14 research outputs found
Homogenization for advection-diffusion in a perforated domain
The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Carlo algorithm for the computation of the volume growth rate of the sausage is introduced and some numerical results are presented for the TaylorāGreen velocity field
Long-Run Accuracy of Variational Integrators in the Stochastic Context
This paper presents a Lie-Trotter splitting for inertial Langevin equations
(Geometric Langevin Algorithm) and analyzes its long-time statistical
properties. The splitting is defined as a composition of a variational
integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the
splitting are geometrically ergodic, the paper proves the discrete invariant
measure of the splitting approximates the invariant measure of inertial
Langevin to within the accuracy of the variational integrator in representing
the Hamiltonian. In particular, if the variational integrator admits no energy
error, then the method samples the invariant measure of inertial Langevin
without error. Numerical validation is provided using explicit variational
integrators with first, second, and fourth order accuracy.Comment: 30 page
On the existence and the applications of\ud modified equations for stochastic differential\ud equations
In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive a SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed
Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDE)
are proposed which (i) are ergodic with respect to the (known) equilibrium
distribution of the SDE and (ii) approximate pathwise the solutions of the SDE
on finite time intervals. Both these properties are demonstrated in the paper
and precise strong error estimates are obtained. It is also shown that the
Metropolized integrator retains these properties even in situations where the
drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for
SDEs typically become unstable and fail to be ergodic.Comment: 46 pages, 5 figure