A variational approach to closure of nonlinear dynamical systems
Wednesday, September 2, 2020
2:30pm – 3:45pm
Virtual via zoom (register here for link), Virginia Tech Campus
The modeling of physical phenomena oftentimes leads to partial differential equations (PDEs) that are usually nonlinear and can also be subject to various uncertainties. Solutions of such equations typically involve multiple spatial and temporal scales, which can be numerically expensive to fully resolve. On the other hand, for many applications, it is large-scale features of the solutions that are of interest. The closure problem of a given PDE system seeks essentially for a smaller system that governs to a certain degree the evolution of such large-scale features, in which the small-scale effects are modeled through various parameterization schemes. This talk will provide an introduction to a new approach for this parameterization problem by adopting a variational framework. We will show that efficient parameterizations can be explicitly determined as parametric deformations of some geometric objects called invariant manifolds. The minimizers are objects, called the optimal parameterizing manifolds, that are intimately tied to the conditional expectation of the original system in which the effects of the unresolved variables are averaged out. We will use Lorenz's low-order primitive equations and also the Kuramoto-Sivashinsky equation to facilitate the discussion.
Honghu Liu joined Virginia Tech in the fall of 2015 as an assistant professor in the Department of Mathematics. Before that he was a postdoctoral researcher in the Theoretical Climate Dynamics group at UCLA from 2013--2015. He earned his Ph.D. degree in Mathematics at Indiana University in 2013. Dr. Liu's research focuses on the design of effective low-dimensional reduced models for nonlinear deterministic and stochastic PDEs as well as delay differential equaitons. Applications in classical and geophysical fluid dynamics are actively pursued. Particular problems that are addressed include bifurcation analysis, phase transition, surrogate systems for optimal control, and stochastic closures for turbulence.