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@article{faucris.308496518,
abstract = {We describe a “discretize-then-relax” strategy to globally
minimize integral functionals over functions *u* in a Sobolev space
subject to Dirichlet boundary conditions. The strategy applies
whenever the integral functional depends polynomially on *u* and its
derivatives, even if it is nonconvex. The “discretize” step uses
a bounded finite element scheme to approximate the integral
minimization problem with a convergent hierarchy of polynomial
optimization problems over a compact feasible set, indexed by the
decreasing size *h* of the finite element mesh. The “relax” step
employs sparse moment-sum-of-squares relaxations to approximate each
polynomial optimization problem with a hierarchy of convex
semidefinite programs, indexed by an increasing relaxation order *ω*.
We prove that, as *ω* tends to infinity and *h* tends to zero, solutions
of such semidefinite programs provide approximate minimizers that
converge in a suitable sense (including in certain *L*^{p} norms) to the
global minimizer of the original integral functional if it is unique.
We also report computational experiments showing that our numerical
strategy works well even when technical conditions required by our
theoretical analysis are not satisfie},
author = {Fantuzzi, Giovanni and Fuentes, Federico},
doi = {10.1137/23M1592584},
faupublication = {yes},
journal = {SIAM Journal on Scientific Computing},
keywords = {Global minimization, calculus of variations, finite-element method, convex relaxation, sparse polynomial optimization, moment-SOS hierarchy},
peerreviewed = {Yes},
title = {{Global} minimization of polynomial integral functionals},
volume = {46},
year = {2024}
}