Parametric Integration

  • Conference paper
  • First Online:
Monte Carlo and Quasi-Monte Carlo Methods 2012

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 65))

Abstract

We study the complexity of Banach space valued integration. The input data are assumed to be r-smooth. We consider both definite and indefinite integration and analyse the deterministic and the randomized setting. We develop algorithms, estimate their error, and prove lower bounds. In the randomized setting the optimal convergence rate turns out to be related to the geometry of the underlying Banach space. Then we study the corresponding problems for parameter dependent scalar integration. For this purpose we use the Banach space results and develop a multilevel scheme which connects Banach space and parametric case.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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.

Similar content being viewed by others

References

  1. Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North Holland, Amsterdam (1993)

    MATH  Google Scholar 

  2. Heinrich, S.: Monte Carlo complexity of global solution of integral equations. J. Complexity 14, 151–175 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Heinrich, S.: Monte Carlo approximation of weakly singular integral operators. J. Complexity 22, 192–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Heinrich, S.: The randomized information complexity of elliptic PDE. J. Complexity 22, 220–249 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Heinrich, S., Milla, B.: The randomized complexity of indefinite integration. J. Complexity 27, 352–382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heinrich, S., Sindambiwe, E.: Monte Carlo complexity of parametric integration. J. Complexity 15, 317–341 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  8. Light, W.A., Cheney, W.: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics 1169. Springer, Berlin (1985)

    Google Scholar 

  9. Maurey, B., Pisier, G.: Series de variables aléatoires vectorielles independantes et propriétés geométriques des espaces de Banach. Studia Mathematica 58, 45–90 (1976)

    MathSciNet  MATH  Google Scholar 

  10. Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics 1349. Springer, Berlin (1988)

    Google Scholar 

  11. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems, Volume 2, Standard Information for Functionals. European Mathematical Society, Zürich (2010)

    Book  MATH  Google Scholar 

  12. Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic, New York (1988)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thomas Daun .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Daun, T., Heinrich, S. (2013). Complexity of Banach Space Valued and Parametric Integration. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_12

Download citation

Publish with us

Policies and ethics