On Lavrentiev Regularization for Ill-Posed Problems in Hilbert Scales

Abstract

In this article we study the problem of identifying the solution x ^† of linear ill-posed problems Ax = y in a Hilbert space X where instead of exact data y noisy data y ^δ ∈ X are given satisfying with known noise level δ. Regularized approximations are obtained by the method of Lavrentiev regularization in Hilbert scales, that is, is the solution of the singularly perturbed operator equation where B is an unbounded self-adjoint strictly positive definite operator satisfying . Assuming the smoothness condition we prove that the regularized approximation provides order optimal error bounds (i) in case of a priori parameter choice for and (ii) in case of Morozov's discrepancy principle for s ≥ p. In addition, we provide generalizations, extend our study to the case of infinitely smoothing operators A as well as to nonlinear ill-posed problems and discuss some applications.

Keywords:

• Ill-posed problems
• Inverse problems
• Lavrentiev regularization
• A priori parameter choice
• Discrepancy principle
• Hilbert scales
• Monotone operators
• AMS Subject Classifications: 65J20
• 65M30

Acknowledgments

J. Janno gratefully acknowledges the support received from the Deutsche Forschungs-gemeinschaft (DFG) to visit the Department of Mathematics and Natural Sciences of the University of Applied Sciences Zittau/Görlitz from June until August 2002, under the grant 436 EST 17/1/02. J. Janno gratefully acknowledges also the support of the Estonian Science Foundation, grant 5006.