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 uncertainty in predictions from mathematical models, such as neural networks?


Such 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 structure: they can all be posed as inverse problems, in which unknown quantities (such as tissue properties or reaction parameters) must be inferred from indirect, noisy observations.
In this talk, I will outline recent advances in sparsity-promoting computational methods for efficiently solving inverse problems. I will also show how adopting a Bayesian perspective—treating unknown quantities probabilistically—enables uncertainty quantification in reconstructions and downstream predictions. This, in turn, supports more trustworthy scientific inference and decision-making. Furthermore, I will outline some recently developed approaches for efficient inference in hierarchical sparsity-promoting Bayesian models, including sparse Bayesian learning, that blend ideas from numerical analysis, optimization, and measure-transport theory.
