Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals

Abstract

The determination of solutions of many inverse problems usually requires a set of measurements which leads to solving systems of ill-posed equations. In this paper, we propose the Landweber iteration of Kaczmarz type with general uniformly convex penalty functional. The method is formulated by using tools from convex analysis. The penalty term is allowed to be non-smooth to include the L1 and total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications. Under reasonable conditions, we establish the convergence of the method. Finally, we present numerical simulations on tomography problems and parameter identification in partial differential equations to indicate the performance.