微分方程与动力系统系列报告
报告题目：A variational approach to closure of nonlinear dynamical systems
报告人：刘鸿鹄助理教授
报告时间：2021/11/24 09:30-10:30
腾讯会议：5318371238
报告摘要：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 low-dimensional 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 to facilitate the discussion. The approach will also be illustrated---in the context of the Kuramoto-Sivashinsky turbulence with many unstable modes---to provide efficient closures without slaving for a cutoff scale placed within the inertial range and the reduced state space just spanned by the unstable modes.
报告人简介：Dr. Honghu Liu is an assistant professor in the Department of Mathematics at Virginia Tech. Prior to his current position, Dr. Liu was a postdoctoral scholar in the Theoretical Climate Dynamics group at University of California, Los Angeles from 2013--2015. He earned his Ph.D. in Mathematics at Indiana University Bloomington in 2013. Dr. Liu's research focuses on the design of dynamics informed low-dimensional reduced models for nonlinear deterministic and stochastic PDEs as well as delay differential equations. 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.