Uncertainty quantification and numerical analysis

The quantification of uncertainty in the results of numerical procedures is a topic receiving a lot of attention, to the point of having uncertainty quantification emerge as a field of applied mathematics on its own. The tools for quantifying the uncertainty in solutions require a rather wide range of mathematical and statistical tools, and in some case have inspired new approaches to solve classical problems. One of my current research interests is to understand how the probabilistic and statistical tools used in uncertainty identification can be applied to state of the art numerical analysis algorithms to improve their performance and assess the reliability of the results produced

As part of this project, I am revisiting Krylov subspace iterative linear solvers from the Bayesian perspective, encoding a priori statistical descriptions of the expected solution and statistical information about the error in the data in the form of left and right preconditioners. One of the goals of this investigation is to understand the spectral properties of the transformed problem change as the sophistication of the probabilistic models increases, and whether the probabilistic point of view can be used to improve the computational schemes. This project is supported in part by an NSF-DMS grant.

The mathematics of human brain

This project started within the mission of the NIH funded Center for Modeling Integrated Metabolic Systems to designing predictive computational models of the metabolic processes occurring in human cells. My current effort is concerned advancing the understanding of the interplay in human brain of biochemistry, electrophysiology and hemodynamics. To this end I am carrying out separate research project in each of these three areas, and at the same time investigating to couple them in a physiologically and anatomically realistic manner. This project requires a lot of mathematical modeling and computational skills, in addition to a good understanding of the brain biochemistry, electrophysiology and hemodynamics. This project is sponsored in part with a Simons Foundation Collaborative Grant.

The MEG inverse problem of meditation

The goal of this project is to employ a Bayesian framework, state of the art computational algorithms and a novel methodology for solving the MEG inverse problem to study how different forms of meditation change the patters of brain activation. In the first part of the project, we have designed a very fast MEG inverse solvers with higher resolution in the deep brain that available software. The computational efficiency of the method is essential to take advantage of the millisecond resolution of MEG while analyzing hours of mediation data. The project is likely to lead to new paradigms for the analysis of large time series, as well as shed some light on how the brain adapts to different resting and meditative states. Part of this project has been carried out while I was a Simons Fellow.

Multiscale modeling of gas transport through gas channels in living cells

This NIH funded project studies the transport of gas across cell membranes via multiscale mathematical models to understand the role of gas channels. The mathematical models span different scales, from submicron to centimeter scale, and the computer simulations require the design of computationally efficient algorithms. Parameter estimation is part of the project, as is the physiological interpretation. This project is carried out in collaboration with Dr. Boron’s lab in Physiology and Biophysics at Case Western and Dr. Tajkhorshid’s lab at the University of Illinois at Urbana-Champain.