Rapid Solution of Minimal Riesz Energy Problems

In R n , n ≥ 2, we compute the solution to both the unconstrained and constrained Gauss variational problem, considered for the Riesz kernel ∥ x − y ∥ α − n of order 1 < α < n and a pair of compact, disjoint, boundaryless ( n − 1)-dimensional C k − 1 , 1 -manifolds Γ i , i = 1 , 2, where k > ( α − 1) / 2, each Γ i being charged with Borel measures with the sign α i := ± 1 prescribed. Such variational problems over a cone of Borel measures can be formulated as minimization problems over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space H − ε/ 2 (Γ), where ε := α − 1 and Γ := Γ 1 ∪ Γ 2 (see H. Harbrecht, W.L. Wendland, and N. Zorii. [ Math. Nachr. 287 (2014) 48–69]). We thus 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 consider- ation is efficiently approximated by means of an H -matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first order optimality system. Numerical results in R 3 are given to demonstrate our approach.