TY - JOUR

T1 - Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

AU - Bourne, David P.

AU - Cristoferi, Riccardo

N1 - Funding Information:
The authors would like to thank Giovanni Leoni, Mark Peletier, Lucia Scardia and Florian Theil for useful discussions, and Steven Roper for producing Figs. , and Table . This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) via the Grant EP/R013527/2 Designer Microstructure via Optimal Transport Theory.
Publisher Copyright:
© 2021, The Author(s).

PY - 2021/9/27

Y1 - 2021/9/27

N2 - We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure fdx by a discrete probability measure ∑imiδzi, subject to a constraint on the particle sizes mi. The locations zi of the particles, their sizes mi, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

AB - We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure fdx by a discrete probability measure ∑imiδzi, subject to a constraint on the particle sizes mi. The locations zi of the particles, their sizes mi, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

UR - http://www.scopus.com/inward/record.url?scp=85115761112&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04216-6

DO - 10.1007/s00220-021-04216-6

M3 - Article

VL - 387

SP - 1549

EP - 1602

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -