Computational Forward and Inverse Radiative Transfer

辐射传输方程（RTE）是一种重要的建模工具，在生物医学成像、临床放射治疗计划、大气成分和结构研究等领域有着广泛的应用。由于其复杂的结构，RTE的分析和仿真提出了挑战。本研究计画旨在发展新颖、精确且有效的计算方法，以求解辐射传递正向与反向问题。更一般地说，在这个项目中获得的见解，预计将提供新的视角解决一类积分方程。正在开发的数学工具和计算算法将广泛传播，以推动科学和技术进步。不同层次的研究与教育相结合将为计算数学家提供培训。与拟议研究有关的监督研究项目和研讨会将提供给大三/大四本科生和研究生。将鼓励代表性不足的团体的成员参加。辐射传输方程的解在不同的区域/制度中表现得非常不同。这些挑战需要精心设计和充分理解的数值算法，可以考虑到特殊的属性和结构的RTE及其解决方案与底层应用程序。虽然研究和发展的数值方法RTE的基础上的微分方程和概率公式已经完成，积分公式为基础的方法是不太了解，并没有得到充分的发展。这个项目将系统地开发和分析基于积分公式的有效数值算法，并探索积分和微分公式的组合。其关键思想是利用降维，奇异性的仔细处理，以及由此产生的稠密矩阵的特殊结构来开发快速求解器。本研究适用于辐射流体动力学的数值模拟和随机介质中波传播的建模，并进一步应用于光学层析成像、光声层析成像、荧光成像等由RTE建模的反问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.