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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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Volume 16, Issue 3-4


Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice

Rami El Haddad
  • Département de Mathématiques, Faculté des Sciences, Université Saint-Joseph, BP 11-514 Riad El Solh, Beyrouth 1107 2050, Lebanon. E-mail:
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/ Christian Lécot
  • Laboratoire de Mathématiques, UMR 5127 CNRS and Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac Cedex, France. E-mail:
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/ Gopalakrishnan Venkiteswaran
  • Department of Mathematics, Birla Institute of Technology and Science, Vidya Vihar Campus, Pilani, 333 031 Rajasthan, India. E-mail:
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Published Online: 2010-10-20 | DOI: https://doi.org/10.1515/mcma.2010.009


We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.

Keywords.: Quasi-Monte Carlo; random walk; low-discrepancy sequences; diffusion equation

About the article

Received: 2009-11-16

Revised: 2010-09-06

Published Online: 2010-10-20

Published in Print: 2010-12-01

Citation Information: Monte Carlo Methods and Applications, Volume 16, Issue 3-4, Pages 211–230, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma.2010.009.

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