Multiscale Geometric Integration of Deterministic and Stochastic Systems
 Molei Tao, CIMS

 As part of a continuous effort in developing upscaling methods for accelerated numerical simulations, we develop multiscale geometric integrators. These integrators employ coarse steps that do not resolve the fast scale in the system; nevertheless, they capture the effective contribution of the fast dynamics --- in fact, accuracies are demonstrated in a sense called two-scale flow convergence. These integrators works for a broad class of systems, including stiff ODEs, SDEs and PDEs, by not requiring an identification of underlying slow variables or processes. They also numerically preserve intrinsic geometric structures (e.g., symplecticity, conservation laws, and invariant distribution), which not only lead to improved long time accuracy, but also a possibility to sample statistical distributions via dynamics. These new properties are due to a new philosophy based on averaging flow maps instead of vector fields.