In this project, the investigators will perform large scale computations to support research in three fundamental branches of mathematics: pure mathematics, applied mathematics, and statistics. One thing that ties these projects together is that each of the research computations can be scaled. This scalability is important, since all three will be leveraging a newly established campus computing resource at Georgia Tech which is designed to serve as a stepping stone to national supercomputing facilities such as the TeraGrid. Hence, there is the potential for this project to raise the visibility of the TeraGrid as a resource for broader mathematical research beyond the traditional areas of scientific computing. Undergraduate and graduate student training in both the computational methods and the underlying mathematical concepts will accompany all of these projects. Moreover, all electronic artifacts(e.g. software packages, datasets, etc.) developed from this work will be shared publicly.
The first project will require substantial computations related to knot theory. We will compute a wide variety of knot invariants for knots with 20 crossings, and an extensive table of colored Jones polynomials for knots with 15 crossings. These invariants will be loaded into a carefully structured database for efficient searching, testing of conjectures, and experimentation. The knot database will be a valuable resource to an entire community of researchers, including topologists, geometers, and even molecular biologists. For the second project in applied mathematics, we will run codes for the evolution of flexible bodies in inviscid fluids. This is a model used in the study of schooling fish, bird formations, and other interacting bodies in fluids. We hope to fill in some of the large gaps in our fundamental understanding of how flexible bodies interact with flowing fluids. The third project in statistics will focus on a stochastic processes defined by stochastic differential equations. The statistical inference for such processes faces major challenges due to their complexity and model observation structure. We will employ modern nonparametric statistical inference methods, which can be very computationally intensive, to build a solid framework and improve our understanding of these processes. Applications of this work includes modeling and forecasting portfolio risk with a more realistic portfolio of a few hundred securities.
In this project, the investigators acquired computing hardware to facilitate mathematical computations. In particular, we purchased 8 compute nodes that had 4 AMD Bulldozer CPUs each (for a total of 64 cores/node) and 256 Gb RAM each. In addition, we added 20TB of external storage. This hardware was all integrated into the campus HPC infrastructure at Georgia Tech (PACE) so that we could leverage existing power, cooling, networking, software, and system administration. The research supported by our computations falls broadly into three fundamental branches of mathematics: pure mathematics, applied mathematics, and statistics. The first project included several substantial computations related to knot theory. We computed a wide variety of knot invariants for all knots up 17 crossings. These invariants, which had never been fully available previously, were analyzed for certain patterns and have been submitted for inclusion in a future version of the Snappea software for others in the community to explore. An undergraduate student helped in the analysis of the knot data, and to date there have been six publications in this area which leveraged the hardware covered by this grant. The second project used the cluster hardware to simulate an axisymmetric bell-shaped swimmer and an axisymmetric vortex sheet for long times and with wide ranges of kinematic parameters. The length of the vortex sheet grows exponentially in time, so we developed a method to approximate the evolving vortex sheets with vortex sheets using a limited number of discretization points. Even with this method, the simulations required multiple days to reach the desired state of quasi-periodic body velocity due to the unsteadiness of high Reynolds number flows. We used a range of power-law kinematics, and determined the kinematics giving maximum swimming speeds and swimming efficiency. The simulations were extremely valuable for their physical insight and in comparisons with an analytical model, as well as (viscous) immersed boundary simulations. To date there has been at least one publication of this work. A third project performed a detailed numerical and computational analysis of the performance of higher order approximations for the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices. This work used several jump-diffusion models and stochastic volatility models with Levy jumps. Since the time to maturity could be very small, only the enhanced power made available by the new equipment truly allowed us to distinguish between first and higher-order approximations. Further computations were done by a graduate student in the area of small-time asymptotics and expansion of option prices. These computations ended up in that student's thesis. The hardware also played a major role in additional statistical work by another co-PI including several results on Jackknife empirical likelihood tests. To date there has been seventeen papers which benefited from the computational power provided by this project.
