$\ell_{1}^{2}-\eta\ell_{2}^{2}$ sparsity regularization for nonlinear ill-posed problems

In this study, we investigate the $\left\|\cdot\right\|_{\ell_{1}}^{2}-\eta\left\|\cdot\right\|_{\ell_{2}}^{2}$ sparsity regularization with $0< \eta\leq 1$, in the context of nonlinear ill-posed inverse problems. We focus on the examination of the well-posedness associated with this regularization approach. Notably, the case where $\eta=1$ presents weaker theoretical outcomes than $0< \eta<1$, primarily due to the absence of coercivity and the Radon-Riesz property associated with the regularization term. Under specific conditions pertaining to the nonlinearity of the operator $F$, we establish that every minimizer of the $\left\|\cdot\right\|_{\ell_{1}}^{2}-\eta\left\|\cdot\right\|_{\ell_{2}}^{2}$ regularization exhibits sparsity. Moreover, for the case where $0<\eta<1$, we demonstrate convergence rates of $\mathcal{O}\left(\delta^{1/2}\right)$ and $\mathcal{O}\left(\delta\right)$ for the regularized solution, concerning a sparse exact solution, under differing yet widely accepted conditions related to the nonlinearity of $F$. Additionally, we present the iterative half variation algorithm as an effective method for addressing the $\left\|\cdot\right\|_{\ell_{1}}^{2}-\eta\left\|\cdot\right\|_{\ell_{2}}^{2}$ regularization in the domain of nonlinear ill-posed equations. Numerical results provided corroborate the effectiveness of the proposed methodology.