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.
We will present 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 geometric objects constructed from dynamically based
analytical formulas. The minimizers 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 parameterizations based on backward-forward auxiliary
systems and short solution data. Concrete examples arising from
geophysical considerations will also be presented to illustrate the
effectiveness of the approach.