Mathematics of Computation

Error and stability estimates for surface-divergence free RBF interpolants on the sphere

Author(s): Edward J. Fuselier; Francis J. Narcowich; Joseph D. Ward; Grady B. Wright.
Journal: Math. Comp. 78 (2009), 2157-2186.
MSC (2000): Primary 41A05, 41A63; Secondary 76M25, 86-08, 86A10
Posted: January 22, 2009
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Abstract: Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in $\mathbb{R}^3$ . In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, $\mathbb{S}^2$ . In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.

1.: Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, second ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
2.: T. C. Chen, Global water vapor flux and maintenance during FGGE, Mon. Wea. Rev. 113 (1985), 1801-1819.
3.: Gregory E. Fasshauer and Jack G. Zhang, On choosing ``optimal'' shape parameters for RBF approximation, Numer. Algorithms 45 (2007), no. 1-4, 345-368. MR 2355993
4.: Thomas A. Foley, Near optimal parameter selection for multiquadric interpolation, J. Appl. Sci. Comput. 1 (1994), no. 1, 54-69. MR 1295335
5.: Bengt Fornberg and Julia Zuev, The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl. 54 (2007), no. 3, 379-398. MR 2338845 (2008e:41002)
6.: W. Freeden, T. Gervens, and M. Schreiner, Constructive approximation on the sphere, Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, 1998, With applications to geomathematics. MR 1694466 (2000e:41001)
7.: Willi Freeden and Theo Gervens, Vector spherical spline interpolation, Multivariate approximation theory, IV (Oberwolfach, 1989), Internat. Ser. Numer. Math., vol. 90, Birkhäuser, Basel, 1989, pp. 157-171. MR 1034306 (91a:41011)
8.: Edward J. Fuselier, Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants, Math. Comp. 77 (2008), no. 263, 1407-1423. MR 2398774
9.: -, Improved stability estimates and a characterization of the native space for matrix-valued RBFs, Adv. Comp. Math. 29 (2008), no. 3, 269-290. MR 2438347
10.: Peter B. Gilkey, The index theorem and the heat equation, Publish or Perish Inc., Boston, Mass., 1974, Notes by Jon Sacks, Mathematics Lecture Series, No. 4. MR 0458504 (56:16704)
11.: A. E. Gill, Atmosphere-ocean dynamics, Academic Press, London, 1982.
12.: T. Gneiting, Correlation functions for atmospheric data analysis, Q. J. R. Meteorol. Soc. 125 (1999), 2449-2464.
13.: Michael Golomb and Hans F. Weinberger, Optimal approximation and error bounds, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Edited by R. E. Langer. Publication No. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, Wis., 1959, pp. 117-190. MR 0121970 (22:12697)
14.: J. R. Holton, An introduction to dynamic meteorology, third ed., Academic Press, San Francisco, 1992.
15.: S. Hubbert, Radial basis function interpolation on the sphere, Ph.D. thesis, Imperial College, 2002.
16.: Armin Iske, Multiresolution methods in scattered data modelling, Lecture Notes in Computational Science and Engineering, vol. 37, Springer-Verlag, Berlin, 2004. MR 2060191 (2005f:94023)
17.: John David Jackson, Classical electrodynamics, second ed., John Wiley & Sons Inc., New York, 1975. MR 0436782 (55:9721)
18.: Kurt Jetter, Joachim Stöckler, and Joseph D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp. 68 (1999), no. 226, 733-747. MR 1642746 (99i:41032)
19.: Q. T. Le Gia, F. J. Narcowich, J. D. Ward, and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory 143 (2006), no. 1, 124-133. MR 2271729 (2007k:41078)
20.: J. Levesley and X. Sun, Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory 133 (2005), no. 2, 269-283. MR 2129483 (2005k:41024)
21.: J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177 (50:2670)
22.: C. L. Mader, Numerical modeling of water waves, CRC Press, Boca Raton, 2004.
23.: W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988), no. 4, 77-89. MR 986343 (90e:41006)
24.: Bertil Matérn, Spatial variation, second ed., Lecture Notes in Statistics, vol. 36, Springer-Verlag, Berlin, 1986, With a Swedish summary. MR 867886 (87m:62038)
25.: Tanya M. Morton and Marian Neamtu, Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels, J. Approx. Theory 114 (2002), no. 2, 242-268. MR 1883408 (2002k:65200)
