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

  • M. Thamban Nair

    Indian Institute of Technics, Madras, Chennai, India
  • Ulrich Tautenhahn

    University of Applied Sciences, Zittau, Germany

Abstract

In this paper we study the problem of identifying the solution of linear ill-posed problems with non-negative and self-adjoint operators on a Hilbert space where instead of exact data noisy data are given satisfying with known noise level . Regularized approximations are obtained by the method of Lavrentiev regularization, that is, is the solution of the singularly perturbed operator equation , and the regularization parameter is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set , we prove that the regularized approximations provide order optimal error bounds on the set . Our results cover the special case of finitely smoothing operators and extend recent results for infinitely smoothing operators. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order 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