Applied Mathematics

Framework for solving and optimization of stochastic partial differential equations governing flow through porous media

My Ph.D. thesis was focused on solving partial differential equations in the presence of uncertainties (parameter, geometric or uncertainty in boundary conditions). I developed a mathematical framework to solve stochastic partial differential equations governing complex systems such as oil or groundwater flow in porous media, thermal heat conduction, manufacturing processes, accounting for uncertainties in geometry, boundary conditions and other parameters. In the framework, I develop a functional space method, that treats the unknowns as varying with not just space and time, but a “stochastic space” as well. A stochastic space can be considered as a sigma algebra with a probability measure. Stochastic models are created using the maximum entropy principle and stochastic gradient descent is used for optimization. We also developed information learning methods, the stochastic counterpart of machine learning, to accelerate generation of Maximum Entropy based models.

Multiscale uncertainty propagation of material properties from metallic polycrystals using the maximum entropy method
An information learning framework (stochastic machine learning using Kullback-Liebler divergence error measure) for generating stochastic micro-structure models.