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 $\Vert F(x_{\hat{I}}{\pm }^{\hat{I}}´)-y^{\hat{I}}´{\Vert }_Y$ does not depend continuously on $\hat{I}\pm$ and it is questionable whether there exists a regularization parameter $\hat{I}\pm$ such that $\ddot{I}{„}_1\hat{I}´\le \Vert F(x_{\hat{I}}{\pm }^{\hat{I}}´)-y^{\hat{I}}´{\Vert }_Y\le \ddot{I}{„}_2\hat{I}´$ $(1\le \ddot{I}{„}_1<\ddot{I}{„}_2)$. In this paper, we prove the existence of $\hat{I}\pm$ under Morozov's discrepancy principle if $\ddot{I}{„}_2\ge (3+2{\hat{I}}^3)\ddot{I}{„}_1$, where ${\hat{I}}^3>0$ 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.

24 pages, 10 figures