  Regularization techniques for the numerical solution of inverse scattering
problems in two space dimensions are discussed. Assuming that the boundary of a
scatterer is its most prominent feature, we exploit as model the class of
cartoon-like functions. Since functions in this class are asymptotically
optimally sparsely approximated by shearlet frames, we consider shearlets as a
means for regularization in a Tikhonov method. We analyze two approaches,
namely solvers for the nonlinear problem and for the linearized problem
obtained by the Born approximation technique. As example for the first class we
study the acoustic inverse scattering problem, and for the second class, the
inverse scattering problem of the Schr\"{o}dinger equation. In both cases, we
derive analytical results for our approaches. Whereas our emphasis for the
linearized problem is more on the theoretical side due to the standardness of
associated solvers, we provide numerical examples for the nonlinear problem
that highlight the effectiveness of our algorithmic approach.
