Spherical $t_\epsilon$-designs for approximations on the sphere
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- by Yang Zhou and Xiaojun Chen PDF
- Math. Comp. 87 (2018), 2831-2855 Request permission
Abstract:
A spherical $t$-design is a set of points on the unit sphere that are nodes of a quadrature rule with positive equal weights that is exact for all spherical polynomials of degree $\le t$. The existence of a spherical $t$-design with $(t+1)^2$ points in a set of interval enclosures on the unit sphere $\mathbb {S}^2 \subset \mathbb {R}^3$ for any $0\le t \le 100$ is proved by Chen, Frommer, and Lang (2011). However, how to choose a set of points from the set of interval enclosures to obtain a spherical $t$-design with $(t+1)^2$ points is not given in loc. cit. It is known that $(t+1)^2$ is the dimension of the space of spherical polynomials of degree at most $t$ in 3 variables on $\mathbb {S}^2$. In this paper we investigate a new concept of point sets on the sphere named spherical $t_\epsilon$-design ($0 \le \epsilon <1$), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree $\le t$. The parameter $\epsilon$ is used to control the variation of the weights, while the sum of the weights is equal to the area of the sphere. A spherical $t_\epsilon$-design is a spherical $t$-design when $\epsilon =0,$ and a spherical $t$-design is a spherical $t_\epsilon$-design for any $0<\epsilon <1$. We show that any point set chosen from the set of interval enclosures (loc. cit.) is a spherical $t_\epsilon$-design. We then study the worst-case error in a Sobolev space for quadrature rules using spherical $t_\epsilon$-designs, and investigate a model of polynomial approximation with $l_1$-regularization using spherical $t_\epsilon$-designs. Numerical results illustrate the good performance of spherical $t_\epsilon$-designs for numerical integration and function approximation on the sphere.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- Congpei An, Xiaojun Chen, Ian H. Sloan, and Robert S. Womersley, Well conditioned spherical designs for integration and interpolation on the two-sphere, SIAM J. Numer. Anal. 48 (2010), no. 6, 2135–2157. MR 2763659, DOI 10.1137/100795140
- Congpei An, Xiaojun Chen, Ian H. Sloan, and Robert S. Womersley, Regularized least squares approximations on the sphere using spherical designs, SIAM J. Numer. Anal. 50 (2012), no. 3, 1513–1534. MR 2970753, DOI 10.1137/110838601
- Kendall Atkinson and Weimin Han, Spherical harmonics and approximations on the unit sphere: an introduction, Lecture Notes in Mathematics, vol. 2044, Springer, Heidelberg, 2012. MR 2934227, DOI 10.1007/978-3-642-25983-8
- Bela Bajnok, Construction of spherical $t$-designs, Geom. Dedicata 43 (1992), no. 2, 167–179. MR 1180648, DOI 10.1007/BF00147866
- Eiichi Bannai, On tight spherical designs, J. Combin. Theory Ser. A 26 (1979), no. 1, 38–47. MR 525085, DOI 10.1016/0097-3165(79)90052-9
- Eiichi Bannai and Etsuko Bannai, A survey on spherical designs and algebraic combinatorics on spheres, European J. Combin. 30 (2009), no. 6, 1392–1425. MR 2535394, DOI 10.1016/j.ejc.2008.11.007
- Eiichi Bannai and Etsuko Bannai, Remarks on the concepts of $t$-designs, J. Appl. Math. Comput. 40 (2012), no. 1-2, 195–207. MR 2965326, DOI 10.1007/s12190-012-0544-1
- Brad J. C. Baxter and Simon Hubbert, Radial basis functions for the sphere, Recent progress in multivariate approximation (Witten-Bommerholz, 2000) Internat. Ser. Numer. Math., vol. 137, Birkhäuser, Basel, 2001, pp. 33–47. MR 1877496
- Andriy Bondarenko, Danylo Radchenko, and Maryna Viazovska, Optimal asymptotic bounds for spherical designs, Ann. of Math. (2) 178 (2013), no. 2, 443–452. MR 3071504, DOI 10.4007/annals.2013.178.2.2
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- Johann S. Brauchart and Josef Dick, A simple proof of Stolarsky’s invariance principle, Proc. Amer. Math. Soc. 141 (2013), no. 6, 2085–2096. MR 3034434, DOI 10.1090/S0002-9939-2013-11490-5
- Johann S. Brauchart and Josef Dick, A characterization of Sobolev spaces on the sphere and an extension of Stolarsky’s invariance principle to arbitrary smoothness, Constr. Approx. 38 (2013), no. 3, 397–445. MR 3122277, DOI 10.1007/s00365-013-9217-z
- Johann S. Brauchart and Kerstin Hesse, Numerical integration over spheres of arbitrary dimension, Constr. Approx. 25 (2007), no. 1, 41–71. MR 2263736, DOI 10.1007/s00365-006-0629-4
- J. S. Brauchart, E. B. Saff, I. H. Sloan, and R. S. Womersley, QMC designs: optimal order quasi Monte Carlo integration schemes on the sphere, Math. Comp. 83 (2014), no. 290, 2821–2851. MR 3246811, DOI 10.1090/S0025-5718-2014-02839-1
