Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general non-linear inverse problem, the discrepancy $\|F(x_{ \alpha}^{ \delta})-y^{ \delta}\|_Y$ does not depend continuously on $\alpha$ and it is questionable whether there exists a regularization parameter $\alpha$ such that $\tau_1\delta\leq \|F(x_{ \alpha}^{ \delta})-y^{ \delta}\|_Y\leq \tau_2 \delta$ $(1\le \tau_1<\tau_2)$. In this paper, we prove the existence of $\alpha$ under Morozov's discrepancy principle if $\tau_2\ge (3+2\gamma)\tau_1$, where $\gamma>0$ is a parameter in a tangential cone condition for the nonlinear operator $F$. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach.
Categories: math.NA cs.NA math.OC