Hybrid couplings of small-scale systems to large-scale dynamics

Markos Katsoulakis
University of Massachusetts Amherst
Mathematics and Statistic

Hybrid systems arise as couplings of stochastic models representing active small scales to deterministic macroscopic equations, and are commonplace in a wide array of applications ranging from catalysis and polymeric flows to stochastic models for tropical and open ocean convection. A major challenge in all these problems arises in the direct numerical simulation of realistic size systems involving both scale and model disparities; furthermore, due to nonlinear interactions across a wide range of scales, the stochasticity inherited from the microscopic model can play a subtle but important role in the dynamic behavior of the overall system.

In this talk we attempt to mathematically formulate and partly address such issues in the context of mathematical prototypes that capture essential features of their realistic counterparts, exhibiting a host of complex phenomena such as nucleation, metastability, random oscillations and strong intermittency. We also demonstrate regimes where the hybrid modeling approach of directly coupling microscopics to macroscopics is inadequate and a more hierarchical approach needs to be undertaken.

Audio (MP3 File, Podcast Ready) Presentation (PDF File)

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