Inverse Problems Arising from Kinetic Theory and Applications

Recovering a hidden cause, which cannot be directly observed or measured, solely from its effected measurements is an important scientific problem. This challenge arises in a wide range of applications, and rich methodologies for reconstructing this unseen information are employed in various fields, including geophysics, medical imaging, biology, solar physics, remote sensing and machine learning. The main goal of this project is to advance the understanding of reconstructing hidden properties from measurable data. This project specifically aims to investigate fundamental questions and devise reconstruction methods to uncover unknown coefficients in kinetic equations, including the Boltzmann equation, which describes the dynamics of dilute gases with binary collisions, and the Fokker-Planck equation, used to describe the dynamics in a plasma. Mathematical tools will be developed to provide a theoretical understanding of emerging topics on these inverse problems. Moreover, the project integrates research and education, providing training and research opportunities for graduate students and postdoctoral researchers. The main objective of this project is to conduct fundamental research on nonlinear kinetic equations and associated partial differential equation models in both the forward and inverse frameworks. The work consists of three primary goals. The first goal centers on analyzing three nonlinear time dependent equations, the Boltzmann equation, the Bhatnagar-Gross-Krook equation, and a transport equation, with the goal of retrieving the unknown collision kernel and coefficients based on measurements taken on the boundaries. The second goal aims to uniquely and stably recover diffusion, absorption, and source coefficients from boundary measurements using the Fokker-Planck equation. The third goal involves an analysis of the Wigner equation and Schrödinger equations, with the purpose of understanding the relationship between quantum and classical mechanics. Due to the wide variety of kinetic equations across applications, novel ideas and analytical techniques will be designed to suit their distinct features, such as transport and interaction. This research will improve the mathematical understanding of inverse kinetic theory and related topics. 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.