An iteration regularization method with general convex penalty for nonlinear inverse problems in Banach spaces

Abstract

This paper investigates the construction, analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in Banach spaces. The method is formulated by combining the homotopy perturbation technique. The penalty term is allowed to be nonsmooth to include L1-norm or total variation-like penalty functionals, especially suitable for detecting the special features of the sought solutions such as sparsity or piecewise constant. By using tools from convex analysis in Banach spaces, we establish the detailed convergence and stability results. Numerical simulations for one-dimensional and two-dimensional parameter identification problems are performed to validate that our approach is competitive in terms of reducing the overall computational time in comparison with the existing Landweber iteration with general convex penalty. Moreover, our approach is used to the non-Gaussian noisy data to illustrate the iterative regularization in Banach spaces.