论文标题
一类最佳位置问题的三角晶格的渐近最优性
Asymptotic optimality of the triangular lattice for a class of optimal location problems
论文作者
论文摘要
我们证明了渐近结晶导致一类非局部粒子系统的二维。确切地说，我们考虑了给定的2-Wasserstein度量的绝对连续概率度量$ f \ mathrm {d} x $，通过离散概率度量$ \ sum_im_iΔ__iδ_{z_i} $，受粒子尺寸尺寸$ m_i $的约束。粒子的$ z_i $，它们的尺寸$ m_i $以及粒子的数量都是问题的未知数。我们研究一个参数的约束家族。这是最佳位置问题（或最佳采样或量化问题）的一个示例，并且在经济学，信号压缩和数值集成中具有应用。随着粒子数量流向无穷大，我们建立了（重新定化的）近似误差的渐近最小值。特别是，我们表明，对于通过离散度量限制了Lebesgue度量的最佳近似，其支持为三角形晶格的离散度量在渐近上是最佳的。此外，我们证明了一个问题的类似结果，即约束被惩罚替换。这些结果也可以看作是六角形瓷砖的渐近最优性，以解决最佳分区问题。他们将Bourne，Peletier和Theil（数学物理学的通信，2014年）的结晶结果从单个粒子系统转变为一类粒子系统，并证明了Bouchitté，Jimenez和Mahadevan的猜想案例（JournaldeMathématiquesPures Pures etaterpiquées，2011年）。最后，我们证明了一个结晶结果，该结果指出，具有接近三角形晶格的能量的最佳构型在几何上接近三角形晶格。
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 measure $f \mathrm{d}x$ by a discrete probability measure $\sum_i m_i δ_{z_i}$, 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 Bouchitté, Jimenez and Mahadevan (Journal de Mathématiques Pures et Appliquées, 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.
