αℓ_1-βℓ_2 sparsity regularization for nonlinear ill-posed problems
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In this paper, we consider the α·_ℓ_1-β·_ℓ_2 sparsity regularization with parameter α≥β≥0 for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where α>β≥0, the results for the case α=β≥0 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·_ℓ_1-β·_ℓ_2 regularization is sparse. For the case α>β≥0, if the exact solution is sparse, we derive convergence rate O(δ^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·_ℓ_1-β·_ℓ_2 regularization problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.
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