Rapid Solution of Minimal Riesz Energy Problems. Issue 6 (22nd March 2016)

• Record Type: Journal Article
• Authors: Harbrecht, Helmut; Wendland, Wolfgang L.; Zorii, Natalia
• Is Part Of: Numerical methods for partial differential equations. Volume 32:Issue 6(2016:Nov.)
• Page Start: 1535
• Page End: 1552
• Publication Date: 2016-03-22
• Subjects: Gauss variational problem -- Riesz kernel -- boundary element method -- active set strategy; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22060
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : 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 Harbrecht et al., 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 consideration 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. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1535–1552, 2016