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Fast Harmonic/Spherical Splines and Parameter Choice Methods

  • Martin GuttingEmail author
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Lösungen zu Randwertproblemen aus den Geowissenschaften, bei denen die Randfläche durch die Erdoberfläche gegeben ist, können mittels harmonischer Splines konstruiert werden. Diese Splines sind lokalisierende Testfunktionen, die aus einem reproduzierenden Kern hervorgehen. Sie erlauben regionale Modelle sowie die lokale Verbesserung globaler Modelle in Teilen der Erdoberfläche.

In bestimmten Fällen dieser reproduzierenden Kerne ist eine schnelle Matrix-Vektor Multiplikation durch das schnelle Multipolverfahren (FMM) verfügbar. Die Kernidee des schnellen Multipolalgorithmus’ besteht aus zwei Teilen: Zum einen die hierarchische Unterteilung des dreidimensionalen Berechnungsgebiets in geschachtelte Würfel. Zum anderen wird eine Approximation anstelle des Kerns für weiter entfernte Punkte benutzt, die es erlaubt viele entfernte Punkte auf ein Mal zu betrachten. Der numerische Aufwand der Matrix-Vektor Multiplikation wird so linear bzgl. der Punkteanzahl bei einer vorgeschriebenen Genauigkeit der Approximation des reproduzierenden Kerns.

Diese schnelle Multiplikation wird bei der Spline-Approximation benutzt, um die auftretenden linearen Gleichungssysteme effizient zu lösen. Spline-Approximation erlaubt es auch verrauschte Daten zu nutzen, erfordert allerdings die Wahl eines Glättungsparameters. Verschiedene Verfahren werden präsentiert, die idealerweise automatisch diesen Parameter wählen – manche mit und manche ohne Kenntnis des Rauschlevels. Da ein schneller Lösungsalgorithmus für die linearen Gleichungssysteme benutzt wird, stehen dabei die gesamte Matrix oder gar ihre Singulärwerte, deren Berechnung einen viel höheren numerischen Aufwand erfordert, nicht zur Verfügung. Die Parameterwahlverfahren müssen diese Situation adäquat widerspiegeln.

Keywords

Boundary value problem Generalized interpolation Harmonic splines Spline approximation Fast multipole method Regularization Ill-posed problems Parameter choice methods 

Abstract

Solutions to boundary value problems in geoscience where the boundary is the Earth’s surface can be constructed in terms of harmonic splines. These are localizing trial functions that make use of a reproducing kernel. Splines allow regional modeling or the improvement of a global model in a part of the Earth’s surface.

For certain cases of the reproducing kernels a fast matrix-vector multiplication using the fast multipole method (FMM) is available. The main idea of the fast multipole algorithm consists of two parts: First, the hierarchical decomposition of the three-dimensional computational domain into cubes. Second, an approximation instead of the actual kernel is used for the more distant points which allows to consider many distant points at once. The numerical effort of the matrix-vector multiplication becomes linear in reference to the number of points for a prescribed accuracy of the approximation of the reproducing kernel.

