Multiscale Modeling in a Stochastic Setting

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. 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, 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.