Regularization wavelets and multiresolution

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Abstract
Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind A F = G. Due to the ill-posedness of the equation, a variety of regularization methods are being considered for an approximate solution, where particular emphasis must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular-value decomposition as a wavelet analysis. This enables us to perform, for example, Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz satellite gravity gradiometry.