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Analysis of the Monte-Carlo Error in a Hybrid Semi-Lagrangian Scheme

Bréhier, Charles-Edouard ; Faou, Erwan

In: Applied Mathematics Research eXpress, 2015, vol. 2015, no. 2, p. 167-203

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    Title
    • Analysis of the Monte-Carlo Error in a Hybrid Semi-Lagrangian Scheme
    Author
    • Bréhier, Charles-Edouard. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, Neuchâtel 2000, Switzerland
    • Faou, Erwan. INRIA & ENS Rennes - IRMAR, Université Rennes 1, Avenue Robert Schuman, F-35170 Bruz, France
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Applied Mathematics Research eXpress, 2015, vol. 2015, no. 2, p. 167-203. Oxford University Press
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:298431
    Summary
    We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. The goal of this paper is two-fold. First we give rigorous convergence estimates for our algorithm: In a simple setting, we show that under an anti-Courant-Friedrichs-Lewy condition on the time step $\delta t$ and on the mesh size $\delta x$ and for a reasonably large number of independent realizations $N$, we control the Monte-Carlo error by a term of order $\mathcal {O}(\sqrt {\delta t /N})$. Then, we show various applications of the numerical method in very general situations (nonlinear, different boundary conditions, higher dimension) and numerical examples showing that the theoretical bound obtained in the simple case seems to persist in more complex situations