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Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

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CiteScore 2016: 0.70

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1569-3961
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Volume 19, Issue 3 (Oct 2013)

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Monte Carlo approximations of the Neumann problem

Sylvain Maire
  • 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
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/ Etienne Tanré
Published Online: 2013-09-03 | DOI: https://doi.org/10.1515/mcma-2013-0010

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.

Keywords: Neumann problem; Monte Carlo methods; spectral methods; local time approximation

About the article

Received: 2013-01-09

Accepted: 2013-08-21

Published Online: 2013-09-03

Published in Print: 2013-10-01


Citation Information: Monte Carlo Methods and Applications, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma-2013-0010.

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© 2013 by Walter de Gruyter Berlin Boston. Copyright Clearance Center

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