Regularization of inverse problems by two-point gradient methods with convex constraints
For researchers in inverse problems, this method offers a new approach to handle non-smooth penalties, but the results are incremental as it builds on existing Landweber iteration and extrapolation.
The paper proposes a two-point gradient method for inverse problems in Banach spaces that incorporates non-smooth penalties like L1 and total variation, demonstrating effectiveness through numerical simulations.
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.