Optimization-Based Moment Models for Multiscale Kinetic Equations

The focus of this proposal is the design, analysis, and implementation of optimization-based moment models for solving collisional kinetic equations with diffusive limits. In the field of computational kinetic theory, moment methods are an effective, yet highly complex tool for discretizing kinetic equations and capturing multiscale phenomena. They reduce the kinetic description of a many-particle system to a set of partial differential equations for velocity averages of the kinetic distribution. Optimization-based moment models are derived via the solution of a physically motivated, convex optimization problem that enforces important properties of the underlying kinetic equation. The driving application for this effort will be neutrino transport in core-collapse supernovae simulations, where the computational requirements for a fully resolved kinetic simulation are well beyond the capabilities of the largest supercomputers.

The design of tractable reduced models for complex systems is both timely and important. Indeed, for a broad range of physical, biological, and social science problems, methodologies are needed to extract relevant macroscopic features from intractable amounts of microscopic data. In this context, the design and implementation of moment models is an important part of understanding the behavior of many-particle systems, for which direct numerical simulation is not possible. Important examples include rarefied gases, radiation, and charged-particle transport---all of which play critical roles in advanced energy and technology applications of national interest. Thus, among other things, the project will provide a rewarding and relevant research experience to students and postdocs.