Simulation of diffusion using quasi-random walk methods

Coulibaly, Ibrahim; Lécot, Christian

doi:10.1016/s0378-4754(98)00100-1

Cited by 12 publications

(7 citation statements)

References 16 publications

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“…Brought to you by | Purdue University Libraries Authenticated Download Date | 5/29/15 1:45 PM A QMC version of the previous scheme has been analyzed in [2]. The basic idea of QMC methods is to replace the pseudo-random points of MC methods by low-discrepancy point sets.…”

Section: Random Walk On a Latticementioning

confidence: 99%

“…This approach may Brought to you by | Purdue University Libraries Authenticated Download Date | 5/29/15 1:45 PM be extended to several dimensions. In [2], convergence of the algorithm is established in the multi-dimensional case; the results of computational experiments indicate that a significant improvement is achieved over standard random walk simulation.…”

Section: Random Walk On a Latticementioning

confidence: 99%

“…An additional technique was first introduced in [12] and consists of sorting the particles according to position at each time step. This technique was applied to the diffusion equation (in spatially homogeneous media) in [2,14,16,21]. In a previous communication [3], we have proposed and tested a QMC algorithm for the simulation of the diffusion equation based on the corrected Gaussian steplength of [4].…”

Section: Introductionmentioning

confidence: 99%

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

Haddad¹,

Lécot²,

Venkiteswaran³

2010

Monte Carlo Methods and Applications

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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.

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Section: Random Walk On a Latticementioning

confidence: 99%

Section: Random Walk On a Latticementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Haddad¹,

Lécot²,

Venkiteswaran³

2010

Monte Carlo Methods and Applications

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“…[19]. On the other hand, some authors did use quasi-random sequences to obtain approximate solutions to partial differential equations simulating quasi-random paths of continuous stochastic processes, as it was done here, see for instance [19,22,27,28], to quote just a few. In [27], it was shown that simulating quasi-random walks can be advantageous even to solve certain nonlinear equations.…”

Section: The Numerical Errors In Computing Nodal Valuesmentioning

confidence: 99%

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Acebrón

Busico²,

Lanucara³

et al. 2005

Journal of Computational Physics

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“…Of primary importance is the even distribution of the points. A review of the development of this area is given in the monograph [3] by Niederreiter. A randomized algorithm for solving the initial value problem for the finite dimensional system y (t) = f t, y(t) , 0 < t < T, (1) y(0) = y 0 (2) was recently proposed by Stengle in [5]. The hypothesis is that f is smooth in space (y) but no more than bounded and measurable in time (t).…”

Section: Introductionmentioning

confidence: 99%

A quasi-randomized Runge-Kutta method

Coulibaly

Lécot

1999

Math. Comp.

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Abstract. We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations y (t) = f(t, y(t)). The function f is smooth in y and we suppose that f and D 1 y f are of bounded variation in t and that D 2 y f is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.

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Simulation of diffusion using quasi-random walk methods

Cited by 12 publications

References 16 publications

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

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

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

A quasi-randomized Runge-Kutta method

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