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. 15351552.
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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:  0749159X 
eISSN:  10982426 
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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