Department of Mathematics
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Item Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice(De Gruyter, 2010-10) Venkiteswaran, G.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.Item Quasi-Monte Carlo Simulation of Diffusion in a Spatially Nonhomogeneous Medium(Springer, 2009-11) Venkiteswaran, G.We propose and test a quasi-Monte Carlo (QMC) method for solving the diffusion equation in the spatially nonhomogeneous case. For a constant diffusion coefficient, the Monte Carlo (MC) method is a valuable tool for simulating the equation: the solution is approximated by using particles and in every time step the displacement of each particle is drawn from a Gaussian distribution with constant variance. But for a spatially dependent diffusion coefficient, the straightforward extension using a spatially variable variance leads to biased results. A correction to the Gaussian steplength was recently proposed and provides satisfactory results. In the present work, we devise a QMC variant of this corrected MC scheme. We present the results of some numerical experiments showing that our QMC algorithm converges better than the corresponding MC method for the same number of particles.Item Extended Latin Hypercube Sampling for Integration and Simulation(Springer, 2013-01) Venkiteswaran, G.We analyze an extended form of Latin hypercube sampling technique that can be used for numerical quadrature and for Monte Carlo simulation. The technique utilizes random point sets with enhanced uniformity over the s-dimensional unit hypercube. A sample of N = n s points is generated in the hypercube. If we project the N points onto their ith coordinates, the resulting set of values forms a stratified sample from the unit interval, with one point in each subinterval [(k−1)/N,k/N). The scheme has the additional property that when we partition the hypercube into N subcubes ∏si=1[(ℓi−1)/n,ℓi/n), each one contains exactly one point. We establish an upper bound for the variance, when we approximate the volume of a subset of the hypercube, with a regular boundary. Numerical experiments assess that the bound is tight. It is possible to employ the extended Latin hypercube samples for Monte Carlo simulation. We focus on the random walk method for diffusion and we show that the variance is reduced when compared with classical random walk using ordinary pseudo-random numbers. The numerical comparisons include stratified sampling and Latin hypercube sampling.Item Quasi-Monte Carlo Simulation of Diffusion in a Spatially Nonhomogeneous Medium(Springer, 2009-11) Venkiteswaran, G.;We propose and test a quasi-Monte Carlo (QMC) method for solving the diffusion equation in the spatially nonhomogeneous case. For a constant diffusion coefficient, the Monte Carlo (MC) method is a valuable tool for simulating the equation: the solution is approximated by using particles and in every time step the displacement of each particle is drawn from a Gaussian distribution with constant variance. But for a spatially dependent diffusion coefficient, the straightforward extension using a spatially variable variance leads to biased results. A correction to the Gaussian steplength was recently proposed and provides satisfactory results. In the present work, we devise a QMC variant of this corrected MC scheme. We present the results of some numerical experiments showing that our QMC algorithm converges better than the corresponding MC method for the same number of particles.