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Measuring the error by anℓ  1{\ell^{1}}   -norm, we analyze under sparsity assumptions anℓ  0{\ell^{0}}   -regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equationsA  ⁢  x   =  y{Ax=y}    with an injective and bounded linear operator  A  mapping betweenℓ  2{\ell^{2}}    and a Banach space  Y  are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between theℓ  0{\ell^{0}}   -term and the complementing convex penalty, the important special case of theℓ  2{\ell^{2}}   -norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

Tikhonov regularization with ℓ 0 -term complementing a~convex penalty: ℓ 1 -convergence under sparsity constraints