Research Page

My current research is in the area of Sensitivity Analysis. I am interested in the design, development and implementation of computational methods for sensitivity calculations. Some logical questions you might want to ask are:

What is a sensitivity?
Mathematicians and other scientists model physical systems using a set of mathematical equations --- these might be algebraic equations or ordinary differential equations or even partial differential equations. These equations will describe how some STATE VARIABLE such as temperature, velocity, pressure or population changes over time. Inevitably, there are also PARAMETERS floating around in the models. These parameters might describe physical properties such as thermal conductivity, permeability, porosity, birth and death rates, parameters that describe the shape of an airfoil or an airplane wing, parameters that describe the position of sensors and actuators within a control system on a MAV, etc. The sensitivity describes how small changes in a parameter value affect the state variables. In short, a sensitivity is a derivative (math word); it's the rate of change of the state variable per unit change in the parameter value.
How does one compute a sensitivity?
There are many ways that scientists attempt to compute sensitivities, and it varies largely with the application that one is considering. My current research focuses on the use of Continuous Sensitivity Equation Methods (CSEMs). This approach begins by "differentiating" the mathematical model in order to derive an equation (or a set of equations) for which the sensitivity is a solution. This equation is called the SENSITIVITY EQUATION, and deriving, solving, analyzing and approximating sensitivity equations has been the focus of my research for the last several years. (more than I care to count at this point)
How might one use a sensitivity?
Sensitivities provide for a direct gradient calculation in some optimal design problems where the constraints are defined by PDEs. They can also be used for parameter prioritization in model analysis, and some of my colleagues have used them in the quanitification of uncertainty in models. For the most part, I am referring to models of a large variety of engineering systems.

DESCRIPTION OF MY CURRENT RESEARCH PROJECT
And I am working with one Ph.D. candidate, Ms. Jennifer Thorenson.
One of my current research projects involves the study of two mathematical models used to describe transcription processes in bacteria, such as E. coli. One of the models is a stochastic model related to the TASEP models that are often used in the literature. The other is a nonlinear hyperbolic PDE in conservation law form. The basic equation was once suggested as a model for traffic flow in the 1950's. Jennifer is using Discontinuous Galerkin Methods to simulate various versions of the model as we try to incorporate realistic biological features such as density dependent velocities and transcriptional pauses. Experimental data suggests that certain types of transcriptional pausing are present in certain types of genes, and it has been hypothesized that the pauses serve as a regulatory mechanism to balance transcription and translation to optimize protein production by the cell. One of the goals of this project is to investigate the effect that these pauses have on overall protein production. We intend to do some sensitivity analysis on the locations, durations and interplay of multiple pauses. However, we are still tuning our model for coordination with realistic experimental data.