This fast multiplication is used in spline approximation for the solution of the occurring linear systems which also allows the treatment of noisy data requires the choice of a smoothing parameter. Several methods are presented which ideally automatically choose this parameter with and without prior knowledge of the noise level. Using a fast solution algorithm we no longer have access to the whole matrix or its singular values whose computation requires a much larger numerical effort. This situation must be reflected by the parameter choice methods.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, Inc., New York (1972)Google Scholar
  2. 2.
    Akaike, H.: Information theory and an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory (Tsahkadsor, 1971), pp. 267–281. Akadémiai Kiadó, Budapest (1973)Google Scholar
  3. 3.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)CrossRefGoogle Scholar
  4. 4.
    Bakushinskii, A.B.: Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. U.S.S.R. Comput. Math. Math. Phys. 24, 181–182 (1984)CrossRefGoogle Scholar
  5. 5.
    Bauer, F.: Some considerations concerning regularization and parameter choice algorithms. Inverse Prob. 23(2), 837–858 (2007)CrossRefGoogle Scholar
  6. 6.
    Bauer, F., Gutting, M., Lukas, M.A.: Evaluation of parameter choice methods for regularization of ill-posed problems in geomathematics. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn., pp. 1713–1774. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  7. 7.
    Bauer, F., Hohage, T.: A Lepskij-type stopping rule for regularized Newton methods. Inverse Prob. 21, 1975–1991 (2005)CrossRefGoogle Scholar
  8. 8.
    Bauer, F., Kindermann, S.: Recent results on the quasi-optimality principle. J. Inverse Ill-Posed Prob. 17(1), 5–18 (2009)Google Scholar
  9. 9.
    Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simul. 81(9), 1795–1841 (2011)CrossRefGoogle Scholar
  10. 10.
    Becker, S.M.A.: Regularization of statistical inverse problems and the Bakushinkii veto. Inverse Prob. 27, 115010, 22pp (2011)CrossRefGoogle Scholar
  11. 11.
    Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics (Theory and Application). Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Reading (1981)Google Scholar
  12. 12.
    Blanchard, G., Mathé, P.: Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration. Inverse Prob. 28, 115011, 23pp (2012)CrossRefGoogle Scholar
  13. 13.
    Carrier, J., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput. 9(4), 669–686 (1988)CrossRefGoogle Scholar
  14. 14.
    Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155, 468–498 (1999)CrossRefGoogle Scholar
  15. 15.
    Choi, C.H., Ivanic, J., Gordon, M.S., Ruedenberg, K.: Rapid and staple determination of rotation matrices between spherical harmonics by direct recursion. J. Chem. Phys. 111(19), 8825–8831 (1999)CrossRefGoogle Scholar
  16. 16.
    Cummins, D.J., Filloon, T.G., Nychka, D.: Confidence intervals for nonparametric curve estimates: toward more uniform pointwise coverage. J. Am. Statist. Assoc. 96(453), 233–246 (2001)CrossRefGoogle Scholar
  17. 17.
    Driscoll, J.R., Healy, D.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)CrossRefGoogle Scholar
  18. 18.
    Edmonds, A.R.: Drehimpulse in der Quantenmechanik. Bibliographisches Institut, Mannheim (1964)Google Scholar
  19. 19.
    Engl, H.W., Gfrerer, H.: A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Appl. Numer. Math. 4(5), 395–417 (1988)CrossRefGoogle Scholar
  20. 20.
    Engl, H.W., Hanke, H., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  21. 21.
    Epton, M.A., Dembart, B.: Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. SIAM J. Sci. Comput. 16(4), 865–897 (1995)CrossRefGoogle Scholar
  22. 22.
    Fengler, M.J.: Vector spherical harmonic and vector wavelet based non-linear Galerkin schemes for solving the incompressile Navier–Stokes equation on the sphere. Ph.D. thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, Shaker, Aachen (2005)Google Scholar
  23. 23.
    Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry – case study: spatial and temporal mass variation in South America. Inverse Prob. 28, 065012, 34pp (2012)CrossRefGoogle Scholar
  24. 24.
    Freeden, W.: On approximation by harmonic splines. Manuscripta Geod. 6, 193–244 (1981)Google Scholar
  25. 25.
    Freeden, W.: On spherical spline interpolation and approximation. Math. Method. Appl. Sci. 3, 551–575 (1981)CrossRefGoogle Scholar
  26. 26.
    Freeden, W.: Interpolation and best approximation by harmonic spline functions. Boll. Geod. Sci. Aff. 1, 105–120 (1982)Google Scholar
  27. 27.
    Freeden, W.: On spline methods in geodetic approximation problems. Math. Method. Appl. Sci. 4, 382–396 (1982)CrossRefGoogle Scholar
  28. 28.
    Freeden, W.: Ein Konvergenzsatz in sphärischer Spline-Interpolation. Z. f. Vermes-sungswes.(ZfV) 109, 569–576 (1984)Google Scholar
  29. 29.
    Freeden, W.: Spherical spline interpolation: basic theory and computational aspects. J. Comput. Appl. Math. 11, 367–375 (1984)CrossRefGoogle Scholar
  30. 30.
    Freeden, W.: Harmonic splines for solving boundary value problems of potential theory. In: Mason, J.C., Cox, M.G. (eds.) Algorithms for Approximation. The Institute of Mathematics and Its Applications, Conference Series, vol. 10, pp. 507–529. Clarendon Press, Oxford (1987)Google Scholar
