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Fast Harmonic/Spherical Splines and Parameter Choice Methods

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Handbuch der Geodäsie

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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.

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.

This chapter is part of the series Handbuch der Geodäsie, volume “Mathematical Geodasy/ Mathematische Geodäsie”, edited by Willi Freeden, Kaiserslautern.

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  1. Department of Biotechnology, Mannheim University of Applied Sciences, 68163, Paul-Wittsack-Straße 10, Mannheim, Germany

    Martin Gutting

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  1. Martin Gutting
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  1. Lehr- und Forschungsgebiet "Geomathemati, TU Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Physikalische Geodäsie, TU München Inst. Astronomische und, München, Germany

    Reiner Rummel

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  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

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Gutting, M. (2018). Fast Harmonic/Spherical Splines and Parameter Choice Methods. In: Freeden, W., Rummel, R. (eds) Handbuch der Geodäsie. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46900-2_106-1

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