${\ell }_1^2-\hat{I}\cdot {\ell }_2^2$ sparsity regularization for nonlinear ill-posed problems

Abstract

In this study, we investigate the ${\Vert \cdot \Vert }_{{\ell }_1}^2-\hat{I}\cdot {\Vert \cdot \Vert }_{{\ell }_2}^2$ sparsity regularization with $0<\hat{I}\cdot \le 1$, 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 $\hat{I}\cdot =1$ presents weaker theoretical outcomes than $0<\hat{I}\cdot <1$, 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 ${\Vert \cdot \Vert }_{{\ell }_1}^2-\hat{I}\cdot {\Vert \cdot \Vert }_{{\ell }_2}^2$ regularization exhibits sparsity. Moreover, for the case where $0<\hat{I}\cdot <1$, we demonstrate convergence rates of $\mathcal{O}\left(\hat{I}{´}^{1/2}\right)$ and $\mathcal{O}\left(\hat{I}´\right)$ 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 ${\Vert \cdot \Vert }_{{\ell }_1}^2-\hat{I}\cdot {\Vert \cdot \Vert }_{{\ell }_2}^2$ regularization in the domain of nonlinear ill-posed equations. Numerical results provided corroborate the effectiveness of the proposed methodology.

33 pages, 3 figures