In the past few years, I have been mostly working on forward and inverse kinetic theory. Kinetic theory is an umbrella term for a set of theories concerning the evolution of systems composed by many particles that follow the same physics laws. One model equation is the Boltzmann equation that characterizes a large number of rarefied gas particles with a statistical mechanics perspective.
In the forward setting, we try to understand the wellposedness of kinetic equations in different settings and design fast solvers to compute them. This leads to my work on boundary layer analysis for linearized kinetic equations, and a class of asymptotic-preserving/numerical homogenization methods. In the inverse setting, we try to understand when and how parameters in kinetic equations can be reconstructed. Numerically, PDE-based inverse problems come down to PDE-constrained optimization/Bayesian inference. We study how to build features of kinetic equations into inverse solvers to speed up the inversion process.
Interestingly, kinetic equation, due to its statistical nature, becomes fairly useful in many machine learning problems, when algorithms call for a large number of players that follow identical rules. Examples include ensemble type Bayesian sampling methods, in which samples interact to reconstruct a target distribution, and neural networks in the mean-field regime, in which neurons interact to find the best presentations of the underlying functions that map input-output data pairs. Kinetic theory, especially the mean-field limit theory, becomes the natural tool for analyzing these algorithms.
I also have a few side projects as interests go. It has been fascinating to collaborate with scientists and engineers to see how math unfolds in the real world. Ultimately, this is why we study math, right?
A lot of research conducted by the group and activities organized by the group members are summarized in
Kinetic At Madison
For publication list, please see
Publication list by topic (updated less frequently)
Publication list by year
