Grad Students/Postdocs

My research involves finding verified, or mathematically guaranteed, numerical solutions of systems of nonlinear ordinary differential equations (ODEs), seeking both dynamic (transient trajectories) and static (equilibrium) solutions. ODE systems of two or more variables may permit limit cycle trajectories, and systems of three or more variables may include solutions that are deterministically chaotic. In my research, the ODEs of interest are often those of dynamics in biochemical kinetics, physiology, and population ecology or epidemiology.

Generally, numerical solutions of systems of equations are approximations to the true solutions, so I seek to improve tools that make use of interval methods and Taylor models to perform computations that mathematically guarantee the existence of the true solution between two (machine number) bounds. The majority of my research involves the Verified Solver for Parametric ODEs (VSPODE) developed by Lin and Stadtherr to provide transient solutions to systems of ODEs with uncertain initial conditions and/or parameters.

I have used VSPODE to investigate the well-studied Rosenzweig-MacArthur model, using artificially-created scenarios with uncertain initial conditions or uncertain parameters that straddled separatrix or bifurcation points in this model. I have also used the program to seek solutions to two-point boundary value problems (TPBVPs), utilizing an interval shooting method. I aim to implement the ideas of probability distribution variables (PDVs) or probability-boxes (p-boxes) in algorithms similar to those of VSPODE to compute probabilistic bounds on uncertain variables, effectively creating a stochastic-deterministic hybrid that is less computationally expensive than Monte Carlo processes. I have completed work in this area for several areas of mathematical research, including physiology, epidemiology, and chemical engineering models.

My other major interest is in education, which I am pursuing more informally through courses and workshops at Notre Dame's Kaneb Center for Teaching and Learning and through opportunities to serve as a teaching assistant, co-instructor, and instructor for chemical engineering courses. As of the fall of 2007, I work for the Kaneb Center as a teaching associate. I am also a participant in the College of Engineering's first Teaching Apprenticeship program, working with faculty across the College in developing the first-year engineering sequence. I also actively work to assist in the department's electronic portfolio initiative for undergraduate students.

My career goal is to use my PhD to obtain a teaching position in chemical engineering, general engineering, or applied mathematics. My CV can be found here.