Adaptive Hybrid Method for Radiative Transfer and Related Applications

这个项目的总体目的是通过发挥现有数值方法的优势，为基于光子的成像应用开发新的科学计算方向。这些应用(如生物医学成像、临床放射治疗计划、安全扫描)的常用模型称为辐射传输方程(RTE)。在计算上，由于RTE的高维性，寻找其精确解是非常具有挑战性的。虽然许多求解RTE的数值方法的研究和发展都是基于微分形式的，但基于积分形式的方法并没有得到完全的理解。在这个项目中，研究人员致力于结合微分和积分公式的优点，开发基于自适应混合公式的高效计算工具。该算法将使一大类正问题和反问题受益。它还可以扩展到基于非局部PDE模型的各种应用。建议的方法将有利于生物医学成像、国家安全和生物燃料开发。将广泛传播开发的数学工具和计算算法，以推动这些领域的科学和技术进步。将为计算数学和相关跨学科领域的未来研究人员提供多层次的教育和培训计划。与这项拟议研究相关的受监督的研究项目和研讨会将提供给初级/高级本科生和研究生。调查员的目标是通过为他们提供更多的机会和可能性，招募代表人数不足的少数群体参与该项目。该项目由计算数学计划和已建立的激励竞争研究计划(EPSCoR)共同资助。该项目将从RTE的积分公式入手，推导和分析其自适应混合数值算法。这一分析将有助于确定短期微分和远程积分算子的平衡组合。在计算上，将开发一种涵盖短程和长程相互作用的混合公式的有效近似，这可能是一种适合于RTE的预条件。基于自适应混合公式的严格分析和快速数值重建算法将被研究用于核共振荧光成像、荧光寿命成像、RTE识别和趋光性导航。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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).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.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.