Asymptotics of greedy energy points
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- by A. López García and E. B. Saff PDF
- Math. Comp. 79 (2010), 2287-2316
Abstract:
For a symmetric kernel $k:X\times X \rightarrow \mathbb {R}\cup \{+\infty \}$ on a locally compact metric space $X$, we investigate the asymptotic behavior of greedy $k$-energy points $\{a_{i}\}_{1}^{\infty }$ for a compact subset $A\subset X$ that are defined inductively by selecting $a_{1}\in A$ arbitrarily and $a_{n+1}$ so that $\sum _{i=1}^{n}k(a_{n+1},a_{i})=\inf _{x\in A}\sum _{i=1}^{n}k(x,a_{i})$. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy $\sum _{i\neq j}^{N}k(a_{i},a_{j})$ as $N\rightarrow \infty$ that is asymptotically the same as $\mathcal {E}(A,N):=\min \{\sum _{i\neq j}k(x_{i},x_{j}):x_{1},\ldots ,x_{N}\in A\}$), and have asymptotic distribution equal to the equilibrium measure for $A$. For the case of Riesz kernels $k_{s}(x,y):=|x-y|^{-s}$, $s>0$, we show that if $A$ is a rectifiable Jordan arc or closed curve in $\mathbb {R}^{p}$ and $s>1$, then greedy $k_{s}$-energy points are not asymptotically energy minimizing, in contrast to the case $s<1$. (In fact, we show that no sequence of points can be asymptotically energy minimizing for $s>1$.) Additional results are obtained for greedy $k_{s}$-energy points on a sphere, for greedy best-packing points (the case $s=\infty$), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.References
- J. Baglama, D. Calvetti, and L. Reichel, Fast Leja points, Electron. Trans. Numer. Anal. 7 (1998), 124–140. Large scale eigenvalue problems (Argonne, IL, 1997). MR 1667643
- A.R. Bausch, M.J. Bowick, A. Cacciuto, A.D. Dinsmore, M.F. Hsu, D.R. Nelson, M.G. Nikolaides, A. Travesset, and D.A. Weitz, Grain boundary scars and spherical crystallography, Science 299 (2003), 1716–1718.
- József Beck, The modulus of polynomials with zeros on the unit circle: a problem of Erdős, Ann. of Math. (2) 134 (1991), no. 3, 609–651. MR 1135879, DOI 10.2307/2944358
- S. V. Borodachov, D. P. Hardin, and E. B. Saff, Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1559–1580. MR 2357705, DOI 10.1090/S0002-9947-07-04416-9
- S. V. Borodachov, D. P. Hardin, and E. B. Saff, Asymptotics of best-packing on rectifiable sets, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2369–2380. MR 2302558, DOI 10.1090/S0002-9939-07-08975-7
- J. S. Brauchart, D. P. Hardin, and E. B. Saff, The Riesz energy of the $N$th roots of unity: an asymptotic expansion for large $N$, Bull. Lond. Math. Soc. 41 (2009), no. 4, 621–633. MR 2521357, DOI 10.1112/blms/bdp034
- M. Bowick, D.R. Nelson, and A. Travesset, Interacting topological defects in frozen topographies, Phys. Rev. B62 (2000), 8738–8751.
- J. S. Brauchart, Optimal logarithmic energy points on the unit sphere, Math. Comp. 77 (2008), no. 263, 1599–1613. MR 2398782, DOI 10.1090/S0025-5718-08-02085-1
- Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760, DOI 10.5802/aif.53
- G. Choquet, Diamètre transfini et comparaison de diverses capacités, Technical report, Faculté des Sciences de Paris (1958).
