"Fixing a Regularity Mismatch in PDE-constrained Boundary Control Problems"

John Urschel

Department of Mathematics

Massachusetts Institute of Technology

## Abstract:

Vector quantization is a classical signal-processing technique with significant applications in data compression, pattern recognition, clustering, and data stream mining. It is well known that for critical points of the quantization energy, the tessellation of the domain is a centroidal Voronoi tessellation. However, for dimensions greater than one, the conditions under which a centroidal Voronoi tessellation for a given density and domain is unique have been elusive, as have the conditions under which this tessellation minimizes the energy of the underlying quantization problem. We prove sufficient conditions for which there exists a unique centroidal Voronoi tessellation, which is both a local and global minimum of the quantization energy. In doing so, we also give conditions for global convergence of Lloyd’s algorithm to the global minimum.