26.: Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin, 1966. MR 0199449 (33:7593)
27.: Francis J. Narcowich, Xingping Sun, Joseph D. Ward, and Holger Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math. 7 (2007), no. 3, 369-390. MR 2335250 (2008i:41023)
28.: Francis J. Narcowich, Xinping Sun, and Joseph D. Ward, Approximation power of RBFs and their associated SBFs: a connection, Adv. Comput. Math. 27 (2007), no. 1, 107-124. MR 2317924
29.: Francis J. Narcowich and Joseph D. Ward, Generalized Hermite interpolation via matrix-valued conditionally positive definite functions, Math. Comp. 63 (1994), no. 208, 661-687. MR 1254147 (95c:41014)
30.: -, Scattered data interpolation on spheres: error estimates and locally supported basis functions, SIAM J. Math. Anal. 33 (2002), no. 6, 1393-1410 (electronic). MR 1920637 (2003j:41021)
31.: -, Scattered-data interpolation on ${\mathbb{R}}\sp n$ : error estimates for radial basis and band-limited functions, SIAM J. Math. Anal. 36 (2004), no. 1, 284-300 (electronic). MR 2083863 (2005g:41011)
32.: Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Refined error estimates for radial basis function interpolation, Constr. Approx. 19 (2003), no. 4, 541-564. MR 1998904 (2004f:41019)
33.: -, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), no. 250, 743-763 (electronic). MR 2114646 (2005k:41051)
34.: -, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx. 24 (2006), no. 2, 175-186. MR 2239119 (2007g:41020)
35.: Francis J. Narcowich, Joseph D. Ward, and Grady B. Wright, Divergence-free RBFs on surfaces, J. Fourier Anal. Appl. 13 (2007), no. 6, 643-663. MR 2350442
36.: Shmuel Rippa, An algorithm for selecting a good value for the parameter in radial basis function interpolation, Adv. Comput. Math. 11 (1999), no. 2-3, 193-210, Radial basis functions and their applications. MR 1731697
37.: E. B. Saff and A. B. J. Kuijlaars, Distributing many points on a sphere, Math. Intelligencer 19 (1997), no. 1, 5-11. MR 1439152 (98h:70011)
38.: R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp. 68 (1999), no. 225, 201-216. MR 1604379 (99d:41037)
39.: Robert Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (1995), no. 3, 251-264. MR 1325034 (96a:41004)
40.: Paul N. Swarztrauber, The approximation of vector functions and their derivatives on the sphere, SIAM J. Numer. Anal. 18 (1981), no. 2, 191-210. MR 612138 (83e:65039)
41.: Holger Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), no. 4, 389-396. MR 1366510 (96h:41025)
42.: -, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724 (2006i:41002)
43.: David L. Williamson, John B. Drake, James J. Hack, Rüdiger Jakob, and Paul N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992), no. 1, 211-224. MR 1177513 (93d:86006)
44.: R. S. Womersley and I. H. Sloan, Interpolation and cubature on the sphere, accessed 2008, http://web.maths.unsw.edu.au/~rsw/Sphere/.
45.: Zong Min Wu and Robert Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), no. 1, 13-27. MR 1199027 (93m:65012)

Edward J. Fuselier
Affiliation: Department of Mathematical Sciences, United States Military Academy, West Point, New York 10996
Email: edward.fuselier@usma.edu

Francis J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: fnarc@math.tamu.edu

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: jward@math.tamu.edu

Grady B. Wright
Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725-1555
Email: wright@diamond.boisestate.edu

DOI: 10.1090/S0025-5718-09-02214-5
PII: S 0025-5718(09)02214-5
Keywords: Sphere, vector fields, incompressible fluids, radial basis functions, numerical modeling, stream function.
Received by editor(s): February 8, 2008
Received by editor(s) in revised form: August 25, 2008
Posted: January 22, 2009
Additional Notes: The second author's research was supported by grant DMS-0504353 from the National Science Foundation.
The third author's research was supported by grant DMS-0504353 from the National Science Foundation.
The fourth author's research was supported by grant ATM-0801309 from the National Science Foundation.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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