- Dan I. Coroian and Peter Dragnev, Constrained Leja points and the numerical solution of the constrained energy problem, J. Comput. Appl. Math. 131 (2001), no. 1-2, 427–444. MR 1835725, DOI 10.1016/S0377-0427(00)00258-2
- Stefano De Marchi, On Leja sequences: some results and applications, Appl. Math. Comput. 152 (2004), no. 3, 621–647. MR 2062764, DOI 10.1016/S0096-3003(03)00580-0
- P. D. Dragnev and E. B. Saff, Riesz spherical potentials with external fields and minimal energy points separation, Potential Anal. 26 (2007), no. 2, 139–162. MR 2276529, DOI 10.1007/s11118-006-9032-2
- Albert Edrei, Sur les déterminants récurrents et les singularités d’une fonction donnée par son développement de Taylor, Compositio Math. 7 (1939), 20–88 (French). MR 1285
- T. Erber and G. M. Hockney, Complex systems: equilibrium configurations of $N$ equal charges on a sphere $(2\leq N\leq 112)$, Advances in chemical physics, Vol. XCVIII, Adv. Chem. Phys., XCVIII, Wiley, New York, 1997, pp. 495–594. MR 1238214
- Bálint Farkas and Béla Nagy, Transfinite diameter, Chebyshev constant and energy on locally compact spaces, Potential Anal. 28 (2008), no. 3, 241–260. MR 2386099, DOI 10.1007/s11118-008-9075-7
- Bent Fuglede, On the theory of potentials in locally compact spaces, Acta Math. 103 (1960), 139–215. MR 117453, DOI 10.1007/BF02546356
- J. Górski, Les suites de points extrémaux liés aux ensembles dans l’espace à $3$ dimensions, Ann. Polon. Math. 4 (1957), 14–20 (French). MR 121770, DOI 10.4064/ap-4-1-14-20
- Mario Götz, On the distribution of Leja-Górski points, J. Comput. Anal. Appl. 3 (2001), no. 3, 223–241. MR 1840565, DOI 10.1023/A:1011597427738
- D. P. Hardin and E. B. Saff, Minimal Riesz energy point configurations for rectifiable $d$-dimensional manifolds, Adv. Math. 193 (2005), no. 1, 174–204. MR 2132763, DOI 10.1016/j.aim.2004.05.006
- A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), no. 2, 523–538. MR 1458327, DOI 10.1090/S0002-9947-98-02119-9
- N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
- F. Leja, Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme, Ann. Polon. Math. 4 (1957), 8–13. MR 100726, DOI 10.4064/ap-4-1-8-13
- A. López García, Ph.D. Dissertation, Vanderbilt University.
- A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov, and E. B. Saff, Asymptotics for minimal discrete Riesz energy on curves in $\Bbb R^d$, Canad. J. Math. 56 (2004), no. 3, 529–552. MR 2057285, DOI 10.4153/CJM-2004-024-1
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Makoto Ohtsuka, On potentials in locally compact spaces, J. Sci. Hiroshima Univ. Ser. A-I Math. 25 (1961), 135–352. MR 180695
- E. B. Saff and A. B. J. Kuijlaars, Distributing many points on a sphere, Math. Intelligencer 19 (1997), no. 1, 5–11. MR 1439152, DOI 10.1007/BF03024331
- Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
- J. Siciak, Two criteria for the continuity of the equilibrium Riesz potentials, Comment. Math. Prace Mat. 14 (1970), 91–99. MR 277742
- N. V. Zoriĭ, Equilibrium problems for potentials with external fields, Ukraïn. Mat. Zh. 55 (2003), no. 10, 1315–1339 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 55 (2003), no. 10, 1588–1618. MR 2073873, DOI 10.1023/B:UKMA.0000022070.73078.7b
- N. V. Zoriĭ, Equilibrium potentials with external fields, Ukraïn. Mat. Zh. 55 (2003), no. 9, 1178–1195 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 55 (2003), no. 9, 1423–1444. MR 2075132, DOI 10.1023/B:UKMA.0000018005.67743.86
Additional Information
- A. López García
- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- Email: abey.lopez@vanderbilt.edu
- E. B. Saff
- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 152845
- Email: edward.b.saff@vanderbilt.edu
- Received by editor(s): December 12, 2008
- Received by editor(s) in revised form: June 27, 2009
- Published electronically: April 16, 2010
- Additional Notes: The results of this paper form a part of the first author’s Ph.D. dissertation at Vanderbilt University
The research of the second author was supported, in part, by National Science Foundation grants DMS-0603828 and DMS-0808093. - © Copyright 2010 A. López García and E. B. Saff
- Journal: Math. Comp. 79 (2010), 2287-2316
- MSC (2010): Primary 65D99, 52A40; Secondary 78A30
- DOI: https://doi.org/10.1090/S0025-5718-10-02358-6
- MathSciNet review: 2684365