Morozov's principle for the augmented Lagrangian method applied to linear inverse problems (Q2903610)

The paper deals with the ill-posed optimization problem \(J(u)\to \min\) subject to \(Ku=g\), where \(J:H_1 \to R\) is a convex functional and \(K:H_1 \to H_2\) is a linear bounded operator between Hilbert spaces \(H_1\) and \(H_2\). It is assumed that instead of exact data \(g \in H_2\) only noise-affected observation \(g^{\delta}\) is available, where \(\| g^{\delta}-g\| \leq \delta\). The authors study the iterative method \(u_n^{\delta} \in \text{{Argmin}}_{u \in H_1} (0.5 \tau_n \| Ku-g^{\delta}\|^2+ J(u)-\langle p_{n-1}^{\delta}, Ku-g^{\delta}\rangle)\),NEWLINENEWLINE \(p_n^{\delta}=p_{n-1}^{\delta}+\tau_n (g^{\delta}-Ku_n^{\delta})\); \(\tau_n>0\), with Morozov's discrepancy principle as a stopping rule.