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Kernel based quadrature on spheres and other homogeneous spaces

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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.

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Notes

  1. See [21] for a discussion of the Riemannian geometry involved, including metrics, tensors, covariant derivatives, etc.

References

  1. Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Lecture Notes in Mathematics. Springer, Berlin (2012)

  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)

  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)

  4. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14, 1–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borodachov, S.V., Hardin, D.P. Saff, E.: Low complexity methods for discretizing manifolds via Riesz energy minimization. Submitted (2012)

  6. Brown, G., Dai, F.: Approximation of smooth functions on compact two-point homogeneous spaces. J. Funct. Anal. 220, 401–423 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Giné, M.E.: The addition formula for the eigenfunctions of the Laplacian. Adv. Math. 18, 102–107 (1975)

    Google Scholar 

  14. Giraldo, F.X.: Lagrange-Galerkin methods on spherical geodesic grids. J. Comput. Phys. 136, 197–213 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. González, A.: Measurement of areas on a sphere using Fibonacci and latitude longitude lattices. Math. Geosci. 42, 49–64 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gräf, M.: A unified approach to scattered data approximation on \(S^3\) and \(SO(3)\). Adv. Comput. Math. 37, 379–393 (2012)

  17. Gräf, M.: Efficient algorithms for the computation of quadrature points on Riemannian manifolds, PhD thesis, Chemnitz University of Technology, Department of Mathematics (2013)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gräf, M., Potts, D.: Sampling sets and quadrature formulae on the rotation group. Numer. Funct. Anal. Optim. 30, 665–688 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hangelbroek, T., Schmid, D.: Surface spline approximation on \(\text{ SO }(3)\). Appl. Comput. Harmon. Anal. 31, 169–184 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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. Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices Am. Math. Soc. 51, 1186–1194 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Hebey, E.: Sobolev spaces on Riemannian manifolds. Lecture Notes in Mathematics, vol. 1635. Springer, Berlin (1996)

  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)

  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. 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)

    Article  Google Scholar 

  30. Keiner, J., Kunis, S., Potts, D.: Fast summation of radial functions on the sphere. Computing 78, 1–15 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  36. Pesenson, I., Geller, D.: Cubature formulas and discrete fourier transform on compact manifolds. arXiv:1111.5900v1 [math.FA] (2011)

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  39. Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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

    Article  Google Scholar 

  42. Sommariva, A., Womersley, R.S.: Integration by rbf over the sphere. Applied Mathematics Report AMR05/17, U. of New South Wales (2005)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  44. Swinbank, R., James Purser, R.: Fibonacci grids: a novel approach to global modelling. Q. J. R. Meteorol. Soc. 132, 1769–1793 (2006)

    Article  Google Scholar 

  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)

  46. Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Scott, Foresman and Co., London (1971)

  47. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  48. Williams, D.R.: Planetary Fact Sheets. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html. Visited Nov. 1, 2012 (2005)

  49. Wright, G.B.: http://math.boisestate.edu/~wright/quad_weights/. Accessed 30 Oct 2012

  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)

    Article  Google Scholar 

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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].

Author information

Authors and Affiliations

  1. Department of Mathematics, High Point University, High Point, NC, 27262, USA

    E. Fuselier

  2. Department of Mathematics, University of Hawaii, Honolulu, HI, 96822, USA

    T. Hangelbroek

  3. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    F. J. Narcowich & J. D. Ward

  4. Department of Mathematics, Boise State University, Boise, ID, 83725, USA

    G. B. Wright

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  1. E. Fuselier
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  2. T. Hangelbroek
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Correspondence to F. J. Narcowich.

Additional information

T. Hangelbroek’s research was supported by grant DMS-1232409 from the National Science Foundation. F. J. Narcowich and J. D. Ward’s research was supported by grant DMS-1211566 from the National Science Foundation. G. B. Wright’s research was supported by grants DMS-0934581and DMS-1160379 from the National Science Foundation.

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Fuselier, E., Hangelbroek, T., Narcowich, F.J. et al. Kernel based quadrature on spheres and other homogeneous spaces. Numer. Math. 127, 57–92 (2014). https://doi.org/10.1007/s00211-013-0581-1

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