My research belongs to the area of partial differential equations. I work on nonlinear problems, applying both analytical theory and numerical experimentation. A common theme in my work has been attempting to answer questions about fluids. Currently I am struggling to understand pattern formation in water waves. I hope to find stable solutions of permanent form that might form the basis of a practical theory for inviscid 3-D water waves. I've also worked on inverse problems . What does this term mean? Typically, math problems are posed as forward problems: Given a well understand map, apply this map to some input and record the result. Inverse problems are not as easily characterized. You might have some output and a reasonable picture of the map, and want to find the associated inputs. Or you might know both the inputs and outputs, and would like to find some specific ingredients that go into the map.
Given a pressure field, is it possible to recover the associated wind velocity?
This problem is related to the Potential Vorticity (PV) inversion problem in atmospheric science. I worked for CIRA, with Mark DeMaria and Wayne Schubert. I explored various inversion methods to apply to the pressure field which could improve the estimated wind velocities. Although we didn't perfect a method, much was learned along the way. Here is review of some of things that we tried ( .pdf). I have a proposal for future work.
In working on inverse problems, I like to first understand the forward problem. I gain intuition rapidly by looking at many solutions of the forward problem. For this I use both Matlab and Femlab. I have found Femlab useful for problems involving multiphysics. A few such problems have been
I worked (sporadically) with Joe von Fischer, a biologist at Colorado State University. We are currently developing an effective model for methane diffusion in partially saturated soils. Joe's lab allows us to check our model continually, and refine it as necessary. The current work-in-progress webpage is here. We will soon be looking at wetland models, which require introducing transport terms.
I work on an unkown ingredient problem for the Richards equation. Richards equation is PDE that is commonly used to describe fluid flow in hydrology. My Ph.D. thesis advisor Paul DuChateau and I were able apply an adjoint method and recover soil parameters from surface observations. My thesis can be found here. A preprint from the initial stage this work is available here.
I would also like to analyze hysteresis, which occurs in many physical systems. I've been thinking about coupling with PDE with an underlying ODE whose solution would provide the action of the hysteretic coefficient. This would, in a sense, be an adaptation of the dynamical systems approach to hysteresis.
My current area of research is in the field of nonlinear waves. This is a relatively new area for me. It's an incredibly rich area, with applications including deep and shallow water waves, optics, Bose-Einstein Condensates and plasmas. The problems that I've been working on are motivated by patterns in three-dimensional surface water waves. Observations, both in nature and in the lab, suggest that waves of permanent form might exist. If this can be established mathematically, a practical theory for inviscid three-dimensional water waves is possible. For our models of surface waves, we use Euler's equation as well as the nonlinear Schrodinger (NLS), Korteweg-deVries (KdV) and the Kadomtsev-Petviashvili (KP) equations. Recently I have been numerically exploring stability of NLS solutions. A preprint of some early work is available here. This work is part of larger whole. The Focused Research Group (of which I was just a small cog) included specialists in the field including: physical experimentation, perturbation theory, scientific computation, algebraic geometry and mathematical analysis. There is a webpage for this effort.
I also continue to work on inverse problems. Lately I've been wondering if it possible to invert surface data (i.e. wave profile data) to gain information about the bottom profile. I would like to spend more time than I have working on this one.
Series Solution to ODE:
I've been working with a group at JMU on the applications of series methods. Recasting as a system of polynomial ODEs makes both the theory and numerical approximation of solution transparent. By projecting into natural coordinate system, the problem becomes one of analytic continutation. Directly linked to Automatic Differentiation and series, the solution is of arbitary order, making this a symplectic method. Parallels with the Cauchy-Kovaleski uniqueness are explored here, with hopes to extend to PDE. This work has led directly to our group's support of missile flight trajectory simulation. Power-series methods are typicallly proving to be one order of magnitude faster than typical Runge-Kutta methods, and require storage that is one or two orders magnitude less. Thanks to a priori error bound, and an efficent and free a posteriori error estimate, control and validation of of our numerically approximate solution is straightforward. The accessibility of iteratively computed series coffiencients, both numerically and symbolically, makes our analysis possible.