New frontiers for Bayesian hierarchical models in inverse problems

The estimation of unknown causes of observed consequences, known as inverse problems, is a task arising in many important real world applications as wells as in science and engineering. Medical imaging, large language models, the structural health of infrastructure, among other areas, all rely heavily on the availability of fast and robust computational methods known as inverse solvers, especially when the data are parsimonious and noisy. This project will advance Bayesian inverse solvers that constitute the mathematical tools to utilize qualitative properties of these unknowns in a natural way while simultaneously providing a measure of the uncertainty associated with the solutions. In particular, this project will develop hierarchical Bayesian methods since they are particularly attractive for finding solutions when the salient information is consolidated economically into few features of the unknowns, a methodology that is referred to as sparse coding, or when the entries need to be of a prescribed type to facilitate the interpretation of the solution. Successful completion of this project has the potential to advance research in biotechnology, health sciences, and artificial intelligence. This project will combine hierarchical Bayesian techniques in inverse problems with novel ideas leveraging data science techniques and state of the art methods to address large scale computing challenges arising in a number of important real world applications. The targeted applications include functional magnetic resonance imaging (fMRI) of the brain, hemorrhagic stroke monitoring by electrical impedance tomography (EIT), muscle control identification in biomechanics and rehabilitation, fingerprinting of resting states in brain by magnetoencephalography (MEG), semantic and linguistics studies through large language models, and investment portfolio planning. The innovative combination of matrix-free techniques with Bayesian and data science methods will be the foundation and building blocks of algorithms that are fast, yield better solutions by taking advantage of cleverly designed priors, and more energy-efficient than approaches based on machine learning and neural networks. 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.