Rapid solution of minimal Riesz energy problems

In $\mathbb{R}^n$, $n\ge 2$ we obtain the numerical solution to both the unconstrained and constrained Gauss variational problems, considered for the Riesz kernel $\|x-y\|^{\alpha - n}$ of order $1 <\alpha< n$ and a couple of compact, disjoint, boundaryless $(n-1)$-dimensional $C^{k-1,1}$-manifolds $\Gamma_i, i=1,2$, where $k > (\alpha-1)/2$ each $\Gamma_i$ being charged with Borel measures with the sign $\alpha_i := \pm 1$ prescribed. Using the fact that such problems over a cone of Borel measures can alternatively be formulated as minimum problems over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space $H^{-\varepsilon/2}(\Gamma)$, where $\varepsilon := \alpha - 1$ and $\Gamma = \Gamma_1 \cup \Gamma_2$ (see [17]), we approximate the sought density by piecewise constant boundary elements and apply the primal-dual active set strategy to impose the desired inequality constraints. The boundary integral operator which is defined by the Riesz kernel under consideration is e ciently approximated by means of an $\mathcal{H}$-matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first order optimality system. Numerical results in $\mathbb{R}^3$ are given to demonstrate our approach.