The 2015 AMMCS-CAIMS Congress
Interdisciplinary AMMCS Conference SeriesWaterloo, Ontario, Canada | June 7-12, 2015
AMMCS-CAIMS 2015 Plenary Talk
Multiscale Modeling in a Stochastic Setting
Eric Vanden-Eijnden (Courant Institute, New York University)
Applications from molecular dynamics, material science, biology, or atmosphere/ocean sciences present new challenges for applied and numerical mathematics. These applications typically involve systems whose dynamics span a very wide range of spatio-temporal scales, and are subject to random perturbations of thermal or other origin. This second aspect especially complicates the modeling and computation of these systems and requires one to revisit standard tools from numerical analysis from a probabilistic perspective. In this talk I will discuss recent advances that have been made in this context. For example, I will show how tools from Freidlin-Wentzell theory of large deviations and potential theoretic approaches to metastability can be used to develop numerical algorithms to accelerate the computations of reactive events arising in metastable systems. I will also explain how averaging theorems for singularly perturbed Markov processes can help develop schemes bridging micro- to macro-scales of description or compute free energies, etc. As illustrations, I will use a selection of examples from molecular dynamics, material sciences, and fluid dynamics and show how the confrontation with actual problems not only profits from the theory but also enriches it.
My main focus is the development of mathematical tools and numerical methods for the analysis of dynamical systems which are both stochastic and multiscale. The particular areas of applications I am interested in include molecular dynamics, chemical and biological networks, materials science, atmosphere-ocean science, and fluids dynamics. My main objectives are to understand the pathways and rate of occurrence of rare events in complex systems; to develop and analyze multiscale algorithms for the simulation of random dynamical systems; and, more generally, to quantify the effects of random perturbations on the systems dynamics.