Coupling PDE-Based Computational Inversion and Learning Via Weighted Optimization

This project aims at developing a mathematically rigorous computational framework to improve classical image reconstruction algorithms in several application areas, such as medical imaging and seismic imaging. Through intelligent utilization of the recent advances in deep learning techniques, the investigator plans to implement and analyze novel computational inversion methods that have the potential to be significantly more efficient than existing image reconstruction methods. This project also involves an integrated educational component that aims at training students at different levels. The investigator intends to organize a reading group for junior researchers to work on simplified problems from this project to enrich their research experiences. Research internships on the topics of this project will be offered to undergraduate students from the investigator's institution as well as some high school students in its surrounding area. The main technical novelty of the project lies in an efficient computational coupling strategy that integrates offline and online reconstruction algorithms to take their complementary advantages. The coupling is implemented with a weighted optimization method, a general framework to approximate novel objective functions such as those based on the Wasserstein metrics induced in optimal transport theory. In a nutshell, the proposed weighted optimization methods make it possible to reliably train neural networks to learn the dominant components (when represented in an appropriate basis) of an inverse operator. This learned approximate inverse is then utilized to improve model-based iterative inversion schemes. Specifically, the primary proposed research efforts in this project include three main components: (i) to develop a systematic understanding of the stability property of different components of inverse operators and design corresponding weighted optimization schemes to focus on such components in the learning process; (ii) to develop a general computational framework to couple the complementary benefits of learning and classical iterative reconstruction methods; and (iii) to characterize the impact of learning uncertainty on the reconstruction results from new data. 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.