Inverse problems are concerned with finding the unknown causes of observed effects. This includes finding the internal structure of an object from external measurements, as in Computerized Tomography and many medical imaging applications, or the parameter values for systems of linear and linear equations that are best aligned with the observed output.
The brain is a complex organ with a variety of functions. My interests in brain modeling span a rather wide range that includes its metabolism, hemodynamics and electrophysiology, and their mutual interactions. There are several mathematical and computational challenges in brain modeling arising from the differences in spatial and temporal scale of its physiological functions
Uncertainty is naturally modeled in probabilistic terms, and while we obviously do not know quantities that we try to estimate, we often have some expectations about them. Bayesian inference is the natural setting for expressing our combination of uncertainty and a priori belief. A large portion of my research combines Bayesian inference and scientific computing.