| Abstract: |
| The modeling of physical phenomena oftentimes leads to 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 often some 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.
In this talk, we discuss an approach for this parameterization problem by adopting a variational framework. We will show that efficient parameterizations can be explicitly determined as parametric deformations of 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 a simple semi-analytic approach to determine such manifolds based on backward-forward auxiliary systems. The approach will be illustrated on the Kuramoto-Sivashinsky equation and a stochastic Burgers equation. |
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