- XiaoJun Chen, ShouQiang Du, and Yang Zhou, A smoothing trust region filter algorithm for nonsmooth least squares problems, Sci. China Math. 59 (2016), no. 5, 999–1014. MR 3484496, DOI 10.1007/s11425-015-5116-z
- Xiaojun Chen, Andreas Frommer, and Bruno Lang, Computational existence proofs for spherical $t$-designs, Numer. Math. 117 (2011), no. 2, 289–305. MR 2754852, DOI 10.1007/s00211-010-0332-5
- Xiaojun Chen and Robert S. Womersley, Existence of solutions to systems of underdetermined equations and spherical designs, SIAM J. Numer. Anal. 44 (2006), no. 6, 2326–2341. MR 2272596, DOI 10.1137/050626636
- F. H. Clarke, Optimization and nonsmooth analysis, 2nd ed., Classics in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1058436, DOI 10.1137/1.9781611971309
- P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388. MR 485471, DOI 10.1007/bf03187604
- Peter J. Grabner and Robert F. Tichy, Spherical designs, discrepancy and numerical integration, Math. Comp. 60 (1993), no. 201, 327–336. MR 1155573, DOI 10.1090/S0025-5718-1993-1155573-5
- Manuel Gräf and Daniel Potts, On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms, Numer. Math. 119 (2011), no. 4, 699–724. MR 2854125, DOI 10.1007/s00211-011-0399-7
- R. H. Hardin and N. J. A. Sloane, McLaren’s improved snub cube and other new spherical designs in three dimensions, Discrete Comput. Geom. 15 (1996), no. 4, 429–441. MR 1384885, DOI 10.1007/BF02711518
- Kerstin Hesse and Ian H. Sloan, Worst-case errors in a Sobolev space setting for cubature over the sphere $S^2$, Bull. Austral. Math. Soc. 71 (2005), no. 1, 81–105. MR 2127668, DOI 10.1017/S0004972700038041
- Kerstin Hesse and Ian H. Sloan, Cubature over the sphere $S^2$ in Sobolev spaces of arbitrary order, J. Approx. Theory 141 (2006), no. 2, 118–133. MR 2252093, DOI 10.1016/j.jat.2006.01.004
- J. Korevaar and J. L. H. Meyers, Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere, Integral Transform. Spec. Funct. 1 (1993), no. 2, 105–117. MR 1421438, DOI 10.1080/10652469308819013
- Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
- S. V. Pereverzyev, I. H. Sloan, and P. Tkachenko, Parameter choice strategies for least-squares approximation of noisy smooth functions on the sphere, SIAM J. Numer. Anal. 53 (2015), no. 2, 820–835. MR 3324977, DOI 10.1137/140964990
- M. Reimer, Quadrature rules for the surface integral of the unit sphere based on extremal fundamental systems, Math. Nachr. 169 (1994), 235–241. MR 1292809, DOI 10.1002/mana.19941690117
- Robert J. Renka, Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software 14 (1988), no. 2, 139–148. MR 946761, DOI 10.1145/45054.45055
- 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
- P. D. Seymour and Thomas Zaslavsky, Averaging sets: a generalization of mean values and spherical designs, Adv. in Math. 52 (1984), no. 3, 213–240. MR 744857, DOI 10.1016/0001-8708(84)90022-7
- Ian H. Sloan and Robert S. Womersley, A variational characterisation of spherical designs, J. Approx. Theory 159 (2009), no. 2, 308–318. MR 2562747, DOI 10.1016/j.jat.2009.02.014
- Ian H. Sloan and Robert S. Womersley, Filtered hyperinterpolation: a constructive polynomial approximation on the sphere, GEM Int. J. Geomath. 3 (2012), no. 1, 95–117. MR 2915950, DOI 10.1007/s13137-011-0029-7
- G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154
- David S. Watkins, Fundamentals of matrix computations, 3rd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2010. MR 2778339
- Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724
Additional Information
- Yang Zhou
- Affiliation: School of Mathematics and Statistics, Shandong Normal University, Jinan, Shangdong, China 250000
- MR Author ID: 272850
- Email: andres.zhou@connect.polyu.hk
- Xiaojun Chen
- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
- MR Author ID: 196364
- Email: maxjchen@polyu.edu.hk
- Received by editor(s): December 28, 2015
- Received by editor(s) in revised form: November 9, 2016, April 26, 2017, and June 14, 2017
- Published electronically: February 5, 2018
- Additional Notes: The first author’s work was supported in part by Department of Applied Mathematics, The Hong Kong Polytechnic University and Hong Kong Research Council Grant PolyU5002/13p and in part by NSFC grant No. 11626147.
The second author’s work was supported in part by Hong Kong Research Council Grant PolyU153001/14p. - © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 2831-2855
- MSC (2010): Primary 65D30, 41A10, 65G30
- DOI: https://doi.org/10.1090/mcom/3306
- MathSciNet review: 3834687