Parameter Choices for Fast Harmonic Spline Approximation

The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience where the boundary 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 occurring 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 multiplication 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 smoothing parameter. We summarize several methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level.