  31. 31.
    Freeden, W.: A spline interpolation method for solving boundary value problems of potential theory from discretely given data. Numer. Methods Partial Differ. Equ. 3, 375–398 (1987)CrossRefGoogle Scholar
  32. 32.
    Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart/Leipzig (1999)Google Scholar
  33. 33.
    Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. Chapman & Hall/CRC, Boca Raton (2013)Google Scholar
  34. 34.
    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere. Oxford University Press, Oxford (1998)Google Scholar
  35. 35.
    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere (With Applications to Geomathematics). Oxford Science Publications, Clarendon (1998)Google Scholar
  36. 36.
    Freeden, W., Glockner, O., Schreiner, M.: Spherical panel clustering and its numerical aspects. J. Geodesy 72, 586–599 (1998)CrossRefGoogle Scholar
  37. 37.
    Freeden, W., Gutting, M.: Special Functions of Mathematical (Geo-)Physics. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  38. 38.
    Freeden, W., Gutting, M.: Integration and Cubature Methods: A Geomathematically Oriented Course. Chapman and Hall/CRC, Boca Raton (2017)CrossRefGoogle Scholar
  39. 39.
    Freeden, W., Michel, V.: Multiscale Potential Theory (With Applications to Geoscience). Birkhäuser, Boston (2004)CrossRefGoogle Scholar
  40. 40.
    Freeden, W., Nashed, M.Z. (eds.): Handbook of Mathematical Geodesy – Functional Analytic and Potential Theoretic Methods. Birkhäuser, Basel (2018)Google Scholar
  41. 41.
    Freeden, W., Nashed, M.Z., Sonar, T. (eds.): Handbook of Geomathematics, 2nd edn. Springer, Heidelberg (2015)Google Scholar
  42. 42.
    Freeden, W., Schreiner, M.: Special functions in mathematical geosciences: an attempt at a categorization. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 1st edn., pp. 925–948. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  43. 43.
    Freeden, W., Schreiner, M., Franke, R.: A survey on spherical spline approximation. Surv. Math. Ind. 7, 29–85 (1997)Google Scholar
  44. 44.
    Gfrerer, H.: An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comput. 49(180), 507–522 (1987)CrossRefGoogle Scholar
  45. 45.
    Girard, D.: A fast “Monte-Carlo cross-validation” procedure for large least squares problems with noisy data. Numer. Math. 56(1), 1–23 (1989)CrossRefGoogle Scholar
  46. 46.
    Glockner, O.: On numerical aspects of gravitational field modelling from SST and SGG by harmonic splines and wavelets (with application to CHAMP data). Ph.D. thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. Shaker, Aachen (2002)Google Scholar
  47. 47.
    Greengard, L.: The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge (1988)Google Scholar
  48. 48.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(1), 325–348 (1987)CrossRefGoogle Scholar
  49. 49.
    Greengard, L., Rokhlin, V.: Rapid evaluation of potential fields in three dimensions. In: Anderson, C., Greengard, L. (eds.) Vortex Methods, pp. 121–141. Springer, Berlin/New York (1988)CrossRefGoogle Scholar
  50. 50.
    Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6 229–269 (1997)CrossRefGoogle Scholar
  51. 51.
    Gutting, M.: Fast multipole methods for oblique derivative problems. Ph.D. thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. Shaker, Aachen (2007)Google Scholar
  52. 52.
    Gutting, M.: Fast multipole accelerated solution of the oblique derivative boundary value problem. GEM Int. J. Geom. 3(2), 223–252 (2012)CrossRefGoogle Scholar
  53. 53.
    Gutting, M.: Fast spherical/harmonic spline modeling. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn., pp. 2711–2746. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  54. 54.
    Gutting, M.: Parameter choices for fast harmonic spline approximation. In: Freeden, W., Nashed, M.Z. (eds.) Handbook of Mathematical Geodesy – Functional Analytic and Potential Theoretic Methods, pp. 605–639. Birkhäuser, Basel (2018)CrossRefGoogle Scholar
  55. 55.
    Gutting, M., Kretz, B., Michel, V., Telschow, R.: Study on parameter choice methods for the RFMP with respect to downward continuation. Front. Appl. Math. Stat. 3, 1–17 (2017).  https://doi.org/10.3389/fams.2017.00010 CrossRefGoogle Scholar
  56. 56.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)CrossRefGoogle Scholar
  57. 57.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  58. 58.
    Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics (Second Reprint). Chelsea Publishing Company, New York (1965)Google Scholar
  59. 59.
    Hutchinson, M.: A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. Comput. 18(3), 1059–1076 (1989)CrossRefGoogle Scholar
  60. 60.
    Keiner, J., Kunis, S., Potts, D.: Fast summation of radial functions on the sphere. Computing 78, 1–15 (2006)CrossRefGoogle Scholar
  61. 61.
    Kellogg, O.D.: Foundation of Potential Theory. Springer, Berlin/Heidelberg/New York (1967)CrossRefGoogle Scholar
  62. 62.
    Kindermann, S., Neubauer, A.: On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Probl. Imaging 2(2), 291–299 (2008)CrossRefGoogle Scholar
  63. 63.
    Lepskij, O.: On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35(3), 454–466 (1990)CrossRefGoogle Scholar
