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Monte Carlo approximations of the Neumann problem

Author

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  • Maire Sylvain

    (Laboratoire des Sciences de l'Information et des Systemes (LSIS), UMR6168, ISITV, Université de Toulon et du Var, Avenue G. Pompidou, BP 56, 83262 La Valette du Var cedex, France)

  • Tanré Etienne

    (INRIA, EPI Tosca, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis, France)

Abstract

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet–Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied.

Suggested Citation

  • Maire Sylvain & Tanré Etienne, 2013. "Monte Carlo approximations of the Neumann problem," Monte Carlo Methods and Applications, De Gruyter, vol. 19(3), pages 201-236, October.
    • Handle: RePEc:bpj:mcmeap:v:19:y:2013:i:3:p:201-236:n:3
    DOI: 10.1515/mcma-2013-0010
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    References listed on IDEAS

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    1. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    2. Hwang, Chi-Ok & Mascagni, Michael & Given, James A., 2003. "A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(3), pages 347-355.
    3. Jiang, George J. & Knight, John L., 1997. "A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model," Econometric Theory, Cambridge University Press, vol. 13(5), pages 615-645, October.
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