Jan Glaubitz: Computationally efficient inference for sparsity-promoting hierarchical Bayesian models

Abstract: How can we reconstruct high-quality images from measurement data? How can we uncover biochemical reaction mechanisms from experiments? And how can we quantify the uncertainty in the predictions of mathematical models such as neural networks?

These questions arise across a wide range of applications—from medical imaging, remote sensing, and data assimilation to computational chemistry, biology, and neuroscience. Despite their differences, these problems share a common mathematical structure: they can all be formulated as inverse problems—in which unknown quantities must be inferred from indirect and noisy observations.

In this talk, I will adopt a Bayesian perspective to inverse problems—modeling unknown quantities probabilistically—enabling uncertainty quantification in reconstructions and downstream predictions. This, in turn, supports more trustworthy scientific inference and informed decision-making. Specifically, I will focus on hierarchical sparsitypromoting priors and discuss recent advances in their computationally efficient inference. To this end, I will draw on techniques from numerical analysis, optimization, and measuretransport theory. Topics will include block-coordinate descent methods for maximum a posteriori (MAP) estimation and accelerated sampling using prior-normalizing transport maps.

Parts of this talk are based on joint work with Youssef Marzouk (MIT), Anne Gelb (Dartmouth College), Jonathan Lindbloom (Dartmouth College), and Mirjeta Pasha (Virginia Tech). Keywords: Bayesian inverse problems, sparse Bayesian learning, uncertainty quantification, MAP estimation, sampling, transport maps.