  64. 64.
    Lu, S., Mathé, P.: Heuristic parameter selection based on functional minimization: optimality and model function approach. Math. Comput. 82(283), 1609–1630 (2013)CrossRefGoogle Scholar
  65. 65.
    Lu, S., Mathé, P.: Discrepancy based model selection in statistical inverse problems. J. Complex. 30(3), 290–308 (2014)CrossRefGoogle Scholar
  66. 66.
    Lukas, M.A.: Convergence rates for regularized solutions. Math. Comput. 51(183), 107–131 (1988)CrossRefGoogle Scholar
  67. 67.
    Lukas, M.A.: On the discrepancy principle and generalised maximum likelihood for regularisation. Bull. Aust. Math. Soc. 52(3), 399–424 (1995)CrossRefGoogle Scholar
  68. 68.
    Lukas, M.A.: Comparisons of parameter choice methods for regularization with discrete noisy data. Inverse Prob. 14(1), 161–184 (1998)CrossRefGoogle Scholar
  69. 69.
    Lukas, M.A.: Robust generalized cross-validation for choosing the regularization parameter. Inverse Prob. 22(5), 1883–1902 (2006)CrossRefGoogle Scholar
  70. 70.
    Lukas, M.A.: Strong robust generalized cross-validation for choosing the regularization parameter. Inverse Prob. 24, 034006, 16pp (2008)CrossRefGoogle Scholar
  71. 71.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Die Grundlehren der mathematischen Wissenschaften, vol. 52, 3rd edn. Springer, New York (1966)Google Scholar
  72. 72.
    Mathé, P., Pereverzev, S.V.: Regularization of some linear ill-posed problems with discretized random noisy data. Math. Comput. 75(256), 1913–1929 (2006)CrossRefGoogle Scholar
  73. 73.
    Michel, V.: Tomography: problems and multiscale solutions. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics. 1st edn., pp. 949–972. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  74. 74.
    Michel, V.: Lectures on Constructive Approximation – Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball. Birkhäuser, Boston (2013)Google Scholar
  75. 75.
    Michel, V.: RFMP – an iterative best basis algorithm for inverse problems in the geosciences. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn., pp. 2121–2147. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  76. 76.
    Moritz, H.: Classical physical geodesy. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 1st edn., pp. 127–158. Springer, Heidelberg (2010)Google Scholar
  77. 77.
    Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Dokl. 7, 414–417 (1966)Google Scholar
  78. 78.
    Phillips, D.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9, 84–97 (1962)CrossRefGoogle Scholar
  79. 79.
    Potts, D., Steidl, G.: Fast summation at nonequispaced knots by NFFTs. SIAM J. Sci. Comput. 24(6), 2013–2037 (2003)CrossRefGoogle Scholar
  80. 80.
    Raus, T.: On the discrepancy principle for the solution of ill-posed problems. Uch. Zap. Tartu. Gos. Univ. 672, 16–26 (1984)Google Scholar
  81. 81.
    Raus, T.: An a posteriori choice of the regularization parameter in case of approximately given error bound of data. In: Pedas, A. (ed.) Collocation and Projection Methods for Integral Equations and Boundary Value Problems, pp. 73–87. Tartu University, Tartu (1990)Google Scholar
  82. 82.
    Raus, T.: About regularization parameter choice in case of approximately given error bounds of data. In: Vainikko, G. (ed.) Methods for Solution of Integral Equations and Ill-Posed Problems, pp. 77–89. Tartu University, Tartu (1992)Google Scholar
  83. 83.
    Robinson, T., Moyeed, R.: Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression. Commun. Stat. Theory Methods 18(2), 523–539 (1989)CrossRefGoogle Scholar
  84. 84.
    Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187–207 (1985)CrossRefGoogle Scholar
  85. 85.
    Shure, L., Parker, R.L., Backus, G.E.: Harmonic splines for geomagnetic modelling. Phys. Earth Planet. Inter. 28, 215–229 (1982)CrossRefGoogle Scholar
  86. 86.
    Tikhonov, A., Arsenin, V.: Solutions of Ill-Posed Problems. Wiley, New York (1977)Google Scholar
  87. 87.
    Tikhonov, A., Glasko, V.: Use of the regularization method in non-linear problems. U.S.S.R. Comput. Math. Math. Phys. 5(3), 93–107 (1965)CrossRefGoogle Scholar
  88. 88.
    Varshalovich, D.A., Moskalev, A.N., Chersonskij, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)CrossRefGoogle Scholar
  89. 89.
    Wahba, G.: Practical approximate solutions to linear operator equations when the data are noisy. SIAM J. Numer. Anal. 14(4), 651–667 (1977)CrossRefGoogle Scholar
  90. 90.
    Wahba, G.: 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
  91. 91.
    Wahba, G.: Spline Models for Observational Data. SIAM, Philadelphia (1990)CrossRefGoogle Scholar
  92. 92.
    White, C.A., Head-Gordon, M.: Rotating around the quartic angular momentum barrier in fast multipole method calculations. J. Chem. Phys. 105(12), 5061–5067 (1996)CrossRefGoogle Scholar
  93. 93.
    Yamabe, H.: On an extension of the Helly’s theorem. Osaka Math. J. 2(1), 15–17 (1950)Google Scholar
  94. 94.
    Yarvin, N., Rokhlin, V.: Generalized Gaussian quadratures and singular value decomposition of integral equations. SIAM J. Sci. Comput. 20(2), 699–718 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of BiotechnologyMannheim University of Applied SciencesMannheimGermany

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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