Regularization of Wavelet Based Fitting of Scattered Data — Some Experiments∗ Short Title: Regularization of Wavelet Based Fitting of Scattered Data

In [3] an adaptive method to approximate unorganized clouds of points by smooth surfaces based on wavelets has been described. The general fitting algorithm operates on a coarse–to–fine basis and selects on each refinement level in a first step a reduced number of wavelets which are appropriate to represent the features of the data set. In a second step, the fitting surface is constructed as the linear combination of the wavelets that minimizes the distance to the data in a least squares sense. This is then followed by a thresholding procedure of the resulting wavelet coefficients to discard those which are too small to contribute much to the surface representation. Here we adapt this strategy to a classically regularized least square functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling in this framework the usual cross validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, both synthetically produced and of geophysical origin, are presented. Moreover, in order to reduce computational costs, we introduce a generalized cross validation technique which exploits the hierarchical setting based on wavelets and illustrates the performance of the new strategy with some geophysical data.