Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice
-
Rami El Haddad
,
Christian Lécot
Abstract
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.
© de Gruyter 2010
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Articles in the same Issue
- Editiorial
- Random packing of hyperspheres and Marsaglia's parking lot test
- Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice
- Improved Halton sequences and discrepancy bounds
- Generalizing Sudoku to three dimensions
- Adaptive integration and approximation over hyper-rectangular regions with applications to basket option pricing
- Exact simulation of Bessel diffusions
- A good permutation for one-dimensional diaphony
- Error bounds for computing the expectation by Markov chain Monte Carlo
- Stochastic iterative projection methods for large linear systems
- Increasing the number of inner replications of multifactor portfolio credit risk simulation in the t-copula model
- A genetic algorithm approach to estimate lower bounds of the star discrepancy
- Random and deterministic fragmentation models
- MCMC imputation in autologistic model
