Advancement: Data-driven reduced-order modeling for stochastic ODEs and PDEs

Speaker Name: 
Catherine Brennan
Speaker Title: 
PhD Student (Advisor: Daniele Venturi)
Speaker Organization: 
Statistics & Applied Mathematics
Start Time: 
Tuesday, December 11, 2018 - 1:00pm
End Time: 
Tuesday, December 11, 2018 - 3:00pm
Engineering 2, Room 553
Daniele Venturi

Abstract:  High-dimensional stochastic dynamical systems and PDEs arise naturally in many areas of engineering, physical sciences and mathematics. Whether it is a physical system being studied in a lab or an equation being solved on a computer, the full state of the system is often intractable to handle in all its complexity. Instead, it is often desirable to reduce such complexity by moving from a full model of the dynamics to a reduced-order model that involves only the observables of interest. The objective of this research is to provide a new general framework to compute the probability density function (PDF) of such quantities of interest. To this end, we will derive formally exact PDF evolution equations and compute the unknown terms based on accurate data-driven closure approximations. The new method relies on estimating conditional expectations from sample paths or experimental data, and it is independent of the dimension of the underlying p! hase space. We also address the important question of whether enough useful data is being injected into the reduced-order model governing the quantity of interest. To this end, we develop a new paradigm to measure the information content of data based on the numerical solution of hyperbolic systems of equations. The effectiveness of the proposed new methods are demonstrated in applications to nonlinear dynamical systems and PDEs evolving from random initial states.