Fast Spherical/Harmonic Spline Modeling

  • Martin  Gutting
Living reference work entry


Spherical and harmonic splines are closely related approaches to solve interpolation/approximation as well as boundary value problems on the sphere and on regular (sphere-like) surfaces, respectively. In any case they lead to a system of linear equations which requires fast summation methods for the kernel sums. The fast multipole method achieves just that and is combined in this paper with a preconditioner using the same decomposition of the computational domain to solve the system of linear equations resulting from spherical/harmonic splines. Due to the localizing nature of splines, regional problems can also be treated with this approach.


Interpolation Problem Fast Multipole Method Spherical Spline Multiplicative Variant Singularity Kernel 
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  1. Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68:337–404MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bauer F, Lukas MA (2011) Comparing parameter choice methods for regularization of ill-posed problems. Math Comput Simul 81(9):1795–1841MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bauer F, Gutting M, Lukas MA (2014) Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  4. Beatson RK, Billings S, Light WA (2000) Fast solution of the radial basis function interpolation equations: domain decomposition methods. SIAM J Sci Comput 22(5):1717–1740MathSciNetCrossRefzbMATHGoogle Scholar
  5. Biedenharn LC, Louck JD (1981) Angular momentum in quantum physics (theory and application). Encyclopedia of mathematics and its applications. Addison-Wesley, ReadingGoogle Scholar
  6. Carrier J, Greengard L, Rokhlin V (1988) A fast adaptive multipole algorithm for particle simulations. SIAM J Sci Stat Comput 9(4):669–686MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chan TF, Mathew TP (1994) Domain decomposition algorithms. Acta Numer 3:61–143MathSciNetCrossRefGoogle Scholar
  8. Cheng H, Greengard L, Rokhlin V (1999) A fast adaptive multipole algorithm in three dimensions. J Comput Phys 155:468–498MathSciNetCrossRefzbMATHGoogle Scholar
  9. Choi CH, Ivanic J, Gordon MS, Ruedenberg K (1999) Rapid and staple determination of rotation matrices between spherical harmonics by direct recursion. J Chem Phys 111(19):8825–8831CrossRefGoogle Scholar
  10. Edmonds AR (1964) Drehimpulse in der Quantenmechanik. Bibliographisches Institut, MannheimGoogle Scholar
  11. Epton MA, Dembart B (1995) Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. SIAM J Sci Comput 16(4):865–897MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fengler MJ (2005) Vector spherical harmonic and vector wavelet based non-linear Galerkin schemes for solving the incompressile Navier–Stokes equation on the sphere. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, Shaker, AachenGoogle Scholar
  13. Freeden W (1981a) On approximation by harmonic splines. Manuscr Geod 6:193–244zbMATHGoogle Scholar
  14. Freeden W (1981b) On spherical spline interpolation and approximation. Math Method Appl Sci 3:551–575MathSciNetCrossRefzbMATHGoogle Scholar
  15. Freeden W (1982a) Interpolation and best approximation by harmonic spline functions. Boll Geod Sci Aff 1:105–120Google Scholar
  16. Freeden W (1982b) On spline methods in geodetic approximation problems. Math Method Appl Sci 4:382–396MathSciNetCrossRefzbMATHGoogle Scholar
  17. Freeden W (1984a) Spherical spline interpolation: basic theory and computational aspects. J Comput Appl Math 11:367–375MathSciNetCrossRefzbMATHGoogle Scholar
  18. Freeden W (1984b) Ein Konvergenzsatz in sphärischer Spline-Interpolation. Z f Vermessungswes.(ZfV) 109:569–576Google Scholar
  19. Freeden W (1987a) A spline interpolation method for solving boundary value problems of potential theory from discretely given data. Numer Methods Partial Differ Equ 3:375–398MathSciNetCrossRefzbMATHGoogle Scholar
  20. Freeden W (1987b) Harmonic splines for solving boundary value problems of potential theory. In: Mason JC, Cox MG (eds) Algorithms for approximation. The institute of mathematics and its applications, conference Series, vol 10. Clarendon Press, Oxford, pp 507–529Google Scholar
  21. Freeden W (1999) Multiscale modelling of spaceborne geodata. B.G. Teubner, Stuttgart/LeipzigzbMATHGoogle Scholar
  22. Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  23. Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, BaselCrossRefzbMATHGoogle Scholar
  24. Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser, Boston/Basel/BerlinCrossRefzbMATHGoogle Scholar
  25. Freeden W, Schreiner M (2014) Special functions in mathematical geosciences: an attempt at a categorization. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  26. Freeden W, Schreiner M, Franke R (1997) A survey on spherical spline approximation. Surv Math Ind 7:29–85MathSciNetzbMATHGoogle Scholar
