Optimal Weighted and Constrained Energy Configurations and Applications

研究者和他的同事研究了离散最小能量问题的理论和应用，这在紧化度量空间的离散化中具有重要意义。通过利用最小加权Riesz能量配置，可以在表面上生成近似于该表面上规定密度函数的点序列。这样的结构在物理和生物科学中有无数可能的应用。具体来说，该团队正在研究(1)分离和填充半径估计，以获得最佳n点加权能量配置；（ii）二维曲面上最优和近似最优加权能量构型的几何性质；（iii）按规定密度快速生成分布在流形上的点的算法；（iv）贪心能量点的性质；(v)地球物理应用中重要问题的近最优离散化。研究者和他的同事们研究了曲面和固体材料上带电粒子构型的稳定（最小能量）状态，特别是当这些粒子的密度受到限制或受到外力影响时。该项目的一个目标是确定产生最佳或接近最佳点电荷分布的有效数值方法，这些方法可用于帮助理解地球地幔的对流，用于地球发电机的建模，并更好地确定地球的引力场。这个项目在几个不同的背景下解决了如何最好地从模拟到数字转换的基本问题。

HardinDMS-1109266 The investigator and his colleagues study the theory and applications of discrete minimum energy problems that are of significance in the discretization of compact metric spaces. By utilizing minimal weighted Riesz energy configurations, one can generate sequences of points on surfaces that approximate prescribed density functions on that surface. Such constructions have a myriad of possible applications to the physical and biological sciences. Specifically, the team is studying (i) separation and fill-radius estimates for optimal N-point weighted energy configurations; (ii) geometrical properties of optimal and near optimal weighted energy configurations on 2-dimensional surfaces; (iii) algorithms for the fast generation of points distributed on a manifold in accordance with a prescribed density; (iv) properties of greedy energy points; and (v) near optimal discretizations for problems of importance in geophysical applications. The investigator and his colleagues study stable (minimum energy) states of charged particle configurations on curved surfaces and solid materials, especially when such particles are restricted in density or under the influence of external forces. One goal of the project is to determine efficient numerical methods for the generation of optimal or near optimal distributions of point charges that can be used to help understand the convection of the Earth's mantle, for the modeling of the geodynamo, and to gain a better determination of the Earth's gravitational field. This project addresses in several different contexts the fundamental problem of how best to convert from analog to digital.