Adaptive Hybrid Method for Radiative Transfer and Related Applications

该项目的总体目的是通过利用现有数值方法的优势，为基于光子的成像应用开发科学计算的新方向。这些应用（例如生物医学成像、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，找到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，但基于积分形式的方法还没有完全理解。在这个项目中，研究人员的重点是开发有效的计算工具的基础上，自适应混合配方结合微分和积分配方的优点。该算法将有利于广泛的一类正问题和反问题。它也可以扩展到基于非局部偏微分方程模型的各种应用。所提出的方法将有利于生物医学成像，国家安全和生物燃料的发展。将广泛传播所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。教育和培训计划在多个层次将提供给未来的研究人员在计算数学和相关的跨学科领域。与本研究相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。调查员的目标是通过为他们提供更多的机会和可能性，招募代表性不足的少数群体参与项目。 本项目由计算数学计划和激励竞争研究计划（EPSCoR）共同资助，将从RTE的积分公式出发，推导和分析RTE的自适应混合数值算法，并通过分析确定短程微分和长程积分算子的平衡组合。 在计算上，一个有效的近似的混合制剂涵盖短期和长期的相互作用，可能是一个合适的预条件RTE将开发。基于自适应混合公式的严格分析和快速数值重建算法将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 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.