Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems
Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems
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DOI：
10.1007/s00220-021-04216-6
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发表时间：
2020-12
影响因子：
2.4
通讯作者：
D. Bourne;R. Cristoferi
中科院分区：
文献类型：
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作者：
D. Bourne;R. Cristoferi
证明了一类非局部粒子系统在二维空间中的渐近结晶结果。为了更准确地说，我们考虑的最佳逼近相对于2-Wasserstein度量的一个给定的绝对连续的概率measureby一个离散的概率measure，受到约束的颗粒尺寸。粒子的位置、大小和数量都是这个问题的未知数。我们研究了一个单参数的家庭的约束。这是一个最佳位置问题（或最佳采样或量化问题）的例子，它在经济学，信号压缩和数值积分中有应用。我们建立的渐近最小值的（重新缩放）近似误差的粒子数趋于无穷大。特别地，我们证明了对于Lebesgue测度的离散测度的约束最佳逼近，其支撑是三角格的离散测度是渐近最优的。此外，我们证明了一个类似的结果的问题，其中的约束被替换为惩罚。这些结果也可以看作是最优划分问题的六边形镶嵌的渐近最优性。他们将Bourne等人（Commun Math Phys，329：117-140，2014）的结晶结果从单粒子系统推广到一类粒子系统，并证明了Bouchitté等人（J Math Pures Appl，95：382-419，2011）的猜想。最后，我们证明了一个结晶的结果，它指出，最佳配置的能量接近的三角形晶格的几何接近的三角形晶格。
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 measureby a discrete probability measure, subject to a constraint on the particle sizes. The locationsof the particles, their sizes, 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.