Transport of Particles in Heterogeneous Media

Heterogeneous environments are ubiquitous, with examples including ocean flows, atmospheric turbulence, oil-bearing sands, and biological tissues. Understanding the overall properties of such media is relevant to virtually every branch of science and engineering, especially computer science, materials science, chemical engineering, geophysics, medical imaging, and fluid dynamics. The first part of this project concerns changes in salinity or chlorofluorocarbon on the surface of the ocean. Oceanic vortices may dramatically change mixing rates of various chemical compounds. The aim of this project is to estimate these rates using simpler mathematical models to illuminate the mechanisms present in the full system. The second goal of this project is related to speeding up Markov Chain Monte Carlo algorithms. Markov Chain Monte Carlo methods are a major approach in machine learning, statistical methods, and scientific computing. This project will also involve the training of graduate student researchers. This work involves mathematical studies of particle propagation in cluttered environments and Monte Carlo methods for Langevin equations. The first project concentrates on the effect of long-range correlations. This represents a novel direction in the theory of propagation in random media, as approximate models of propagation in slowly decorrelating media have not yet been developed. The goal is to explain how long-range correlations lead to time-scale separation of various phenomena. The second project concentrates on speeding up the convergence towards stationary measures, which is fundamental for the development of effective Markov Chain Monte Carlo algorithms. This is particularly relevant to Statistical Physics and understanding machine learning algorithms. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.