  In this paper, we consider the minimization of a Tikhonov functional with an
$\ell_1$ penalty for solving linear inverse problems with sparsity constraints.
One of the many approaches used to solve this problem uses the Nemskii operator
to transform the Tikhonov functional into one with an $\ell_2$ penalty term but
a nonlinear operator. The transformed problem can then be analyzed and
minimized using standard methods. However, by the nature of this transform, the
resulting functional is only once continuously differentiable, which prohibits
the use of second order methods. Hence, in this paper, we propose a different
transformation, which leads to a twice differentiable functional that can now
be minimized using efficient second order methods like Newton's method. We
provide a convergence analysis of our proposed scheme, as well as a number of
numerical results showing the usefulness of our proposed approach.
