An optimal parameter choice for regularized ill-posed problems (Q1110786)

The paper deals with the modification of Schock's criterion (1984), leading to an optimal convergence rate for Tikhonov regularization. Let K be a bounded linear operator from a Hilbert space to another Hilbert space. The numerical instability of solutions of \(Kx=g\) in terms of pseudoinverse is circumvented by Tikhonov by minimizing the functional \(\| Kx-g\|^ 2+\alpha \| x\|^ 2\), \(0<\alpha\) and using the minimizer \(x^{\delta}_{\alpha}\). The Schock's criterion \(\| Kx^{\delta}_{\alpha}-g^{\delta}\| =\delta^ p/\alpha^ q\), \(0<p,q\), gives the error close to the optimal rate \(O(\delta^{2/3})\) for large q. The author uses the case, \(p=q=2\) to show that Schock's criterion gives the optimal order.