Research

My research focuses on inverse problems and uncertainty quantification, incorporating elements of applied and computational mathematics, statistics, and scientific computing.  Broadly speaking, inverse problems involve finding the unknown causes of observed effects.  In the problems that I consider, these causes are typically the unknown inputs (or parameters) of a system, and the effects are some partial, noisy observations of the system components.  The systems are typically represented using mechanistic models governed by differential equations but can also involve more data-driven modeling approaches. 

The main goals of my research are to: (i) design efficient and robust numerical algorithms for system parameter estimation, and (ii) apply these algorithms to analyze real-world data.  My work has primarily focused on developing sequential algorithms within a Bayesian inference framework to assimilate time series observations but has also involved building new mathematical models and applying sensitivity analysis and parameter identification techniques.  While this work is widely applicable in many fields of science and engineering (e.g., signal processing, geophysics, numerical weather prediction), much of my focus has been on applications to biology and medicine.  I enjoy working on problems in these areas due to the potential for positive societal impacts in health care as well as the interdisciplinary scientific challenges presented.  These are challenging problems not only from the mathematical side but also from the life sciences perspective, and I appreciate the opportunity to collaborate with scientists across disciplines to tackle these challenges. 

While system parameters tend to be modeled as constants, certain systems stemming from real-world applications may involve unmeasured parameters that vary with time.  This project focuses on the development of new computational filtering techniques to estimate time-varying parameters (TVPs)  in systems of differential equations from limited, noisy observations of the system states.  

This work has been supported by grant number NSF/DMS-1819203 from the National Science Foundation.

When applying lasers in surgical procedures (e.g., for ablation or photothermal therapy), it is vital to understand and control the interactions between the laser light and the tissue being treated in order to achieve desired clinical outcomes.  We have developed a novel approach that uses thermal sensor measurements (e.g., from a thermal infrared camera) on the tissue surface and ensemble Kalman filtering to estimate the physical properties of the tissue (including the light absorption and scattering coefficients). 

This work is in collaboration with the COMET Lab at WPI.

Stroke is one of the leading causes of death and disability worldwide, and 87% of all strokes are classified as ischemic, i.e., caused by blockage in a blood vessel that results in oxygen deprivation to the brain and cell death in the affected brain area.  Our work aims to understand the role of microglia in this setting, focusing on the activation of microglial cells and the effects of bidirectional microglia phenotype switching (between the beneficial M2 phenotype and detrimental M1 phenotype) on the neuroinflammatory process post-ischemic stroke.