Presenter: Beth Wingate, Professor of Mathematics, University of Exeter
Description: Motivated by using exascale-type computer architectures for high resolution weather and climate simulations, we have developed phase averaging techniques for taking a large, accurate time step with time-parallel numerical methods in highly oscillatory PDEs. The techniques are inspired by the method of fast singular limits that feature a mapping and averaging procedure, but for numerical methods we phase averaging over a few oscillations using a smooth averaging kernel. We first briefly introduce time-parallel time stepping methods, then discuss a proof of 'parareal' convergence whose ingredients include bounds from fast singular limits in the case when epsilon -> 0 and arguments developed by Gander and Hairer. Second, we discuss the more general case, when epsilon finite (and not necessarily small) where we show how the combination of mapping and phase averaging works as a coarse propagator for parareal. We give a few examples. Finally, we end with a brief outline of 3 new lines of inquiry: a proof of convergence for multi-level parareal (with Juliane Rosemeier), alternate mapping that includes a mean correction to the phase averaging (with Tim Andrews). Finally, we show the first explorations of this type of mapping and averaging on the sphere (with Hiroe Yakamazi and Colin Cotter).
Bio: Beth Wingate is a Professor of Mathematics at the University of Exeter. She did her Ph.D. work at the University of Michigan studying numerics, waves and ocean fluid dynamics and spent many years at the Los Alamos National Laboratory in New Mexico, USA. Her research interests are mainly in geophysical and astrophysical fluid mechanics, mathematics, and numerics for high-performance computing. Her recent research is focused on nonlinear resonance, Asymptotic Parallel-in-Time methods for climate modeling and High-Performance Computing, and the fluid mechanics of the slow/fast manifolds.
Hosted by: Professor Marcella Gomez
Zoom link: https://ucsc.zoom.us/j/99978793177?pwd=eG9iR21RdEltUElvKzQ0TjJRUldUQT09