Current Research
Determining symmetry within a collection of spatially oriented points is a problem that occurs in many fields. Often, large amounts of data are collected, and it is desirable to approximate this data with a compressed representation. In many situations, this representation is calculated using the singular value decomposition (SVD). However, there are many situations where it is profitable to also preserve symmetry of the data in the compressed approximation. Therefore, I extend the SVD to provide a symmetry preserving singular value decomposition (SPSVD). This factorization produces the best symmetric low rank approximation to a matrix with respect to the Frobenius and matrix-2 norms.
Calculating an SPSVD is a two-step process. In the first step, a matrix representation for the symmetry of a given data set must be determined. I present this process as a novel iterative reweighting method: a scheme which is rapidly convergent in practice and seems to be extremely effective in ignoring outliers (points that do not respect the symmetry) of the data. Then, in the second step, the best approximation that maintains the symmetry calculated from the first step is computed. This approximation is designated the SPSVD of the data set.
In addition, I provide a means to evaluate just the dominant portion of the SPSVD (the best symmetric low rank approximation) that is well suited to large scale computation such as those involved in molecular dynamics. This computation only requires matrix-vector products involving the point set represented as a matrix, and is no more expensive than constructing the leading terms of the SVD of the full set of points without the symmetry constraint. The ARPACK software can be used to make these calculations.
I use the parallel version of ARPACK, PARPACK, to compute the SPSVD of molecular dynamic trajectories of proteins. Currently, researchers utilize the SVD to calculate the major modes of motion that best describe the flexibility of a protein. However, if a protein is symmetric, this symmetry may be lost in the calculation of the modes. The SPSVD yields the symmetric major modes of motion that best describe the symmetric movements of the protein. Moreover, the SPSVD may average out noise that enters the trajectory during the molecular dynamics simulations. Scientists may use the information from the modes to assist in drug design.
I am interested in extending my thesis work in three areas: generalizing the SPSVD to arbitrary symmetry groups, creating an efficient iterative update for the SPSVD, and extending the applications of the SPSVD.
Other Research
In the past I have worked in two other research areas. My first area of research modeled groundwater contamination. This study began at a Research Experience for Undergraduates (REU) program at Washington State University. My work combined an advective-dispersive equation, which described one-dimensional solute transport in soils under a steady water flow, with nonlinear sorption that was represented using the Freundlich isotherm. This research segued into my Master's thesis from Emory University. Participating in the REU program showed me the importance of undergraduate research. I would be interested in creating and/or participating in such research programs for undergraduates in the future.
The second area of research dealt with preconditioners of saddle point problems. Specifically, I concentrated on the effects of using right verses left preconditioning. I began by creating a bound that shows that if the condition number of the preconditioner is large, then there would be a large disrepancy between the left and right preconditioning residuals. Next, I proved that if there exist two similar matrices, where one system has slow GMRES convergence while the other is fast, then there exist a matrix, M, and preconditioner, P, such that MP-1 has fast convergence, while P-1M has slow convergence. I followed these theorems with abstract examples illustrating these effects. I am currently searching for a real-world cases that show the differences that may occur from using right verses left preconditioning. This work demonstrates the importance of using caution when choosing a preconditioner, and I would like to extend this research by searching for other classes of matrices that exhibit relationships between left and right preconditioning.