A Variational Approach to Small-Scale Parameterization for Nonlinear and Stochastic Dynamical Systems

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 primary 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. In this talk, we will discuss an approach to parameterize the unresolved small-scale dynamics using the resolved large scales for forced dissipative systems. We will show that efficient parameterizations can be explicitly determined as parametric deformations of dynamic/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. We will highlight, within a variational framework, a simple semi-analytic approach to determine such manifolds based on backward-forward auxiliary systems and short DNS data. Concrete examples arising from geophysical considerations will also be presented to illustrate the effectiveness of the approach.