Applications and analysis of discrete energy and polarization

该奖项支持主要研究人员的研究计划，主要涉及“离散能量问题”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导电壳上通过相互静电排斥相互作用的有限数量点电荷的配置-但与几个现代大规模科学计算应用有关。 通过大量但有限数量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（如气候变化和地幔中的热扩散）非常有用。 粒子通过两两排斥相互作用的最小能量方法将被用来产生这样的近似点配置。 研究生和博士后研究员将在研究计划中发挥积极作用，这将涉及与国家大气研究中心的科学家合作。调查小组正在（i）开发一个渐近分析的离散能量问题与外部领域的不可积Riesz内核，并应用该理论开发计算效率高的算法，以产生大-规模节点集建模指定的非均匀密度;（ii）分析能量渐近的周期性问题和它们的连接到最佳晶格（晶体）结构;（iii）调查的几何性质的最佳和接近最佳的极化配置流形上，特别是大量的点。

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.