$\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization for nonlinear ill-posed problems

In this paper, we consider the $\alpha\| \cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $\alpha\geq\beta\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $\alpha>\beta\geq0$, the results for the case $\alpha=\beta\geq0$ are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain condition on the nonlinearity of $F$, we prove that every minimizer of $\alpha\| \cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$ regularization is sparse. For the case $\alpha>\beta\geq0$, if the exact solution is sparse, we derive convergence rate $O(\delta^{\frac{1}{2}})$ and $O(\delta)$ of the regularized solution under two commonly adopted conditions on the nonlinearity of $F$, respectively. In particular, it is shown that the iterative soft thresholding algorithm can be utilized to solve the $\alpha\| \cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$ regularization problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.