NDLI: Multilevel regularization of wavelet based fitting of scattered data

Content Provider	Springer Nature Link
Author	Castaño, Daniel Kuth, Angela
Copyright Year	2005
Abstract	In [6], 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. It 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 which minimizes the distance to the data in a least squares sense. This is followed by a thresholding procedure on the wavelet coefficients to discard those which are too small to contribute much to the surface representation. In this paper, we firstly generalize this strategy to a classically regularized least squares functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling the usual cross-validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, synthetically produced and of geophysical origin, are presented. In order to reduce computational costs, we then introduce a multilevel generalized cross-validation technique which goes beyond the Sobolev formulation and exploits the hierarchical setting based on wavelets. We illustrate the performance of the new strategy on some geophysical data.
Starting Page	81
Ending Page	96
Page Count	16
File Format	PDF
ISSN	10171398
Journal	Numerical Algorithms
Volume Number	39
Issue Number	1-3
e-ISSN	15729265
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2005-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	wavelets scattered data least squares approximation regularization fractional Sobolev norms generalized multilevel cross-validation Theory of Computation Numeric Computing Mathematics Algebra Algorithms
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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