Seminario IMC: Global minimization of polynomial integral functionals: from calculus of variations to real algebraic geometry
21 de Noviembre, 2025
El Instituto de Ingeniería Matemática y Computacional (IMC) invita al seminario que se dictará este miércoles 26 de noviembre.
Miércoles 26 de noviembre de 2025, 13:40 hrs. (Presencial en auditorio Edificio San Agustín. Link Zoom disponible escribiendo a imc@uc.cl)
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.
BIO
Federico Fuentes is an Assistant Professor at the Institute for Mathematical and Computational Engineering (IMC) of the Pontificia Universidad Católica de Chile. He held an H.C. Wang Assistant Professorship in the Department of Mathematics of Cornell University from 2018-2021. His PhD (2018) is in computational and applied mathematics from the Oden Institute for Computational Engineering and Sciences of The University of Texas at Austin. He previously obtained a MSc in computational methods in aeronautical engineering from Imperial College London, and BSc degrees in both mathematics and mechanical engineering from the Universidad de los Andes (Colombia). His research interests lie in the interdisciplinary intersection of mathematics, physics, engineering and computational science, mostly in finite element discretizations and optimization techniques used to study and solve partial differential equations (PDEs) which arise from continuum mechanics.
