On the adaptive selection of the parameter in regularization of ill-posed problems

We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction

Regularized approximation methods with perturbations for ill-posed operator equations

We are concerned with a parameter choice strategy for the Tikhonov regularization $(\tilde{A}+\alpha I)\tilde{x}$ = T* $\tilde{y}$+ w where $\tilde{A}$ is a (not necessarily selfadjoint) approximation of T*T and T*$\tilde y$+ w is a perturbed form of the (not exactly computed) term T*y. We give conditions for convergence and optimal convergence rates

Toying with Jordan matrices

It is shown that an important resolvent estimate is unstable under small perturbations

Toying with Jordan matrices

It is shown that an important resolvent estimate is unstable under small perturbations