Interpolation on the sphere: a fast solution technique

We present a fast solution technique for the problem of interpolation on the sphere, using radial basis functions and multiplicative Schwarz methods. This problem has applications in geodesy and earth science. A bound for the condition number of the preconditioned matrix is proved. Since approximation using radial basis functions is a meshless method, the proof technique is novel compared to that used in finite element methods. Numerical experiments on relatively large sets of scattered data points taken from MAGSAT satellite data are presented. The article illustrates how interpolation of scattered data on the sphere can be efficiently performed. References P. Alfeld, M. Neamtu, and L. L. Schumaker. Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math., 73:5--43, 1996. doi:10.1016/0377-0427(96)00034-9. R. K. Beatson, W. A. Light, and S. Billings. Fast solution of the radial basis function interpolation equations: domain decomposition method. SIAM J. Sci. Comput., 22:1717--1740, 2000. doi:10.1137/S1064827599361771. D. Chen, V. A. Menegatto, and X. Sun. A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131:2733--2740, 2003. doi:10.1090/S0002-9939-03-06730-3. G. E. Fasshauer and L. L. Schumaker. Scattered data fitting on the sphere. In M. Dahlen, T. Lyche, and L. L. Schumaker, editors, Mathematical Methods for Curves and Surfaces II, pages 117--166, Nashville, 1998. Vanderbilt University Press. K. Hesse. Domain Decomposition Methods in Multiscale Geopotential Determination from SST and SGG. PhD thesis, University of Kaiserslautern, Germany, 2002. Q. T. {Le Gia}, I. H. Sloan, and T. Tran. Overlapping additive {S}chwarz preconditioners for elliptic {PDE}s on the unit sphere. Math. Comp., 78:79--101, 2009. doi:10.1090/S0025-5718-08-02150-9. J. Levesley, Z. Luo, and X. Sun. Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions. Proc. Amer. Math. Soc., 127:2127--2134, 1999. doi:10.1090/S0002-9939-99-04683-3. C. Mueller. Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer--Verlag, Berlin, 1966. F. J. Narcowich, X. Sun, J. D. Ward, and H. Wendland. Direct and inverse {S}obolev error estimates for scattered data interpolation via spherical basis functions. Found. Comput. Math., 7:369--390, 2007. doi:10.1007/s10208-005-0197-7. F. J. Narcowich and J. D. Ward. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 33:1393--1410, 2002. doi:10.1137/S0036141001395054. I. J. Schoenberg. Positive definite function on spheres. Duke Math. J., 9:96--108, 1942. doi:10.1215/S0012-7094-42-00908-6. B. Smith, P. Bj\OT1\o rstad, and W. Gropp. Domain Decomposition---Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, 1996. A. Toselli and O. Widlund. Domain Decomposition Methods---Algorithms and Theory, volume 34 of Springer Series in Computational Mathematics. Springer--Verlag, Berlin, 2005. T. Tran, Q. T. {Le Gia}, and I. H. Sloan. Overlapping additive Schwarz preconditioners for interpolation on the unit sphere by spherical radial basis functions. In preparation. http://www.maths.unsw.edu.au/ qlegia/interp_pcg3.pdf. H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. Y. Xu and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116:977--981, 1992. doi:10.2307/2159477