Extremal Eigenvalue Problems in Optics, Geometry, and Data Analysis

Extremal Eigenvalue Problems in Optics, Geometry, and Data Analysis

Braxton Osting

Department of Mathematics, University of California Los Angeles (UCLA)

Date and time: 

Tuesday, November 19, 2013 - 11:30am

Location: 

ECCR 257

Abstract: 

Since Lord Rayleigh conjectured that the disk should minimize the first eigenvalue of the Laplace-Dirichlet operator among all shapes of equal area more than a century ago, extremal eigenvalue problems have been an active research topic. In this talk, I'll demonstrate how extremal eigenvalue problems arise in a variety of contexts, including optics, geometry, and data analysis, and present some recent analytical and computational results in these areas. One of the results I'll discuss is a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet energies of the partition components. With  intuition gained from an analogous continuous problem, we introduce a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations.