I am primarily interested in symbolic computation, computational topology, validated computing, optimization, and numerical analysis. Below is a summary of my research.
|2013||Solving the continuous nonlinear resource allocation problem with an interior point method
Abstract: The sparsity and algebraic structure of convex resource allocation problems have given rise to highly specialized methods. We show that the sparsity structure alone yields a closed-form Newton step for the generic primal-dual interior point method. Computational tests indicate that this makes the interior point method competitive with the leading special-purpose method on several important problem classes for which the latter can exploit additional algebraic structure. Moreover, the interior point method consistently outperforms the specialized method when no additional algebraic structure is used or available.
|2009||On the Solution to the Nonlinear Resource Allocation Problem Using Variable Fixing and Interior-Point Methods
Abstract: We exploit the sparsity and algebraic structure of the convex resource allocation problem to derive a method that is competitive with previous solutions. The method makes use of the sparsity structure in the Newton step of an interior point method, with the addition of Tapia indicators. Numerical results comparing the methods are included. More info.
|2007|| The SIAM 100.0000000-Digit Challenge: A Study In High Accuracy Numerical Computing Using Interval Analysis and Mathematica
Abstract: This Independent Study concerns itself with computational mathematics with an emphasis on three specific problems posed during the SIAM 100-Dollar, 100-Digit Challenge. The difficulty of the challenge is solving the problems with enough precision to get answers accurate to 10 digits. Most solutions put forth by contestants offered no proof of correctness for their answers. By using theory from the field of interval analysis, it will be possible to develop algorithms that will provide a solution within a desired accuracy and that is verifiably correct. To do so, I will be implementing these algorithms in Mathematica and using interval analysis to create computer-assisted proofs for the solutions. More info.
|2006|| Self-Similar Tilings of Nilpotent Lie Groups
Abstract: We construct self-similar fractal tilings on rationally graded nilpotent Lie groups. Specific examples and graphs of fractal tilings in the Heisenberg group are given. More info.
Talks / Posters
|2012||Connectivity in Semialgebraic Sets||ECCAD 12, Oakland University, Rochester, MI||Poster|
|2011||Connectivity in Semialgebraic Sets||SIAM AG11, North Carolina State University, Raleigh, NC||Minisymposium Talk|
|2009||On the Solution to the Nonlinear Resource
Allocation Problem Using Variable Fixing and Interior Point Methods
|Miami University, Oxford, OH||Master’s Project Defense|
|2007||The SIAM 100.0000000-Digit Challenge: A Study in High Accuracy Numerical Computing Using Interval Analysis and Mathematica||College of Wooster, Wooster, OH||Senior Thesis Defense|
|2007||Self-Similar Tilings of Nilpotent Lie Groups||College of Wooster, Wooster, OH||Colloquium Talk|
|2007||Self-Similar Tilings of Nilpotent Lie Groups||JMM 2007, New Orleans, LA||Special Session Talk|