Title: Stochastic Partial Differential Equations for Geophysical
Flows-A Computational and Dynamical Systems Approach

Speaker Dr. Jinqiao Duan,
Department of Applied Mathematics, Illinois Institute of Technology

There is growing recognition of a role for the inclusion of stochastic
effects in the mathematical modeling of multiscale phenomena in
complex systems such as geophysical flows. Macroscopic models for
these phenomena in the form of partial differential equations may
contain such randomness as stochastic forcing, uncertain parameters,
random sources, and random boundary conditions. On the one hand, this
has led to new mathematical problems at the interface of dynamical
systems, numerical analysis, stochastic analysis and partial
differential equations. On the other hand, problems arising in the
mathematical modeling of nonlinear phenomena under stochastic
influences have inspired interesting research about the interactions
between uncertainty, nonlinearity and multiple scales, and about
efficient numerical methods for simulating random phenomena.  We
present recent techniques and results for understanding nonlinear
dynamics under uncertainty, and for efficient and reliable scientific
computing by taking advantage of random dynamical behavior. These
techniques involve cocycles, invariant measures, invariant manifolds,
asymptotic degrees of freedom, random invariant sets and
ergodicity. They are presented in the context of the coupled
atmosphere-ocean system. This investigation also demonstrates delicate
interactions between deterministic and stochastic components of the
system, as well as the impact of boundary noise on the evolution of
the whole system.