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Zwanzig-type PDF Equations for Nonlinear Systems with Parametric Uncertainty

Daniele Venturi
Division of Applied Mathematics, Brown University


The determination of the statistical properties of the solution to a system of stochastic differential equations (SDEs) is a problem of major interest in many areas of science. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, possible discontinuities in probability space and random frequencies. Among different uncertainty quantification approaches, methods that model the probability density function (PDF) of the state variables via deterministic equations have proved to be effective in predicting the statistical properties of various random dynamical systems. In this talk we will present some recent developments on PDF methods, at both theoretical and numerical levels, addressing the question of dimensionality of the solution to SDEs. In particular, we will describe a projection operator technique of Zwanzig-type that allows us to determine closed (exact) PDF equations for goal-oriented low-dimensional functionals of the solution to high-dimensional stochastic problems with parametric uncertainty (e.g., the Nusselt number in stochastic convection phenomena subject to Dirichlet random boundary conditions). Numerical examples will be presented for nonlinear oscillators driven by random noise, stochastic advection-reaction and Burgers equations.

Tuesday, December 11, 2012
3:00PM AP&M 6402