Professor Rustum Choksi
Department of Mathematics and Statistics, McGill University

Title: Optimal Quantization: Gersho's Conjecture in 3D and Navigating the Energy Landscape

Abstract: Nonconvex and nonlocal variational problems are pervasive in energy-driven pattern formation. Two central issues are:

(Q1) can one conjecture and prove asymptotic statements on the (geometric) nature of global minimizers.

(Q2) can one develop systematic, hybrid numerical algorithms to navigate (or probe) the energy landscape and access low energy states whose basin of attraction might be "tiny";

In this talk, we explore both (Q1) and (Q2) in the context of the simple, yet rich, paradigm of optimal quantization and centroidal Voronoi tessellations (CVT). We begin with (Q1) via the 3D Gersho's conjecture. Gersho's conjecture may be viewed as a crystallization conjecture and asserts the periodic structure, as the number of generators tends to infinity, of the optimal CVT. In joint work with Xin Yang Lu (Lakehead University), we present certain bounds which, combined with a 2D approach introduced by Fejes Toth and Gruber, reduce the resolution of the 3D Gersho's conjecture to a finite (albeit very large) computation of an explicit convex problem in finitely many variables.

In ongoing work with Ivan Gonzales and JC Nave (McGill University), we address (Q2) by presenting a new hybrid algorithm which alternates gradient descent (or Lloyd's method) with movement away from the closest generator.
We also address and partially explain some interesting observations about defect structures and optimal CVTs for a small number of generators on the 2D square torus.

Biography: Rustum Choksi received his Ph.D. in mathematics from Brown University in 1994. He then held post-doctoral positions at Carnegie Mellon University and the Courant Institute. From 1997 to 2010, he was a Faculty Member with the Department of Mathematics, Simon Fraser University. In 2010, he joined McGill University where he is currently a full professor in the Department of Mathematics and Statistics. His main research interests are in interaction of the calculus of variations and partial differential equations with pattern formation. He is currently the SIAM Program Director for the Activity Group on Mathematical Aspects of Materials Science and on the editorial board of the Journal of Nonlinear Science.