Node Generation on Surfaces and Bounds on Minimal Riesz Energy

Abstract

Discretizing a manifold is a far reaching subject throughout the mathematical and physical sciences. This thesis has two principal foci. We present and analyze a variety of algorithms for generating point configurations on d-dimensional sphere and the torus, as well introduce a generic strategy for generating locally quasi-uniform points of variable density on any full dimensional subset of Euclidean space. The methods and algorithms are concentrated on construction and computation, though we also prove some properties of distribution and mesh ratio. For the variable density nodes, we consider the particular application to atmospheric modeling with radial basis functions. We implement a parallelizable algorithm to initialize good starting configurations for efficient modeling.

Secondly, we prove a lower bound on the asymptotic Riesz minimal energy in the hypersingular case based off of the linear programming method. This general framework for obtaining lower bounds for minimal energy configurations on the d-dimensional sphere was developed by Yudin and based on the Delsart-Goethals-Seidel bounds on spherical designs. Combining these methods with Levenshtein's work on maximal spherical codes, explicit universal lower bounds are established depending only on the potential function for any monotone potential. We extend this to the asymptotic case as N approaches infinity. In addition, we apply this method to infinite Gaussian potentials on Euclidean space.