Riesz energy functionals and their applications

Abstract

First-order asymptotics of a class of functionals with the Riesz kernel $ |s x - s y|^{-s} $, equipped with an external field and a weight, is obtained. Characterization of the weak$ ^* $ limit of the discrete minimizers of such functionals on rectifiable sets is given, and expressed in terms of the $ Gamma $-convergence of functionals on the space of probability measures. Optimality of local covering and separation of the discrete minimizers is proved. Several properties of the minimizers of the unweighted energy functional are established in the case of optimization on a self-similar fractal, rather than a rectifiable set; in particular, the weak$ ^* $ limit of a sequence with the lowest asymptotics is characterized, and the structure of sequences achieving different asymptotics on a self-similar fractal with equal scaling coefficients is described.

A number of numerical examples involving distribution of discrete configurations on surfaces are given, as well as inside domains of full dimension, with both smooth and non-smooth boundaries. Applications of the Riesz energy functionals to constructing RBF-FD stencils are described and characterized in several cases, including an instance of a configuration for geo-scale atmospheric modeling.