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Combining Space-Time Multigrid Techniques with Multilevel Monte Carlo Methods for SDEs

  • Martin NeumüllerEmail author
  • Andreas ThalhammerEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)

Abstract

In this work we combine multilevel Monte Carlo methods for time-dependent stochastic differential equations with a space-time multigrid method. The idea is to use the space-time hierarchy from the multilevel Monte Carlo method also for the solution process of the arising linear systems. This symbiosis leads to a robust and parallel method with respect to space, time and probability. We show the performance of this approach by several numerical experiments which demonstrate the advantages of this approach.

References

  1. 1.
    A. Andersson, R. Kruse, S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 4, 113–149 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Barth, A. Lang, Simulation of stochastic partial differential equations using finite element methods. Stochastics 84(2–3), 217–231 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1992)Google Scholar
  4. 4.
    M. Gander, M. Neumüller, Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M.B. Giles, Multilevel Monte Carlo path simulation. Oper. Res. 56, 607–617 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Lang, A note on the importance of weak convergence rates for SPDE approximations in multilevel Monte Carlo schemes, in Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, ed. by R. Cools, D. Nuyens (Springer, Cham, 2016), pp. 489–505Google Scholar
  7. 7.
    G.J. Lord, C.E. Powell, T. Shardlow, An Introduction to Computational Stochastic PDEs Cambridge. Texts in Applied Mathematics (Cambridge University Press, New York, 2014)Google Scholar
  8. 8.
    G.N. Milstein, M.V. Tretyakov, Stochastic Numerics for Mathematical Physics. Scientific Computation (Springer, Berlin, 2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Johannes Kepler UniversityInstitute of Computational MathematicsLinzAustria
  2. 2.Johannes Kepler UniversityDoctoral Program “Computational Mathematics” and Institute for StochasticsLinzAustria

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