Multi-parameter Tikhonov regularization with the ℓ0 sparsity constraint

Using a sparsity promoting penalty term in the Tikhonov regularization scheme has been shown to be efficient in reconstructing solutions which own a sparse structure. These penalty terms usually take the form of ℓp (quasi-)norm with p ∈ (0, 2). At the same time, some examples have been shown where there is no regularizing property for the extreme penalty term with p = 0. Here, we examine the multi-parameter Tikhonov regularization with both ℓ0 and ℓ2 penalty terms for nonlinear ill-posed problems. We will show, in the following, that the proposed multi-parameter Tikhonov functional has a minimizer as well as the convergence result when the noise level tends to 0. Error estimates for the discrepancy principle are verified under appropriate source conditions and nonlinearity assumptions. Finally, two numerical examples shed light on the appropriateness of the proposed method. (paper)