Metric-Dependent Strategies for Inverse Problem Analysis and Computation

This project will develop novel approaches to solving inverse problems, which are pivotal in many scientific fields, including biology, geophysics, and medical imaging. Inverse problems often involve deducing unknown parameters from observed data, a task complicated by issues such as sensitivity to measurement noise and complex modeling procedures. The broader significance of this research lies in its potential to significantly enhance the accuracy and efficiency of computational methods used in critical applications such as electrical impedance tomography (EIT), inverse scattering, and cryo-electron microscopy (cryo-EM). For instance, improvements in cryo-EM computation will accelerate breakthroughs in molecular biology and aid in rapid drug development, directly benefiting medical research and public health. Additionally, this project will also (1) engage undergraduate and graduate students in research to foster a new generation of computational mathematicians, and (2) promote STEM careers among K-12 students through outreach activities. The technical focus of this project will be on the development of metric-dependent strategies to improve the stability and computational efficiency of solving inverse problems. Lipschitz-type stability will be established by selecting metrics tailored to the data and unknown parameters to facilitate more robust algorithmic solutions. A key highlight of the project will be the investigation of the stochastic inverse problem's well-posedness. Sampling methods inspired by metric-dependent gradient flows will serve as the novel computational tool for the practical solution of stochastic inverse problems. These analytical and computational strategies will be designed to handle the randomness inherent in many practical scenarios, shifting the traditional deterministic approach for solving inverse problems to a probabilistic framework that better captures the intricacies of real-world data. This research has the promise to not only advance theoretical knowledge in studying inverse problems but also to develop practical, efficient tools for a wide range of applications in science and engineering. 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.