The authors consider non-smooth minimization problems involving a sum of a convex quadratic data misfit term and a convex non-smooth penalty term. They propose a fully explicit iterative algorithm to solve the optimization problem. Each step in the iteration uses four matrix-vector multiplications and a single thresholding. The averages of the first \(n\) iterates of the proposed generalized soft-thresholding algorithm are proven to have \(1/n\) rate on the functional. The proposed algorithm is applied to a stylized numerical example in seismic tomography in which a simple synthetic 2D input model defined on the sphere from given data is reconstructed.