Georgia Institute of Technology College of Sciences

Recently there has been an outburst of parallelization techniques to speed up optimization algorithms, particularly in applications in statistical learning and structured linear programs. Motivated by these developments, we seek for theoretical explanations of provable improvements (or the lack thereof) in performance obtained by parallelizing optimization algorithms. In 1994, Nemirovski proved that for low-dimensional optimization problems there is a very limited improvement that could be obtained by parallelization, and furthermore conjectured that no acceleration should be achievable by these means. In this talk, I will present new results showing that in high-dimensional settings no acceleration can be obtained by parallelization, providing strong evidence towards Nemirovski's conjecture. This is joint work with Jelena Diakonikolas (UC Berkeley).

We consider one or more volumes of a liquid or semi-molten material sitting on a substrate, while the vapor above is assumed to have the same medium in suspension. There may be both evaporation and condensation to move mass from one cell to another. We explore possible equilibrium states of such configurations. Our examples include a single sessile drop (or cell) on the plate, connected clusters of cells of the material on the plate, as well as a periodic configuration of connected cells on the plate. The shape of the configurations will depend on the type of energy that we take into consideration, and in settings with a vertical gravitational potential energy the clusters are shown to exhibit a preferred granular scale. The majority of our results are in a lower dimensional setting, however, some results will be presented in 3-D.