Examples of saturated convergence rates for Tikhonov regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data ( both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u(+) (i.e. the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u(+) is an element of im(A* A)(nu). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with nu only for some values of nu, in general not greater than one. his phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.