A Tree Algorithm for Isotropic Finite Elements on the Sphere

The earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0.4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modeling of geoscientifically relevant data on the sphere plays an important role. In this paper, we deal with isotropic kernel functions showing a local support (briefly called isotropic finite elements) for reconstructing square-integrable functions on the sphere. An essential tool is the concept of multiresolution analysis by virtue of the spherical up function. Because the up function is built by an infinite convolution product, we do not know an explicit representation of it. However, the tree algorithm for the multiresolution analysis based on the up functions can be formulated by convolutions of isotropic kernels of low-order polynomial structure. For these kernels, we are able to find an explicit representation, so that the tree algorithm can be implemented efficiently.

Keywords:

• Locally supported radial basis functions
• Multiresolution analysis
• Spherical
• Tree algorithms
• Up functions

AMS Subject Classification:

• 41A30
• 41A58
• 42C15
• 42C40
• 44A35
• 45E10
• 65T60

ACKNOWLEDGMENTS

Financial support by the Graduiertenkolleg “Mathematik und Praxis,” University of Kaiserslautern, and by the “Stiftung Rheinland-Pfalz für Innovation” is gratefully acknowledged by the first and second authors.