Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions
M. Thamban Nair
Indian Institute of Technics, Madras, Chennai, IndiaUlrich Tautenhahn
University of Applied Sciences, Zittau, Germany

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.
Cite this article
M. Thamban Nair, Ulrich Tautenhahn, Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions. Z. Anal. Anwend. 23 (2004), no. 1, pp. 167–185
DOI 10.4171/ZAA/1192