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

Abstract

AbstractMeasuring the error by anℓ1{\ell^{1}}-norm, we analyze under sparsity assumptions anℓ0{\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 operatorAmapping betweenℓ2{\ell^{2}}and a Banach spaceYare 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 theℓ0{\ell^{0}}-term and the complementing convex penalty, the important special case of theℓ2{\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.