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We consider the ill-posed operator equation $Ax=y$ with an injective andbounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$,possessing the unique solution \linebreak $x^\dag=\{x^\dag_k\}_{k=1}^\infty$.For the cases that sparsity $x^\dag \in \ell^0$ is expected but often slightlyviolated in practice, we investigate in comparison with the$\ell^1$-regularization the elastic-net regularization, where the penalty is aweighted superposition of the $\ell^1$-norm and the $\ell^2$-norm square, underthe assumption that $x^\dag \in \ell^1$. There occur two positive parameters inthis approach, the weight parameter $\eta$ and the regularization parameter asthe multiplier of the whole penalty in the Tikhonov functional, whereas onlyone regularization parameter arises in $\ell^1$-regularization. Based on thevariational inequality approach for the description of the solution smoothnesswith respect to the forward operator $A$ and exploiting the method ofapproximate source conditions, we present some results to estimate the rate ofconvergence for the elastic-net regularization. The occurring rate functioncontains the rate of the decay $x^\dag_k \to 0$ for $k \to \infty$ and theclassical smoothness properties of $x^\dag$ as an element in $\ell^2$.

Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions