Abstract
The asymptotic approximation of continuous minimal s-Riesz energy by the discrete minimal energy of systems of n points on regular sets in \(\mathbb{R}^{3}\) is studied. For this purpose an optimization framework for the numerical solution of the corresponding Gauß variational problem based on an interior point method is developed. Moreover, numerical results for ellipsoids and tori are presented.
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References
J.S. Brauchart, About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case. Integral Transform. Spec. Funct. 17, 321–328 (2006)
J.S. Brauchart, D.P. Hardin, E.B. Saff, The support of the limit distribution of optimal Riesz energy points on sets of revolution in \(\mathbb{R}^{3}\). J. Math. Phys. 48(12), 24 (2007)
__________ , Riesz energy and sets of revolution in \(\mathbb{R}^{3}\). Contemp. Math. 481, 47–57 (2009)
A. Conn, N. Gould, P. Toint, Trust Region Methods (SIAM, Philadelphia, 2000)
L. Giomi, M.J. Bowick, Defective ground states of toroidal crystals. Phys. Rev. E 78, 010601 (2008)
D.P. Hardin, E.B. Saff, Discretizing manifolds via minimum energy points. Not. AMS 51, 647–662 (2004)
__________ , Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds. Adv. Math. 193, 174–204 (2005)
D.P. Hardin, E.B. Saff, H. Stahl, Support of the logarithmic equilibrium measure on sets of revolution in \(\mathbb{R}^{3}\). J. Math. Phys. 48(2), 022901 (2007)
M. Jaraczewski, M. Rozgic̀, M. Stiemer, Numerical determination of extremal points and asymptotic order of discrete minimal Riesz energy for regular compact sets, in Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol. 83 (Springer, New York, 2014), pp. 219–238
A.B.J. Kuijlaars, E.B. Saff, Asymptotics for minimal discrete energy on the sphere. Trans. Am. Math. Soc. 350(2), 523–538 (1998)
N.S. Landkof, Foundations of Modern Potential Theory (Springer, Berlin/New York, 1972)
J. Lee, Introduction to Smooth Manifolds. Graduate Texts in Mathematics (Springer, New York, 2006)
MATLAB, Version 8.2.0.701 (R2013b) (The MathWorks, Natick, 2013)
J. Nocedal, S.J. Wright, Numerical Optimization. Springer Series in Operations Research (Springer, New York, 1990)
M. Ohtsuka, On various definitions of capacity and related notions. Nagoya Math. J. 30, 121–127 (1967)
G. Pólya, G. Szegő, Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. J. Reine Angew. Math. 165, 4–49 (1931)
Q. Rajon, T. Ransford, J. Rostand, Computation of capacity via quadratic programming. J. Math. Pure. Appl. 94, 398–413 (2010)
__ , Computation of weighted capacity. J. Approx. Theory 162(6), 1187–1203 (2010)
E.A. Rakhmanov, E.B. Saff, Y.M. Zhou, Minimal discrete energy on the sphere. Math. Res. Lett. 1, 647–662 (1994)
M. Rozgic̀, M. Jaraczewski, M. Stiemer, Inner point methods: On necessary optimality conditions of various reformulations of a constrained optimization problem, Technical report, Universitätsbibliothek der Helmut-Schmidt-Universität (2014)
E.B. Saff, V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin/Heidelberg, 1997)
E. Saff, A. Kuijlaars, Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)
A. Wächter, L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear pogramming. Math. Program. 106, 25–57 (2006)
G. Wagner, On means of distances on the surface of a sphere (lower bounds). Pac. J. Math. 144(2), 389–398 (1990)
J. Wermer, Potential Theory. Lecture Notes in Mathematics (Springer, Berlin/Heidelberg, 1974)
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Jaraczewski, M., Rozgic̀, M., Stiemer, M. (2015). Determination of Extremal Points and Weighted Discrete Minimal Riesz Energy with Interior Point Methods. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_50
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DOI: https://doi.org/10.1007/978-3-319-10705-9_50
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