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The determination of solutions of many inverse problems usually requires aset of measurements which leads to solving systems of ill-posed equations. Inthis paper we propose the Landweber iteration of Kaczmarz type with generaluniformly convex penalty functional. The method is formulated by using toolsfrom convex analysis. The penalty term is allowed to be non-smooth to includethe $L^1$ and total variation (TV) like penalty functionals, which aresignificant in reconstructing special features of solutions such as sparsityand piecewise constancy in practical applications. Under reasonable conditions,we establish the convergence of the method. Finally we present numericalsimulations on tomography problems and parameter identification in partialdifferential equations to indicate the performance.

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