Past Event: Babuška Forum

Sparse polynomial approximation via nonlinear approximations of high dimensional functions

Clayton G. Webster, Senior Scientist, Oden Institute, UT Austin; Distinguished Research Fellow, Lirio AI Research; and Department of Mathematics, Virginia Tech and Auburn University.

Abstract

In this presentation, we will present both convex and non-convex minimization techniques for approximating complex functions in high dimensions. Of particular interest is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we develop a novel weighted minimization procedure with a precise choice of weights, and a modification of the iterative hard thresholding method, for imposing the downward closed preference. Moreover, the recovery of the corresponding best approximation using our methods is established through an improved bound for the restricted isometry property and a new theory for non convex optimization. We will also present theoretical results that reveal our new computational approaches possess a provably reduced sample complexity compared to existing compressed sensing, least squares, and interpolation techniques. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the new weighted minimization method.

Biography

Clayton Webster is a Senior Scientist in the Oden Institute for Computational Engineering and Sciences at The University of Texas and a Distinguished Research Fellow at Lirio AI Research.  He is also jointly appointed in the Department of Mathematics at Virginia Tech and Auburn University.  Before these appointments, he was a Distinguished Professor in the Department of Mathematics at The University of Tennessee and a Distinguished Scientist and Group Leader in the Computational and Applied Mathematics Group at Oak Ridge National Laboratory.  Previously, Dr. Webster has served as the Director of Quantitative Trading at NextEra Energy Resources, Power Trading LLC., and the John von Neumann Fellow at Sandia National Laboratories.  In addition, his worked has earned him numerous accolades, including the Department of Energy Career Award as well as being appointed as a Frontiers of Science Fellow, by the National Academy of Sciences. Dr. Webster’s research has been supported by a variety of organizations, including the: US Department of Energy, US Department of Defense, National Science Foundational, and several US corporations.  Clayton currently serves as Editor-in-Chief or Numerical Methods for PDEs and several national and international conference organizing committees as well as numerous editorial boards.  He received his Ph.D. under the supervision of Prof. Max Gunzburger, in Mathematics from Florida State University in 2007.  He also earned a M.Sc. and B.Sc. from McMaster University in 2003 and 2001 respectively.