research
I am interested in interdisciplinary research problems at the interface between mathematics, science and engineering. In particular, I study of wave propagation in random media with applications to biomedical optical imaging of tissues and wireless communications. Mathematically, these research areas include the study of differential and integral equations, asymptotic analysis, scientific computing and inverse problems. I have studied also nonlinear dynamical systems in the context of mode-locking fiber lasers. Below is a list of research projects I am currently pursuing. Numerical solution of the radiative transport equation. We are developing direct numerical simulation methods for solving boundary value problems for the radiative transport equation. High frequency wave propagation in random media, such as light propagation in tissues, is governed by the theory of radiative transport. The radiative transport equation takes into account absorption and scattering due to inhomogeneities in the propagating medium. It is a partial differential-integral equation. Analytical solutions of the radiative transport equation are known only for relatively simple problems. For practical problems, we need to compute numerical solutions. Numerical solutions are challenging to compute due to the large number of independent variables (three spatial variables + two angle variables + time). Parameter identification and estimation for reflectance optical spectroscopy. We seek to identify the key optical parameters of tissues that provide insight into tissue health and develop methods to estimate these parameters from measurements of backscattered light. The key challenge here is developing a comprehensive understanding of how optical properties such as absorption, scattering, particle size distribution, etc are manifest in reflectance measurements and how the determination of these optical properties provide useful information about tissue structure and composition. Image reconstruction for diffuse optical tomography. We seek analytical and computational methods to solve inverse problems for the radiative transport equation that compute tomographic reconstructions of the optical properties of tissues from measurements of scattered light. The key challenge here is to use the minimum data needed to reconstruct quality images with an understanding of the maximum resolution possible. Time reversal for ultra-wideband wireless communications. We are incorporating analytical and computational methods from wave propagation in random media, especially time reversal, into wireless communications system design and analysis. For wireless systems, multiple scattering adds space and angle diversity into the channel. Furthermore, broad bandwidths introduce more time and frequency diversity. Time reversal exploits multiple scattering to focus broadband signals in space and time. It provides a method to exploit all of the diversity introduced by the channel. Through this work, we seek to design next-generation systems with faster transmission rates, greater coverage and more reliability.