Past Event: Oden Institute Seminar

Global minimization of polynomial integral functionals

Federico Fuentes, Assistant Professor, Institute for Mathematical and Computational Engineering (IMC), Pontificia Universidad Católica de Chile

Abstract

Many nonlinear integral energy functionals found in practice in continuum mechanics, such as strain energies, which we are interested in *globally* minimizing, are nonconvex, with multiple local minima and a complicated energy landscape. This makes the computation of their global minima very challenging and, except in select cases, far from guaranteed. The usual methods typically only ensure finding approximations of a local minimum through gradient descent or some version of Newton’s method on the Euler-Lagrange partial differential equations (PDEs) associated with the functional, but say nothing of whether the solution is a global minimum.

In this talk, for energy functionals with polynomial nonlinearity, we present an algorithm that provably converges to a global minimum and its corresponding minimizer, and can be applied to problems in nonlinear elasticity, fluid mechanics, pattern formation and PDE analysis. We do this by discretizing functions with finite element discretizations, and then leverage results from approximation theory of Sobolev spaces, calculus of variations, and most importantly, powerful representation theorems of sum-of-squares (SOS) polynomials coming from real algebraic geometry in the field of sparse polynomial optimization. More precisely, we show that as the mesh is refined and a relaxation parameter (associated to a polynomial degree) is raised, the computed results of a semidefinite program (SDP) converge to the global minimum.

We present numerical examples which result in excellent approximations to the global minima of different nonlinear functionals, including the pattern-forming Swift-Hohenberg free energy in two spatial dimensions, and talk about the extension to PDE-constrained minimization of such functionals, and how to use these methods in practice to produce "warm" initial guesses for Newton methods.

Biography

Federico Fuentes is an Assistant Professor at the Institute for Mathematical and Computational Engineering (IMC) of the Pontificia Universidad Católica de Chile in Santiago. He obtained his PhD from the Oden Institute at the University of Texas at Austin, an MSc degree from Imperial College London, and BSc degrees from the Universidad de los Andes in Colombia. Prior to his position, he held an HC Wang Assistant Professorship at Cornell University. His research interests lie between engineering, physics and mathematics, in the numerical analysis of partial differential equations, particularly finite element methods, applied functional analysis and the use of polynomial optimization techniques in physical systems with nonlinear dynamics.