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

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- by Edward J. Fuselier, Francis J. Narcowich, Joseph D. Ward and Grady B. Wright PDF
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**78**(2009), 2157-2186 Request permission

## 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.## References

- Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR**1894376**, DOI 10.1007/978-1-4757-3658-8 - T. C. Chen,
*Global water vapor flux and maintenance during FGGE*, Mon. Wea. Rev.**113**(1985), 1801–1819. - 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**, DOI 10.1007/s11075-007-9072-8 - Thomas A. Foley,
*Near optimal parameter selection for multiquadric interpolation*, J. Appl. Sci. Comput.**1**(1994), no. 1, 54–69. MR**1295335** - 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**, DOI 10.1016/j.camwa.2007.01.028 - 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** - 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** - 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**, DOI 10.1090/S0025-5718-07-02096-0 - Edward J. Fuselier,
*Erratum: “Improved stability estimates and a characterization of the native space for matrix-valued RBFs” [Adv. Comput. Math. 29 (2008), no. 3, 269–290; MR2438345]*, Adv. Comput. Math.**29**(2008), no. 3, 311–313. MR**2438347**, DOI 10.1007/s10444-008-9091-6 - Peter B. Gilkey,
*The index theorem and the heat equation*, Mathematics Lecture Series, No. 4, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks. MR**0458504** - A. E. Gill,
*Atmosphere-ocean dynamics*, Academic Press, London, 1982. - T. Gneiting,
*Correlation functions for atmospheric data analysis*, Q. J. R. Meteorol. Soc.**125**(1999), 2449–2464. - Michael Golomb and Hans F. Weinberger,
*Optimal approximation and error bounds*, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Publication of the Mathematics Research Center, U.S. Army, the University of Wisconsin, no. 1, University of Wisconsin Press, Madison, Wis., 1959, pp. 117–190. Edited by R. E. Langer. MR**0121970** - J. R. Holton,
*An introduction to dynamic meteorology*, third ed., Academic Press, San Francisco, 1992. - S. Hubbert,
*Radial basis function interpolation on the sphere*, Ph.D. thesis, Imperial College, 2002. - Armin Iske,
*Multiresolution methods in scattered data modelling*, Lecture Notes in Computational Science and Engineering, vol. 37, Springer-Verlag, Berlin, 2004. MR**2060191**, DOI 10.1007/978-3-642-18754-4 - John David Jackson,
*Classical electrodynamics*, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR**0436782** - 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**, DOI 10.1090/S0025-5718-99-01080-7 - 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**, DOI 10.1016/j.jat.2006.03.007 - 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**, DOI 10.1016/j.jat.2004.12.005 - J.-L. Lions and E. Magenes,
*Non-homogeneous boundary value problems and applications. Vol. I*, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR**0350177** - C. L. Mader,
*Numerical modeling of water waves*, CRC Press, Boca Raton, 2004. - 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** - Bertil Matérn,
*Spatial variation*, 2nd ed., Lecture Notes in Statistics, vol. 36, Springer-Verlag, Berlin, 1986. With a Swedish summary. MR**867886**, DOI 10.1007/978-1-4615-7892-5 - 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**, DOI 10.1006/jath.2001.3642 - Claus Müller,
*Spherical harmonics*, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR**0199449**, DOI 10.1007/BFb0094775 - 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**, DOI 10.1007/s10208-005-0197-7 - 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**, DOI 10.1007/s10444-005-7506-1 - 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**, DOI 10.1090/S0025-5718-1994-1254147-6 - Francis J. Narcowich and Joseph D. Ward,
*Scattered data interpolation on spheres: error estimates and locally supported basis functions*, SIAM J. Math. Anal.**33**(2002), no. 6, 1393–1410. MR**1920637**, DOI 10.1137/S0036141001395054 - Francis J. Narcowich and Joseph D. Ward,
*Scattered-data interpolation on ${\Bbb R}^n$: error estimates for radial basis and band-limited functions*, SIAM J. Math. Anal.**36**(2004), no. 1, 284–300. MR**2083863**, DOI 10.1137/S0036141002413579 - 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**, DOI 10.1007/s00365-002-0529-7 - Francis J. Narcowich, Joseph D. Ward, and Holger Wendland,
*Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting*, Math. Comp.**74**(2005), no. 250, 743–763. MR**2114646**, DOI 10.1090/S0025-5718-04-01708-9 - Francis J. Narcowich, Joseph D. Ward, and Holger Wendland,
*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**, DOI 10.1007/s00365-005-0624-7 - 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**, DOI 10.1007/s00041-006-6903-2 - Shmuel Rippa,
*An algorithm for selecting a good value for the parameter $c$ in radial basis function interpolation*, Adv. Comput. Math.**11**(1999), no. 2-3, 193–210. Radial basis functions and their applications. MR**1731697**, DOI 10.1023/A:1018975909870 - 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 - R. Schaback,
*Improved error bounds for scattered data interpolation by radial basis functions*, Math. Comp.**68**(1999), no. 225, 201–216. MR**1604379**, DOI 10.1090/S0025-5718-99-01009-1 - Robert Schaback,
*Error estimates and condition numbers for radial basis function interpolation*, Adv. Comput. Math.**3**(1995), no. 3, 251–264. MR**1325034**, DOI 10.1007/BF02432002 - 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**, DOI 10.1137/0718015 - 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**, DOI 10.1007/BF02123482 - Holger Wendland,
*Scattered data approximation*, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR**2131724** - 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**, DOI 10.1016/S0021-9991(05)80016-6 - R. S. Womersley and I. H. Sloan,
*Interpolation and cubature on the sphere*, accessed 2008, http://web.maths.unsw.edu.au/~rsw/Sphere/. - 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**, DOI 10.1093/imanum/13.1.13

## Additional Information

**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
- MR Author ID: 129435
- Email: fnarc@math.tamu.edu
**Joseph D. Ward**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 180590
- Email: jward@math.tamu.edu
**Grady B. Wright**- Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725-1555
- Email: wright@diamond.boisestate.edu
- Received by editor(s): February 8, 2008
- Received by editor(s) in revised form: August 25, 2008
- Published electronically: 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 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**78**(2009), 2157-2186 - MSC (2000): Primary 41A05, 41A63; Secondary 76M25, 86-08, 86A10
- DOI: https://doi.org/10.1090/S0025-5718-09-02214-5
- MathSciNet review: 2521283