Computational Forward and Inverse Radiative Transfer

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The radiative transfer equation (RTE) is an important modeling tool with applications in biomedical imaging, clinical radiation therapy treatment planning, study of the composition and structure of atmosphere, and other fields. Because of its complicated structure, the RTE presents challenges to analysis and simulation. This research project aims at developing novel, accurate, and efficient computational methods for both forward and inverse radiative transfer problems. More generally, the insights obtained in this project are expected to provide new perspectives on solving a class of integral equations. The mathematical tools and computational algorithms under development will be disseminated broadly for advancing scientific and technological progress. Integration of research with education at different levels will provide training for computational mathematicians. Supervised research projects and seminars related to the proposed research will be available to junior/senior undergraduates and graduate students. Participation of members of underrepresented groups will be encouraged.Solutions to radiative transfer equations behave very differently in different regions/regimes. These challenges require well-designed and well-understood numerical algorithms that can take into account special properties and structures of the RTE and its solution associated with the underlying application. Although studies and developments of numerical methods for RTE based on differential and probabilistic formulations have been done, integral-formulation-based approaches are less understood and not fully developed. This project will systematically develop and analyze efficient numerical algorithms based on integral formulation and also explore a combination of integral and differential formulations. The key idea is to utilize dimension reduction, careful treatment of singularity, and the special structures of the resulting dense matrix to develop fast solvers. This research is applicable to numerical simulation of radiation hydrodynamics and modeling of wave propagation in random media.Further applications in inverse problems that are modeled by RTE, such as optical tomography, photoacoustic tomography, fluorescence imaging, etc., will be studied as well.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.