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.