Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.