Date: Friday, December 02, 2016
Location: 1084 East Hall (3:00 PM to 4:00 PM)
Title: Bounding average quantities in dynamical systems
Abstract: I will discuss the task of proving bounds on average quantities in dissipative dynamical systems, including time averages in finitedimensional systems and spatiotemporal averages in PDE systems. In the finitedimensional case, I will describe computerassisted methods where bounds are proven by constructing nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proven by constructing Lyapunov functions. Nonnegativity of polynomials is enforced by requiring them to be representable as sums of squares, a condition that can be checked using the convex optimization technique of semidefinite programming. Rigorous bounds are obtained by supplementing numerical computations with either interval arithmetic or symbolic computation. These methods are illustrated using the Lorenz equations, where they produce novel bounds on various average quantities. In the PDE case, I will describe bounds for fluid dynamical models proven using pencilandpaper analysis. Differences from the finitedimensional case will be discussed, as well as the possibility of improving results for PDEs using computerassisted methods like those described for the finitedimensional case.
Files:
Speaker: David Goluskin
Institution: University of Michigan
Event Organizer: AIM Seminar Organizers millerpd@umich.edu
