NSF Award
2309530
The overall purpose of this project is to develop new directions of scientific computing for photon-based imaging applications by bringing the advantages of existing numerical methods. The commonly used model for these applications (e.g. biomedical imaging, clinical radiotherapy treatment planning, security scan) is known as the radiative transport equation (RTE). Computationally, finding an accurate solution to RTE is very challenging due to its high dimensionality. Although many studies and developments of numerical methods solving RTE are based on differential formulation, the methods that are based on integral formulation are not fully understood. In this project, the investigator focuses on developing efficient computational tools based on the adaptive hybrid formulation by combining the advantages of both differential and integral formulations. The algorithm will benefit a broad class of forward and inverse problems. It could also be extended to various applications based on nonlocal PDE models. The proposed methods will benefit biomedical imaging, national security, and biofuel development. The developed mathematical tools and computational algorithms will be disseminated broadly to advance scientific and technological progress in these areas. Education and training plans at multiple levels will be provided for future researchers in computational mathematics and related interdisciplinary areas. Supervised research projects and seminars related to this proposed research will be available to junior/senior undergraduates and graduate students. The investigator aims to recruit underrepresented and minority groups to participate in the project by providing them with more opportunities and possibilities. This project is jointly funded by the Computational Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).<br/><br/>The project will start with the integral formulation of RTE to derive and analyze the adaptive hybrid numerical algorithm for it. The analysis will help to determine a balanced combination of short-range differential and long-range integral operators. Computationally, an efficient approximation of the hybrid formulation covering both short and long range interactions that could be a suitable preconditioner for RTE will be developed. Rigorous analysis and fast numerical reconstruction algorithms based on the adaptive hybrid formulation will be investigated for nuclear resonance fluorescence imaging, fluorescent lifetime imaging, RTE identification, and phototaxis navigation.<br/><br/>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.
