Theoretical and Computational Investigations of Minimal Energy Problems

Abstract

Let A be a d-dimensional compact subset of p-dimensional Euclidean space. For s in (0,d) the Riesz s-equilibrium measure is the unique Borel probability measure that minimizes the Riesz s-energy over the set of all Borel probability measures supported on A. In this paper we show that if A is a strictly self-similar d-fractal or a strongly Hausdorff d-rectifiable set, then the s-equilibrium measures converge in the weak-star topology to normalized Hausdorff measure restricted to A as s approaches d from below.

Additionally we describe numerical experiments on the 2-sphere involving discrete energies mediated by the Riesz s-kernel. These experiments provide approximate discrete minimal energies for N=20,...,200 and s=0,...,3 where, in the case s=0, the Riesz kernel becomes the logarithmic kernel. These experiments corroborate several conjectures regarding the asymptotic expansion as N goes go infinity of the minimal N-point energies. Further, the number of stable configurations observed as a function of N and s is reported. Finally, two algorithms used in this experiment are presented. The first minimizes the effect of roundoff error when computing sums of many terms, the second uses graph theory to speed the identification of isometries between collections of on the 2-sphere.