Asymptotic optimality of the triangular lattice for a class of optimal location problems

TL;DR

This paper proves that, in the limit of many particles, the triangular lattice configuration is asymptotically optimal for a class of particle approximation problems involving the 2-Wasserstein metric, with applications in economics and signal processing.

Contribution

It extends crystallization results to a broader class of particle systems, establishing the asymptotic optimality of the triangular lattice in optimal location problems.

Findings

Triangular lattice is asymptotically optimal for best approximation of measures.

Results generalize previous crystallization findings to new particle systems.

Optimal configurations with near-minimal energy are geometrically close to a triangular lattice.

Abstract

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 $f \mathrm{d}x$ by a discrete probability measure $\sum_i m_i \delta_{z_i}$, subject to a constraint on the particle sizes $m_i$. The locations $z_i$ of the particles, their sizes $m_i$, 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, Peletier and Theil (Communications in Mathematical Physics, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitt\'{e}, Jimenez and Mahadevan (Journal de Math\'ematiques Pures et Appliqu\'ees, 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.