On the existence of a variational regularization parameter under Morozov's discrepancy principle for nonlinear inverse problems

Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general nonlinear inverse problem, the discrepancy \|F(xαδ)-yδ\|Y does not depend continuously on α and it is questionable whether there exists a regularization parameter α such that τ₁δ≤ \|F(xαδ)-yδ\|Y≤ τ₂ δ (1≤ τ₁<τ₂). In this paper, we prove the existence of α under Morozov's discrepancy principle if τ₂≥ (3+2γ)τ₁ or τ₁≤ τ₂/(3+2γ), where γ>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.