Mathematical Analysis for Kinetic Equations and Elliptic Equations

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This project addresses several fundamental questions in reconstructing unknown properties through nondestructive methods. These methods allow one to recover hidden parameters, which are usually unseen in nondestructive evaluation, from external observations. Related applications appear everywhere from the medical imaging in our daily lives, to the study of dynamics of our solar system and beyond, including the detection of tumor tissues in medical imaging, finding cracks and interfaces within materials, and the study of the Earth and solar interior. One primary component of this project aims to study central mathematical questions that arise from the investigation of the dynamics of dilute charged particles, and performance optimization for semiconductor devices. The methodologies developed in this project will excite innovative applications of the nondestructive method in scientific investigations. This project will integrate the research component with the educational training of graduate students, and will particularly address the involvement of underrepresented groups. This project will investigate inverse problems for the kinetic theory and elliptic equations. The major goal will be focused on fundamental questions and important applications related to these equations, with the aim of developing mathematical theories for the reconstruction of significant information from the given data. Specifically, the first part of this project is to study several kinetic equations in both forward and inverse settings with applications in plasma physics, semiconductor, and medical imaging. The topics include the identification of unknown properties in Boltzmann equations, which model the dynamics of dilute charged particles, and the investigation of material parameters and complex collision effects from measurable data. The second part of the project centers around the inverse boundary value problems for elliptic operators. The goal is to reconstruct unknown coefficients in linear and nonlinear elliptic equations that arise naturally in many physical phenomena in a bounded region from partial or full data on the boundary. In particular, the investigator will study uniqueness and stability issues in the reconstruction process. 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.