Department of Mathematics
Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920
Browse
5 results
Search Results
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.Item Deterministic Particle Methods for High Dimensional Fokker-Planck Equations(Springer, 2006) Venkiteswaran, G.We consider a mathematical model for polymeric liquids which requires the solution of high-dimensional Fokker-Planck equations related to stochastic differential equations. While Monte-Carlo (MC) methods are classically used to construct approximate solutions in this context, we consider an approach based on Quasi- Monte-Carlo (QMC) approximations. Although QMC has proved to be superior to MC in certain integration problems, the advantages are not as pronounced when dealing with stochastic differential equations. In this article, we illustrate the basic difficulty which is related to the construction of QMC product measures.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.