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Rapid Solution of Minimal Riesz Energy Problems

Harbrecht, Helmut and Wendland, Wolfgang L. and Zorii, Natalia. (2016) Rapid Solution of Minimal Riesz Energy Problems. Numerical Methods for Partial Differential Equations, 32 (6). pp. 1535-1552.

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Official URL: http://edoc.unibas.ch/44256/

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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 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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
UniBasel Contributors:Harbrecht, Helmut
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Wiley
ISSN:0749-159X
e-ISSN:1098-2426
Note:Publication type according to Uni Basel Research Database: Journal article -- The final publication is available at Wiley, see DOI link.
Language:English
Identification Number:
Last Modified:09 Oct 2017 07:57
Deposited On:06 Oct 2016 11:59

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