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.

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.

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.

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.