My main field of interest is inverse problems. Usually, mathematical models are predicting consequences based on known causes. In inverse problems, one moves against the causality, seeking to solve the causes based on observed consequences. Inverse problems are characterized by their ill-posedness: A solution may not exist, or if it does, it may be non-unique, and small errors in data may be amplified to huge errors in the solution. These difficulties reflect the fact that in causal processes, information is often lost, as the second law of thermodynamics tells.

Probability theory, and Bayesian probability in particular, provides a natural framework to address inverse problems both theoretically and computationally. The idea dates back to the classical papers of Pierre-Simon Laplace and his "inverse probability". Moreover, the concept of prior provides a tool to augment ill-posed inverse problems by extra information that often render the problems well posed. Current computational resources make Bayesian methods practical and extremely useful, yet mathematically challenging. Bayesian methods constitute one of the central building blocks of the field of uncertainty quantification (UQ), which is one of my active research areas.

Mathematical modeling plays a central role in understanding the functioning of complex biological systems that are hard or impossible to observe directly. I am particularly interested in modeling the human brain, requiring a good understanding of different phenomena such as electrophysiology, energy metabolism, and hemodynamics, and the complex interplay between them. A particular challenge for a modeler is the multiscale nature of the phenomena, and matching models of different scales. Different imaging modalities give us indirect information about the functioning of the brain on a macroscopic level, while detailed models starting from cell physiology and biochemistry explain the functions on a microscopic level, making the bridging between models and observations a tremendous task.