This project is concerned with the computational study of some imaging, optimal design and inverse problems related to systems of kinetic equations that model the propagation of particles in complex heterogeneous media. The objective is to develop fast and robust numerical algorithms by intelligently utilizing information on the structures of both the model equations and the inverse problems that we obtained through computational and mathematical studies of them, sometimes in simplified settings. Precisely, the proposed research include (i) to develop fast forward and inversion algorithms by preconditioning the computation with computationally less expensive models; (ii) to develop efficient computational strategies for uncertainty quantification and variance reduction in the imaging and inverse problems involve randomness; and (iii) to develop and validate direct non-iterative methods for imaging in extended targets in heterogeneous media.
The main motivation for the study is the application of the problems in (i) optimizing the delivery of radiation therapy to caner patients; (ii) designing efficient nano-scale semiconductor devices; and (iii) imaging extended targets in highly-scattering random environments. The proposed research is expected to have long-term impacts in the practices of intensity-modulated radiation treatment planning for cancer treatment, semiconductor solar cell design for energy harvesting, and imaging in heterogeneous environment for homeland security applications. The project involves a significant education component that aims at training both undergraduate and graduate students. The ideas and techniques developed in this project will be incorporated into a graduate level class on numerical methods for imaging and inverse problems which will benefit graduates who are interested in applying advanced mathematical and computational techniques to solve real-world problems. The lecture notes for this graduate class will be made accessible to the general public through the PI's webpage hosted by the PI's institution.
