An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems
The paper is concerned with nonlinear ill-posed operator equations \(F(x)=y\), where \(F\) is a mapping between real Banach spaces \(X\) and \(Y\). The exact element \(y\) is given by its approximation \(y^{\delta}\), \(\| y^{\delta}-y\|_Y \leq \delta\). The authors establish rate of convergence estimates for variational regularization methods associated with the Tikhonov type functional \(T_{\alpha}^{\delta}(x)=\psi(\| F(x)-y^{\delta}\|_Y)+\alpha \Omega(x)\). The estimates are given in terms of the Bregman distance and a generalized source condition presented as a variational inequality.
scientific article; zbMATH DE number 5835147
scientific article