Parameter Choices for Fast Harmonic Spline Approximation

Abstract

The approximation by harmonic trial functions allows the construc- tion of the solution of boundary value problems in geoscience where the bound- ary is often the known surface of the Earth itself. Using harmonic splines such a solution can be approximated from discrete data on the surface. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth’s surface is possible with splines. Fast multipole methods have been developed for some cases of the oc- curring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector mul- tiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. In combination with an iterative solver this provides a fast computation of the spline coefficients. The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smooth- ing parameter. We summarize several methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level.