  27. Freeden W, Gervens T, Schreiner M (1998a) Constructive approximation on the sphere (with applications to geomathematics). Oxford Science Publications, Clarendon Press, OxfordzbMATHGoogle Scholar
  28. Freeden W, Glockner O, Schreiner M (1998b) Spherical panel clustering and its numerical aspects. J Geodesy 72:586–599CrossRefzbMATHGoogle Scholar
  29. Glockner O (2002) On numerical aspects of gravitational field modelling from SST and SGG by harmonic splines and wavelets (with application to CHAMP data). PhD thesis, Geomathenatics Group, Department of Mathematics, University of Kaiserslautern. Shaker, AachenGoogle Scholar
  30. Greengard L (1988) The rapid evaluation of potential fields in particle systems. MIT, CambridgezbMATHGoogle Scholar
  31. Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73(1):325–348MathSciNetCrossRefzbMATHGoogle Scholar
  32. Greengard L, Rokhlin V (1988) Rapid evaluation of potential fields in three dimensions. In: Anderson C, Greengard L (eds) Vortex methods. Springer, Berlin/Heidelberg/New York, pp 121–141CrossRefGoogle Scholar
  33. Greengard L, Rokhlin V (1997) A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer 6:229–269MathSciNetCrossRefGoogle Scholar
  34. Gutting M (2007) Fast multipole methods for oblique derivative problems. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. Shaker, AachenGoogle Scholar
  35. Gutting M (2012) Fast multipole accelerated solution of the oblique derivative boundary value problem. GEM Int J Geom 3(2):223–252MathSciNetCrossRefzbMATHGoogle Scholar
  36. Hastings D, Row LW III (1997) TerrainBase global Terrain model summary documentation. National Geodetic Data Center, BoulderGoogle Scholar
  37. Hesse K (2002) Domain decomposition methods for multiscale geopotential determination from SST and SGG. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, Shaker, AachenGoogle Scholar
  38. Hobson EW (1965) The theory of spherical and ellipsoidal harmonics (second reprint). Chelsea Publishing Company, New YorkGoogle Scholar
  39. Keiner J, Kunis S, Potts D (2006) Fast summation of radial functions on the sphere. Computing 78:1–15MathSciNetCrossRefzbMATHGoogle Scholar
  40. Kellogg OD (1967) Foundation of potential theory. Springer, Berlin/Heidelber/New YorkCrossRefGoogle Scholar
  41. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP-1998-206861, NASA Goddard Space Flight Center, GreenbeltGoogle Scholar
  42. Michel V (2013) Lectures on constructive approximation – Fourier, spline, and wavelet methods on the real line, the sphere, and the ball. Birkhäuser, BostonzbMATHGoogle Scholar
  43. Michel V (2014a) RFMP – an iterative best basis algorithm for inverse problems in the geosciences. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  44. Michel V (2014b) Tomography: problems and multiscale solutions. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  45. Moritz H (2014) Classical physical geodesy. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  46. Potts D, Steidl G (2003) Fast summation at nonequispaced knots by NFFTs. SIAM J Sci Comput 24(6):2013–2037MathSciNetCrossRefzbMATHGoogle Scholar
  47. Rakhmanov EA, Saff EB, Zhou YM (1994) Minimal discrete energy on the sphere. Math Res Lett 1:647–662MathSciNetCrossRefzbMATHGoogle Scholar
  48. Rokhlin V (1985) Rapid solution of integral equations of classical potential theory. J Comput Phys 60:187–207MathSciNetCrossRefzbMATHGoogle Scholar
  49. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  50. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869MathSciNetCrossRefzbMATHGoogle Scholar
  51. Shure L, Parker RL, Backus GE (1982) Harmonic splines for geomagnetic modelling. Phys Earth Planet Inter 28:215–229CrossRefGoogle Scholar
  52. Smith BF, Bjørstad PE, Gropp WD (1996) Domain decomposition (parallel multilevel methods for elliptic partial differential equations). Cambridge University Press, CambridgezbMATHGoogle Scholar
  53. Vars̆alovic̆ DA, Moskalev AN, Chersonskij VK (1988) Quantum theory of angular momentum. World Scientific, SingaporeGoogle Scholar
  54. Wahba G (1981) Spline interpolation and smoothing on the sphere. SIAM J Sci Stat Comput 2:5–16. Also errata: SIAM J Sci Stat Comput 3:385–386 (1981)Google Scholar
  55. Wahba G (1990) Spline models for observational data. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  56. White CA, Head-Gordon M (1996) Rotating around the quartic angular momentum barrier in fast multipole method calculations. J Chem Phys 105(12):5061–5067CrossRefGoogle Scholar
  57. Yamabe H (1950) On an extension of the Helly’s theorem. Osaka Math J 2(1):15–17MathSciNetzbMATHGoogle Scholar
  58. Yarvin N, Rokhlin V (1998) Generalized Gaussian quadratures and singular value decomposition of integral equations. SIAM J Sci Comput 20(2):699–718MathSciNetCrossRefGoogle Scholar
  59. Zhou X, Hon YC, Li J (2003) Overlapping domain decomposition method by radial basis functions. Appl Numer Math 44:241–255MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Geomathematics Group, University of SiegenSiegenGermany

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