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In this paper we study a perturbative approach to the problem of quantization of probability distributions in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view [10,12,15], we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a new mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.
Keywords: Parabolic systems of PDEs, Gradient flow, Quantization of probability distributions, Wasserstein distance
@article{AIHPC_2018__35_6_1531_0,
author = {Caglioti, Emanuele and Golse, Fran\c{c}ois and Iacobelli, Mikaela},
title = {Quantization of probability distributions and gradient flows in space dimension 2},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1531--1555},
publisher = {Elsevier},
volume = {35},
number = {6},
year = {2018},
doi = {10.1016/j.anihpc.2017.12.003},
mrnumber = {3846235},
zbl = {1394.35219},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.003/}
}
TY - JOUR AU - Caglioti, Emanuele AU - Golse, François AU - Iacobelli, Mikaela TI - Quantization of probability distributions and gradient flows in space dimension 2 JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1531 EP - 1555 VL - 35 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.003/ DO - 10.1016/j.anihpc.2017.12.003 LA - en ID - AIHPC_2018__35_6_1531_0 ER -
%0 Journal Article %A Caglioti, Emanuele %A Golse, François %A Iacobelli, Mikaela %T Quantization of probability distributions and gradient flows in space dimension 2 %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1531-1555 %V 35 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.003/ %R 10.1016/j.anihpc.2017.12.003 %G en %F AIHPC_2018__35_6_1531_0
Caglioti, Emanuele; Golse, François; Iacobelli, Mikaela. Quantization of probability distributions and gradient flows in space dimension 2. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1531-1555. doi: 10.1016/j.anihpc.2017.12.003
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