On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization
The authors study the iterated Tikhonov regularization scheme \(x_{\alpha,n}^{\delta}=(T^*T+\alpha I)^{-1}(T^* y^{\delta}+\alpha x_{\alpha,n-1}^{\delta})\), \(x_{\alpha,0}^{\delta}=0\) for linear ill-posed problems \(Tx=y^{\delta}\), where \(\| y^{\delta}-y \| \leq \delta\), \(T\in L(X,Y)\), and \(X,Y\) are Hilbert spaces. For a special error free parameter choice rule \(\alpha=\alpha(y^{\delta})\), the authors derive conditions on \(y^{\delta}-y\) that ensure the convergence of the approximations \(x_{\alpha(y^{\delta}),n}^{\delta}\) to a solution of the unperturbed equation as \(\delta\to 0\). Estimates of the rate of convergence are also established.
scientific article; zbMATH DE number 5500622