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

TL;DR

This paper introduces a Kaczmarz-type Landweber iteration incorporating general convex, possibly non-smooth penalties like L1 and TV, with proven convergence and demonstrated effectiveness in tomography and PDE parameter identification.

Contribution

It develops a novel iterative method for inverse problems that handles non-smooth convex penalties, broadening applicability to sparsity and piecewise constant solutions.

Findings

Convergence of the proposed method is established under reasonable conditions.

Numerical simulations show effective reconstruction in tomography.

Method successfully handles non-smooth penalties like L1 and TV.

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 $L^1$ and total variation (TV) 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.