CAREER: Applicable Kinetic Computation with Boundaries and Rough Media

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Kinetic theory describes the dynamics of a large number of particles in a statistical manner. The theory is central to statistical mechanics and naturally connects the micro-world described by Newton's law and the macro-world described by the Navier-Stokes equations. Its applications arise from aerospace engineering, mechanical engineering, nuclear engineering, and atmospheric science. In all these engineering fields, rarified gas, photons, neutrons, and many other types of particles are modeled by kinetic equations. This research project aims to advance kinetic theory in both abstract and practical ways, developing novel theory that is widely applicable to systems of central interest in science and engineering. This research project concentrates on three main questions in kinetic theory: 1. theoretically and numerically understanding particles' interactions with boundaries/interfaces, with the focus placed on understanding the boundary layer behavior; 2. analytically and numerically relaxing homogeneity assumptions for the media, with both highly oscillatory and rough media considered; 3. studying uncertainties and sensitivities of the problem in both forward and backward manner, that is, tracing the propagation of perturbations in the forward setting to understand how error gets enlarged in the associated inverse problem. Due to the multi-scale multi-physics nature of kinetic theory, the theoretical results will largely be obtained from performing asymptotic analysis on the formal level and regularity theory on the rigorous level. To tackle the challenges in computation, the PI intends to incorporate solution structure given by partial-differential-equation analysis, and to explore randomized solvers that have seen success in data science and compressed sensing. Throughout the project, the PI intends to collaborate with engineers to ensure the techniques under development are truly applicable.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.