Applications and analysis of discrete energy and polarization

该奖项支持首席研究人员的研究计划，他们主要关注“离散能源问题”的数学和计算方面。这些都是经典根的问题--最简单的例子是确定通过相互静电斥力相互作用的导体壳层上有限数量的点电荷的配置--但与几个现代大规模科学计算应用有关。用大量但有限的点表示表面和体积对于数据采样方法和模拟复杂的地球物理过程很有用，例如气候变化和地幔中的热扩散。粒子通过成对排斥相互作用相互作用的最小能量方法将被用来产生这样的近似点构型。研究生和博士后将在研究计划中发挥积极作用，这将涉及到与国家大气研究中心的科学家合作。研究团队正在(I)为不可积Riesz核开发具有外场的离散能量问题的渐近分析，并正在应用该理论开发计算高效的算法，以生成模拟特定非均匀密度的大规模节点集；(Ii)分析周期问题的能量渐近性及其与最优晶格(晶体)结构的联系；(Iii)研究流形上最优和接近最优极化配置的几何性质，特别是对于大量的点。

This award supports the research program of the Principal Investigators concerned primarily with mathematical and computational aspects of "discrete energy problems." These are problems with classical roots -- the simplest example is that of determining the configuration of a finite number of point charges on a conducting shell interacting via mutual electrostatic repulsion -- yet are relevant to several modern large-scale scientific computing applications. Representing surfaces and volumes by a large but finite number of points is useful for data sampling methods and for simulating complex geophysical processes such as climate changes and heat diffusion in Earth's mantle. Minimal energy methods for particles interacting via pairwise repulsive interactions will be used to generate such approximating point configurations. Graduate students and post-doctoral fellows will play an active role in the research program, which will involve collaboration with scientists at the National Center for Atmospheric Research.The investigative team is (i) developing an asymptotic analysis of discrete energy problems with an external field for nonintegrable Riesz kernels and is applying the theory to develop computationally efficient algorithms for generating large-scale node sets modeling a specified nonuniform density; (ii) analyzing energy asymptotics for periodic problems and their connection to optimal lattice (crystalline) structures; (iii) investigating the geometrical properties of optimal and near optimal polarization configurations on manifolds, especially for large numbers of points.