Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

Abstract

Abstract Measuring the error by an {\ell^{1}} -norm, we analyze under sparsity assumptions an {\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 equations {Ax=y} with an injective and bounded linear operator A mapping between {\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 the {\ell^{0}} -term and the complementing convex penalty, the important special case of the {\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.