Probability density function methods for uncertainty propagation in stochastic dynamical systems
Computing the statistical properties of nonlinear stochastic dynamical systems and stochastic partial differential equations is an important problem in many areas of physical sciences, engineering and mathematics. In this talk I will address such problem by using probability density function (PDF) methods. To this end, I will first review the mathematical theory that allows us to transform arbitrary nonlinear systems of SODEs and SPDEs into linear transport equations for probability density functions and probability density functionals, respectively. I will also discuss state-of-the-art algorithms to perform numerical simulations of such high-dimensional (possibly infinite-dimensional) transport equations. In the second part of the talk, I will present recent developments on dimension reduction techniques for high-dimensional PDF equations. In particular, I will discuss the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) technique and the Mori-Zwanzig (MZ) approach. I will also address the question of approximation of BBGKY and MZ equations by data-driven methods, perturbation series and operator cumulant resummation. Throughout the talk I will provide numerical examples and applications of the proposed methods to prototype stochastic problems.
(Refreshments will be served.)