Department of Mathematics
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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 A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids(Elsevier, 2005) Venkiteswaran, G.A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker–Planck equation on . In this article, we present QMC methods for the approximate solution of the Fokker–Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23–41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(IDEAS is a RePEc, 2005) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is N−1/2 accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve N−1 convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(Elsevier, 2005-02) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.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 A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids(Elsevier, 2005-02) Venkiteswaran, G.A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker–Planck equation on . In this article, we present QMC methods for the approximate solution of the Fokker–Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23–41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(Elsevier, 2005-02) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.