Discrete Energy and Polarization Problems on Manifolds with Applications

在一个曲面上进行计算的点的结构会影响人们对曲面函数或曲面本身形状的近似程度。优化与点的排列相关的能量概念的配置可以导致更好的近似。该研究项目探索了这种最佳配置的有效实现，为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）提供了改进的工具。能量优化问题对各种曲面和体积的离散化具有重要意义。例如，由适当加权的Riesz能量产生的点配置可以近似于物体上的规定密度。研究小组(i)通过探索计算流形上接近最优能量配置的有效算法，在数值上实现了主要研究人员开发的理论，（ii）发展了最优能量问题的周期性版本的渐近理论，特别是它们与晶格（晶体）结构的联系。本科生和研究生以及博士后在研究项目中发挥着积极的作用。

The configuration of points on a surface at which computations are carried out affects how well one can approximate functions of the surface or the shape of the surface itself. Configurations that optimize some notion of energy associated with the arrangement of points can lead to better approximations. This research project explores efficient implementation of such optimal configurations, to provide improved tools for data sampling methods and for simulating complex geophysical processes such as climate changes and heat diffusion in Earth's mantle.Optimization problems for energy are of significance in discretizing various surfaces and volumes. For example, point configurations generated by a suitably weighted Riesz energy can approximate a prescribed density on the object. The investigative team is (i) implementing numerically the theory developed by the principal investigators by exploring efficient algorithms for the computation of near optimal energy configurations on manifolds, and (ii) developing the asymptotic theory of periodic versions of optimal energy problems and especially their connection to lattice (crystalline) structures. Undergraduate and graduate students as well as post-doctoral fellows play an active role in the research program.