Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

Abstract

We introduce an intrinsic notion of Hölder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decompositions in characterizing Hölder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated We study the problem of identifying the solution x* of linear ill-posed problems Ax = y with non-negative and self-adjoint operators A on a Hilbert space X where instead of exact data y noisy data yδ in X are given satisfying ||y - yδ|| ≤ δ with known noise level δ. Regularized approximations xαδ are obtained by the method of Lavrentiev regularization, that is, xαδ is the solution of the singularly perturbed operator equation Ax + αx = yδ , and the regularization parameter α is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximations provide order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extend recent results for infinitely smoothing operators. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order m and discuss a special ill-posed problem arising in inverse heat conduction.