Discrete Energy and Polarization Problems on Manifolds with Applications

进行计算的表面上的点的配置影响人们如何近似表面的函数或表面本身的形状。优化与点的排列相关的能量概念的优化可以导致更好的近似。本研究项目旨在探索这种优化配置的有效实现，为数据采样方法和模拟复杂的地球物理过程（如气候变化和地幔热扩散）提供改进的工具。能量优化问题在离散各种表面和体积方面具有重要意义。例如，由适当加权的Riesz能量生成的点配置可以近似于对象上的规定密度。 该研究小组正在（i）通过探索用于计算流形上接近最佳能量配置的有效算法，在数值上实现主要研究人员开发的理论，以及（ii）开发最佳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）结构的连接。 本科生和研究生以及博士后研究员在研究计划中发挥积极作用。

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.