On existence of a variational regularization parameter under Morozov's discrepancy principle

Abstract

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 $\Vert F(x_{\alpha }^{\delta })-y^{\delta }{\Vert }_Y$ does not depend continuously on $\alpha$ and it is questionable whether there exists a regularization parameter $\alpha$ such that ${\tau }_1\delta \le \Vert F(x_{\alpha }^{\delta })-y^{\delta }{\Vert }_Y\le {\tau }_2\delta$ $(1\le {\tau }_1<{\tau }_2)$. 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.

arXiv comment24 pages, 10 figures