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

TL;DR

This paper investigates the existence of a regularization parameter satisfying Morozov's discrepancy principle in nonlinear inverse problems, proving existence under certain conditions and demonstrating convergence and numerical efficiency.

Contribution

It establishes conditions for the existence of a regularization parameter under Morozov's principle in nonlinear problems, which was previously uncertain.

Findings

Existence of regularization parameter proven under specific conditions.

Convergence of regularized solutions shown.

Numerical results demonstrate method efficiency.

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 $\|F(x_{\alpha}^{\delta})-y^{\delta}\|_Y$ 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.