My research is in analysis/PDE's in an area called "Inverse Problems." These problems usually have their origin in some physical phenomena like medical imaging, but the mathematics involved has a theoretical beauty as well. Beyond my graduate research, I am passionate about applied mathematics as a whole. I am intrigued by the ideas in numerical linear algebra as well as machine learning. I try to balance my interests in both pure and applied mathematics as I value them both. |
The electrical conductivity of the human body varies depending on which organs and tissues are involved. Exploiting this fact, electrical impedance tomography uses current and voltage measurements from nodes placed around the patient's chest to reconstruct the conductivity, giving an image of the cross-section. The mathematics behind this process comes from an inverse problem first proposed by Calderón in 1980. In 1996, Adrian Nachman provided a reconstruction algorithm for the conductivity, assuming it was smooth enough. However, Nachman's regularity assumptions rule out the discontinuous conductivities that are expected to occur in medical imaging applications since different tissues have distinct conductivities. Working with minimal assumptions, Astala and Päivärinta were able to reconstruct the conductivity in their 2006 paper. In my study of the inverse conductivity problem, I am starting with an approximation of a realistic conductivity by a sequence of smooth functions and studying how the techniques of Nachman and Astala-Päivärinta can be combined to get an accurate reconstruction.
In the summer of 2012, I participated in the NSF Research Experience for Undergraduates at George Mason University in Fairfax, VA. Under the guidance of my advisors Dr. J.E. Lin and Dr. Padmanabhan Seshaiyer and graduate student Maziar Raissi, I attempted to apply the Group Finite Element Method to the Shallow Water Equations. During the summer, we were able to get qualitative results akin to a proper method of characteristics solution.