Collaborative Research: Breaking the 1D barrier in radiative transfer: Fast, low-memory numerical methods for enabling inverse problems and machine learning emulators

辐射传递方程出现在许多重要的应用中，如医学成像、天体物理学、天气和气候。例如，它描述了太阳光线在大气中传播并被云层吸收或散射时的行为。在这些应用中，经常使用计算机模拟来获得辐射传递方程的解。然而，由于需要大量的维度来描述每个空间位置和每个可能的传播方向（东西、南北、上下）的辐射强度，因此在这些模拟中出现了一个重大挑战。大量的维度需要大量的计算机内存和计算时间。由于这种高计算费用，通常使用简化，例如在天气和气候应用中使用一维（1D）近似或两流近似。该项目旨在克服这种一维屏障并解决完整的辐射传递方程，并通过快速，低内存的计算机模拟来实现。计算方法、对这些方法的理论理解以及软件工具的开发将提高对气候、天气和医学成像的理解，从而影响社会中个人的福祉。对数学和大气科学方面的博士后研究员和学生进行跨学科培训也是该项目的重要组成部分。通过向当地K-12学校开展外展活动，指导和扩大代表性不足群体学生的参与也将是该项目的一部分。该项目旨在开发快速、低内存的数值方法，以克服一维障碍并求解完整的辐射传输方程，这些方法包括用于其低内存特性的不连续伽辽金谱元方法，以及hp-自适应网格细化（hp-AMR），以处理医学成像或大气中云产生的陡峭梯度。除了解决给定大气状态下的辐射传递方程（即解决正向问题）之外，还将解决逆问题，其中使用辐射测量来推断大气状态。逆问题在医学成像、遥感和天气预报数据同化等方面有着重要的应用。hp- adaptive的目标导向版本将用于克服逆问题中出现的一些独特挑战。最后，基于机器学习的模拟器将使用上述方法实现的合成数据进行训练。为了更好地理解大气科学中的三维辐射效应，将分析来自观测和/或大涡模拟的云景数据。该奖项反映了美国国家科学基金会的法定使命，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 arises in many important applications, such as medical imaging, astrophysics, weather and climate. It describes, for example, the behavior of the sun's rays as they propagate through the atmosphere and are absorbed or scattered by clouds. In these applications, computer simulations are often used to obtain solutions to the radiative transfer equation. However, a substantial challenge arises in these simulations due to the large number of dimensions needed to describe the radiant intensity at each spatial location, and in each possible direction of propagation (east-west, north-south, up-down). The large number of dimensions requires a large amount of computer memory and computing time. Due to this high computational expense, it is common to use simplifications, such as a one-dimensional (1D) approximation or two-stream approximation in weather and climate applications. This project aims to overcome this 1D barrier and solve the full radiative transfer equation, and do so with fast, low-memory computer simulations. The computational methods, the theoretical understanding of these methods, and the development of software tools will improve understanding of climate, weather, and medical imaging, and thus influence the well-being of individuals in society. The interdisciplinary training of a postdoctoral researcher and students in mathematics and atmospheric science is also an important component of the project. Mentoring and broadening the participation of students from underrepresented groups, with outreach activities to local K-12 schools will also be part of the project.This project aims to develop fast, low-memory numerical methods that overcome the 1D barrier and solve the full radiative transfer equation, The methods include discontinuous Galerkin spectral element methods used for their low-memory properties, and hp-adaptive mesh refinement (hp-AMR) to handle steep gradients that arise in medical imaging or from clouds in the atmosphere. In addition to solving the radiative transfer equation for a given atmospheric state (i.e., solving the forward problem), the inverse problem will also be solved, where measurements of the radiation are used to infer the state of the atmosphere. The inverse problem has important applications in medical imaging, remote sensing and data assimilation for weather forecasting. A goal-oriented version of hp-adaptivity will be used to overcome some of the unique challenges that arise for the inverse problem. Finally, machine-learning-based emulators will be trained using synthetic data that is made possible by the methods above. To better understand 3D radiative effects in atmospheric science, data will be analyzed from cloud scenes from observations and/or large eddy simulations.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.