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$α\ell_{1}-β\ell_{2}$ sparsity regularization for nonlinear ill-posed problems

In this paper, we consider the $α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $α\geqβ\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $α>β\geq0$, the results for the case $α=β\geq0$ 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 $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization is sparse. For the case $α>β\geq0$, if the exact solution is sparse, we derive convergence rate $O(δ^{\frac{1}{2}})$ and $O(δ)$ 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 $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.