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

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.