About
Much of my research is centered around numerical algorithms. An ongoing effort is to derive solvers of the Navier-Stokes equation at high Reynolds numbers. The incompressible Navier-Stokes equation captures a great variety of phenomena and develops turbulent solutions at even low Reynolds numbers. Computing turbulent solutions at high Reynolds numbers requires grids with more than a billion points. Such large simulations are now possible on even a single computer or two. It is a common belief that computer simulations become more accurate as grids become finer. This belief is true most of the time, but when the grids become extremely fine, rounding errors and not discretization errors begin to dominate. Rounding errors increase as grids become finer. I have been deriving solvers that contain the effect of rounding errors and allow accurate simulations with finer and finer grids.
A part of my research has dealt with the interface to dynamical systems theory. Here I have developed methods that exhibit the fractal structure of the iconic Lorenz attractor in detail. Another contribution is a method for finding periodic orbits and other solutions of partial differential equations that are applicable to transitional turbulence.
In the past, I have worked on problems in probability theory. One result in this area deals with the rate of growth of random Fibonacci sequences. I have also worked on problems related to mixing times of Markov chains. A significant result related to card shuffling was obtained by Mark Conger who discovered a way to deal cards in bridge that appears to be a substantial improvement of the usual cyclic deal.
I have a general interest in fundamental problems of numerical analysis, which includes interpolation and approximation of functions and their derivatives. Most recently, I have become interested in connections to the Vapnik-Smale approach to statistical learning algorithms.
