Abstract
Because physical phenomena on Earth’s surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can “zoom in” on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in ℝd from Wendland, Numer. Math. 116 (2010). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Antoine, J.P., Vandergheynst, P.: Wavelets on the 2-sphere: a group-theoretical approach. Appl. Comput. Harmon. Anal. 7, 262–291 (1999)
Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 131, 2733–2740 (2003)
Filbir, F., Themistoclakis, W.: Polynomial approximation on the sphere using scattered data. Math. Nachr. 281, 650–668 (2008)
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)
Gräf, M., Kunis, S., Potts, D.: On the computation of nonnegative quadrature weights on the sphere. Appl. Comput. Harmon. Anal. 27, 124–132 (2009)
Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Polyharmonic and related kernels on manifolds: interpolation and approximation. Found. Comput. Math. 12, 625–670 (2012)
Hesse, K., Sloan, I.H., Womersley, R.S.: Numerical integration on the sphere. In: Freeden, W., Nashed, Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1187–1220. Springer Verlag (2010)
Hubbert, S., Morton, T.M.: A Duchon framework for the sphere. J Approx. Theory 129, 28–57 (2004)
Le Gia, Q.T., Narcowich, F.J., Ward, J.D., Wendland, H.: Continuous and discrete least-square approximation by radial basis functions on spheres. J. Approx. Theory 143, 124–133 (2006)
Le Gia, Q.T., Mhaskar, H.: Quadrature formulas and localized linear polynomial operators on the sphere. SIAM Numer. Anal. 47, 440–466 (2008)
Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48, 2065–2090 (2010)
Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere. Appl. Comput. Harmon. Anal. 32, 401–412 (2012)
Le Gia, Q.T., Sloan, I.H., Wendland, H.: Data compression on the sphere using multiscale radial basis functions. Adv. Comp. Math. 40, 923–943 (2014)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, New York (1972)
Mhaskar, H.N.: On the representation of smooth functions on the sphere using finitely many bits. Appl. Comput. Harmon. Anal. 18, 215–233 (2005)
Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comp. 70, 1113–1130 (2001)
Müller, C.: Spherical Harmonics, Volume 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966)
Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math Anal. 38, 574–594 (2006)
Narcowich, F.J., Sun, X., Ward, J.D.: Approximating power of RBFs and their associated SBFs: a comparison. Adv. Comp. Math. 27, 107–124 (2007)
Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33, 1393–1410 (2002)
Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Springer, New York (1994)
Saff, E.B., Rakhmanov, E.A., Zhou, Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1, 647–662 (1994)
Schoenberg, I.J.: Positive definite function on spheres. Duke Math. J. 9, 96–108 (1942)
Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. Computer Graphics Proceedings (SIGGRAPH ’95) (1995)
Shewchuk, J.R.: An introduction to the conjugate gradient method without the agonizing pain. School of Computer Science, Carnegie Mellon University, Pittsburgh. http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf
Triebel, H.: Interpolation Theory, Function Spaces and Differential Operators. North-Holland, Amsterdam (1978)
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains. Numer. Math. 116, 493–517 (2010)
Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Amer. Math Soc. 116, 977–981 (1992)
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Communicated by: John Lowengrub
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Le Gia, Q.T., Sloan, I.H. & Wendland, H. Zooming from global to local: a multiscale RBF approach. Adv Comput Math 43, 581–606 (2017). https://doi.org/10.1007/s10444-016-9498-4
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DOI: https://doi.org/10.1007/s10444-016-9498-4