  In this paper, we propose and analyze a two-point gradient method for solving
inverse problems in Banach spaces which is based on the Landweber iteration and
an extrapolation strategy. The method allows to use non-smooth penalty terms,
including the L^1 and the total variation-like penalty functionals, which are
significant in reconstructing special features of solutions such as sparsity
and piecewise constancy in practical applications. The design of the method
involves the choices of the step sizes and the combination parameters which are
carefully discussed. Numerical simulations are presented to illustrate the
effectiveness of the proposed method.
