Convergence and Optimality of Adaptive Regularization for Ill-posed Deconvolution Problems in Infinite Spaces

Abstract

The adaptive regularization method is first proposed by Ryzhikov et al. in [6] for the deconvolution in elimination of multiples which appear frequently in geoscience and remote sensing. They have done experiments to show that this method is very effective. This method is better than the Tikhonov regularization in the sense that it is adaptive, i.e., it automatically eliminates the small eigenvalues of the operator when the operator is near singular. In this paper, we give theoretical analysis about the adaptive regularization. We introduce an a priori strategy and an a posteriori strategy for choosing the regularization parameter, and prove regularities of the adaptive regularization for both strategies. For the former, we show that the order of the convergence rate can approach $O{\left( {{\left\| n \right\|}^{{\frac{{4v}} {{4v + 1}}}} } \right)}$ for some 0 < ν < 1, while for the latter, the order of the convergence rate can be at most $O{\left( {{\left\| n \right\|}^{{\frac{{2v}} {{2v + 1}}}} } \right)}$ for some 0 < ν < 1.