Rapid Solution of Minimal Riesz Energy Problems

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• 作者：Helmut Harbrecht ; Wolfgang L. Wendland and Natalia Zorii
• 刊名：Numerical Methods for Partial Differential Equations
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：32
• 期：6
• 页码：1535-1552
• 全文大小：1516K
• ISSN：1098-2426

文摘

In ℝn, 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 Ck−1,1-manifolds Γi, i=1,2, where k>(α−1)/2, each Γi being charged with Borel measures with the sign αi:=±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:=Γ1∪Γ2 (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 ℝ3 are given to demonstrate our approach.