Advertisement

Numerische Mathematik

, Volume 127, Issue 1, pp 57–92 | Cite as

Kernel based quadrature on spheres and other homogeneous spaces

  • E. Fuselier
  • T. Hangelbroek
  • F. J. Narcowich
  • J. D. Ward
  • G. B. Wright
Article

Abstract

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate—optimally so in many cases—and stable under an increasing number of nodes and in the presence of noise, provided the set \(X\) of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to \(\mathbb S ^n\), oblate spheroids for instance. The weights are obtained by solving a single linear system. For \(\mathbb S ^2\), and the restricted thin plate spline kernel \(r^2\log r\), these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.

Mathematics Subject Classification (2010)

65D32 46E22 41A55 41A05 

Notes

Acknowledgments

We thank Professor Doug Hardin from Vanderbilt University for providing us with code for generating the quasi-minimum energy points used in the numerical examples based on the technique described in [5].

References

  1. 1.
    Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Lecture Notes in Mathematics. Springer, Berlin (2012)Google Scholar
  2. 2.
    Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampère equations. Grundlehren der Mathematischen Wissenschaften, vol. 252 [Fundamental Principles of Mathematical Sciences]. Springer, New York (1982)Google Scholar
  3. 3.
    Baxter, B.J.C., Hubbert, S.: Radial basis functions for the sphere. In: Recent progress in multivariate approximation (Witten-Bommerholz 2000). International Series Numerical Mathematics, vol. 137, pp. 33–47. Birkhäuser, Basel (2001)Google Scholar
  4. 4.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14, 1–137 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Borodachov, S.V., Hardin, D.P. Saff, E.: Low complexity methods for discretizing manifolds via Riesz energy minimization. Submitted (2012)Google Scholar
  6. 6.
    Brown, G., Dai, F.: Approximation of smooth functions on compact two-point homogeneous spaces. J. Funct. Anal. 220, 401–423 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Carter, R., Segal, G., Macdonald, I.: Lectures on Lie groups and Lie algebras. London Mathematical Society Student Texts, vol. 32. Cambridge University Press, Cambridge. With a foreword by Martin Taylor (1995)Google Scholar
  8. 8.
    Faul, A., Powell, M.J.D.: Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions. Adv. Comput. Math. 11, 183–192 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Filbir, F., Mhaskar, H.N.: A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16, 629–657 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Flyer, N., Lehto, E., Blaise, S., Wright, G.B., St-Cyr, A.: A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere. J. Comput. Phys. 231, 4078–4095 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Flyer, N., Wright, G.B.: A radial basis function method for the shallow water equations on a sphere. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 1949–1976 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fuselier, E.J., Hangelbroek, T., Narcowich, F.J., Ward, J.D., Wright G.B.: Localized bases for kernel spaces on the unit sphere. SIAM J. Numer. Anal. 51, 2538–2562 (2013)Google Scholar
  13. 13.
    Giné, M.E.: The addition formula for the eigenfunctions of the Laplacian. Adv. Math. 18, 102–107 (1975)Google Scholar
  14. 14.
    Giraldo, F.X.: Lagrange-Galerkin methods on spherical geodesic grids. J. Comput. Phys. 136, 197–213 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    González, A.: Measurement of areas on a sphere using Fibonacci and latitude longitude lattices. Math. Geosci. 42, 49–64 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gräf, M.: A unified approach to scattered data approximation on \(S^3\) and \(SO(3)\). Adv. Comput. Math. 37, 379–393 (2012)Google Scholar
  17. 17.
    Gräf, M.: Efficient algorithms for the computation of quadrature points on Riemannian manifolds, PhD thesis, Chemnitz University of Technology, Department of Mathematics (2013)Google Scholar
  18. 18.
    Gräf, M., Kunis, S., Potts, D.: On the computation of nonnegative quadrature weights on the sphere. Appl. Comput. Harmon. Anal. 27, 124–132 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gräf, M., Potts, D.: Sampling sets and quadrature formulae on the rotation group. Numer. Funct. Anal. Optim. 30, 665–688 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hangelbroek, T., Narcowich, F.J., Sun, X., Ward, J.D.: Kernel approximation on manifolds II: the \(L_\infty \) norm of the \(L_2\) projector. SIAM J. Math. Anal. 43, 662–684 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Kernel approximation on manifolds I: bounding the lebesgue constant. SIAM J. Math. Anal. 42, 175–208 (2010)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Polyharmonic and related kernels on manifolds: interpolation and approximation. Found. Comput. Math. 12, 625–670 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hangelbroek, T., Schmid, D.: Surface spline approximation on \(\text{ SO }(3)\). Appl. Comput. Harmon. Anal. 31, 169–184 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hannay, J.H., Nye, J.F.: Fibonacci numerical integration on a sphere. J. Phys. A Math. Gen. 37, 11591–11601 (2004)Google Scholar
