Skip to main content
SpringerLink
Log in

A Monte-Carlo Method with Inherent Parallelism for Numerical Solving Partial Differential Equations with Boundary Conditions

  • Conference paper
  • First Online:
Parallel Computation (ACPC 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1557))

  • 443 Accesses

Abstract

One can construct the representation of solutions to various problems for parabolic differential equation with boundary conditions in the framework of classical stochastic analysis. This fact yields Monte Carlo methods for solving PDE’s numerically. Instead of solving a PDE by common numeric techniques, one can simulate a stochastic system which allows for a simple parallelism with linear speed up.

A number of numerical schemes exists for solving stochastic differential equations, i.e. the Euler scheme. If reflection is concerned, most methods have some shortcomings. In this article an efficient algorithm for simulating a reflected stochastic differential equation is developed. Results of numerical experiments are referenced revealing significant reduction in computational time gained by parallelization.

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. P. Altevogt and A. Linke. An algorithm for dynamic load balancing of synchronous Monte Carlo simulations on multiprocessor systems. Comput. Phys. Commun., 79(3), pp. 373–380, 1994.

    Article  MATH  Google Scholar 

  2. V.C. Bhavsar and J.R. Isaac. Design and analysis of parallel Monte Carlo algorithms. SIAM J. Sci. Stat. Comput., 8, pp. 73–95, 1987.

    Article  MathSciNet  Google Scholar 

  3. R. Blumenthal. Excursions of Markov Processes. Birkhäuser Berlin, 1992.

    MATH  Google Scholar 

  4. K. Burdzy. Multidimensional Brownian excursions and potential theory, volume 164 of Pitman research notes in mathamtics series. Longman Scientific and Technical, 1987.

    Google Scholar 

  5. P. Coddington. Random Number Generators for Parallel Computers. NHSE Review, Second Issue, Northeast Parallel Architectures Center, 1996. Available at:http://nhse.cs.rice.edu/NHSEreview/RNG/.

  6. C. Costantini, B. Pacchierotti, and F. Sartoretto. Numerical Approximation for Functionals of Reflecting Diffusion Processes. (submitted) May 1995.

    Google Scholar 

  7. M.J. Durst. Using linear congruential generators for parallel random number generation. In E.A. MacNair, K.J. Musselman, and P. Heidelberger, editors, Proceedings of the 1989 Winter Simulation Conference, pp. 462–466, 1989.

    Google Scholar 

  8. A. Geist. PVM3 beyond network computation. In J. Volkert, editor, Parallel Computing, volume 73 of Lectures Notes in Computer Sience, pp. 194–203. Springer Verlag, 1993.

    Google Scholar 

  9. D. Hafermann and M. Kuntz. Parallelization of a direct simulation Monte Carlo code for workstation cluster. In Parallel computational fluid dynamics: new trends and advances, pp. 81–88, 1995.

    Google Scholar 

  10. R. Hanxleden and L. Scott. Correctness and determinism of parallel Monte Carlo processes. Parallel Comput., 18(2), pp. 373–380, 1994.

    Google Scholar 

  11. E. Hausenblas. Numerical Approximation for Reflecting Diffusion Processes using Ito-Excursion and Local Time. To appear in Osaka Journal of Mathematics.

    Google Scholar 

  12. N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981.

    MATH  Google Scholar 

  13. P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Application of Mathematics. Springer Verlag, 1992.

    Google Scholar 

  14. Slominski L. On the existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting boundary. Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 29(2), pp. 163–198, 1993.

    MATH  MathSciNet  Google Scholar 

  15. P.L. Lions and A.S. Sznitzman. Stochastic Differential Equations with Reflecting Boundary Conditions. Communications on Pure and Applied Mathematics, 37, pp. 511–537, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Mastrangelo, V. Mastrangelo, D. Gassilloud, and F. Simon. Stochastic modelling of diffusion equations on a parallel machine. Comput. Phys. Commun., 76(2–3), pp. 159–183, 1993.

    Article  MATH  Google Scholar 

  17. J. Menaldi. Stochastic variational inequality for reflected difiusion. Indiana University Journal of Mathematics, 32(5), pp. 733–743, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  18. W.P. Petersen. Numerical simulation of Ito stochastic differential equations on supercomputers. Random media, IMA Vol. Math. Appl., 7, pp. 215–225, 1987.

    Google Scholar 

  19. A.V. Skorohood. Stochastic Equation for Diffusion Processes in a Bounded Region. Theory of Probability and its Applications, 6(2), pp. 264–274, 1975.

    Google Scholar 

  20. D. Talay. Elements of Probabilistic Numerical Methods for Partial Differential Equations. In Probabilistic Models for Nonlinear Partial Differential Equations, volume 1627 of Lectures Notes in Mathematics, pp. 148–196. D. Talay, L.Tubaro, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Salzburg, Austria

    Erika Hausenblas

Authors
  1. Erika Hausenblas
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Forschungsinstitut für Softwaretechnologie, Hellbrunnerstr. 34, A-5020, Salzburg, Austria

    Peter Zinterhof

  2. Slovac Academy of Sciences, Institute of Mathematics, Laboratory for Informatics, Dubravska 9, P.O.Box 56, 840 00, Bratislava, Slovakia

    Marian Vajteršic

  3. Universität Salzburg, RIST++, Hellbrunnerstr. 34, A-5020, Salzburg, Austria

    Andreas Uhl

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hausenblas, E. (1999). A Monte-Carlo Method with Inherent Parallelism for Numerical Solving Partial Differential Equations with Boundary Conditions. In: Zinterhof, P., Vajteršic, M., Uhl, A. (eds) Parallel Computation. ACPC 1999. Lecture Notes in Computer Science, vol 1557. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49164-3_12

Download citation

  • DOI: https://doi.org/10.1007/3-540-49164-3_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65641-8

  • Online ISBN: 978-3-540-49164-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation