On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales (Q616674)

For solving linear ill-posed operator equations \(A_0 x=y_0\), where \(A_0\) acts between Hilbert spaces \(X\) and \(Y\), the method of Tikhonov regularization in Hilbert scales \(\min_{x \in X} \{ \| A_h x-y_{\delta}\|^2+\alpha \| B^s x \|^2 \}\) \((\alpha>0, s\geq 0)\) is investigated. Here, \(B: D(B) \subset X \to X\) is an unbounded densely defined selfadjoint strictly positive operator, \(\| A_h-A_0 \| \leq h\), \(\| y_{\delta}-y\| \leq \delta\). To adjust the regularization parameter \(\alpha\) with error levels \(\delta,h\), the generalized discrepancy principle \(\| A_h x_{\alpha}^{\delta,h}-y_{\delta}\|=\delta +h \| B^s x_{\alpha}^{\delta,h} \|\) is exploited. Under certain smoothness assumptions, the authors provide order optimal error bounds that characterize the accuracy of the regularized solution \(x_{\alpha(\delta,h)}^{\delta,h}\). Theoretical results are illustrated by numerical experiments.