  25. 25.
    Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices Am. Math. Soc. 51, 1186–1194 (2004)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Hebey, E.: Sobolev spaces on Riemannian manifolds. Lecture Notes in Mathematics, vol. 1635. Springer, Berlin (1996)Google Scholar
  27. 27.
    Helgason, S.: Groups and geometric analysis. Mathematical Surveys and Monographs, vol. 83 . American Mathematical Society, Providence, RI (2000) (Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original)Google Scholar
  28. 28.
    Hesse, K., Sloan, I.H., Womersley, R.S.: Numerical integration on the sphere. In: Freeden, W., Nashed, Z.M., Sonar, T. (eds.) Handbook of Geomathematics. Springer, Berlin (2010)Google Scholar
  29. 29.
    Hüttig, C., Stemmer, K.: The spiral grid: a new approach to discretize the sphere and its application to mantle convection. Geochem. Geophys. Geosyst. 9, Q02018 (2008)CrossRefGoogle Scholar
  30. 30.
    Keiner, J., Kunis, S., Potts, D.: Fast summation of radial functions on the sphere. Computing 78, 1–15 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Majewski, D., Liermann, D., Prohl, P., Ritter, B., Buchhold, M., Hanisch, T., Paul, G., Wergen, W., Baumgardner, J.: The operational global icosahedral-hexagonal gridpoint model GME: description and high-resolution tests. Mon. Wea. Rev. 130, 319–338 (2002)CrossRefGoogle Scholar
  32. 32.
    Mhaskar, H.N., Narcowich, F.J., Prestin, J., Ward, J.D.: \(L^p\) Bernstein estimates and approximation by spherical basis functions. Math. Comp. 79, 1647–1679 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comp. 70, 1113–1130 (2001) (Corrigendum: Math. Comp. 71 (2001), 453–454)Google Scholar
  34. 34.
    Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Narcowich, F.J., Sun, X., Ward, J.D., Wendland, H.: Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions. Found. Comput. Math. 7, 369–390 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Pesenson, I., Geller, D.: Cubature formulas and discrete fourier transform on compact manifolds. arXiv:1111.5900v1 [math.FA] (2011)Google Scholar
  37. 37.
    Ringler, T.D., Heikes, R.P., Randall, D.A.: Modeling the atmospheric general circulation using a spherical geodesic grid: a new class of dynamical cores. Mon. Wea. Rev. 128, 2471–2490 (2000)CrossRefGoogle Scholar
  38. 38.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7, 856–869 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Shankar, V., Wright, G.B., Fogelson, A.L., Kirby, R.M.: A study of different modeling choices for simulating platelets within the immersed boundary method. Appl. Numer. Math. 63, 58–77 (2013)Google Scholar
  41. 41.
    Slobbe, D., Simons, F., Klees, R.: The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid. J. Geod. 86, 609–628 (2012). doi: 10.1007/s00190-012-0543-x CrossRefGoogle Scholar
  42. 42.
    Sommariva, A., Womersley, R.S.: Integration by rbf over the sphere. Applied Mathematics Report AMR05/17, U. of New South Wales (2005)Google Scholar
  43. 43.
    Stuhne, G.R., Peltier, W.R.: New icosahedral grid-point discretizations of the shallow water equations on the sphere. J. Comput. Phys. 148, 23–53 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Swinbank, R., James Purser, R.: Fibonacci grids: a novel approach to global modelling. Q. J. R. Meteorol. Soc. 132, 1769–1793 (2006)CrossRefGoogle Scholar
  45. 45.
    Vilenkin, N.J.: Special functions and the theory of group representations. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, vol. 22. American Mathematical Society, Providence (1968)Google Scholar
  46. 46.
    Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Scott, Foresman and Co., London (1971)Google Scholar
  47. 47.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  48. 48.
    Williams, D.R.: Planetary Fact Sheets. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html. Visited Nov. 1, 2012 (2005)
  49. 49.
    Wright, G.B.: http://math.boisestate.edu/~wright/quad_weights/. Accessed 30 Oct 2012
  50. 50.
    Wright, G.B., Flyer, N., Yuen, D.: A hybrid radial basis function—pseudospectral method for thermal convection in a 3D spherical shell. Geochem. Geophys. Geosyst. 11, Q07003 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • E. Fuselier
    • 1
  • T. Hangelbroek
    • 2
  • F. J. Narcowich
    • 3
  • J. D. Ward
    • 3
  • G. B. Wright
    • 4
  1. 1.Department of MathematicsHigh Point UniversityHigh PointUSA
  2. 2.Department of MathematicsUniversity of HawaiiHonoluluUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Department of MathematicsBoise State UniversityBoiseUSA

Personalised recommendations