Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals (Q2866184)

This article studies the Landweber iteration of Kaczmarz type for nonlinear inverse problems. The Landweber method is popular for nonlinear inverse problems due to its straightforward implementation and robustness with respect to noise, but it can converge slowly and the classical variant may suffer from oversmoothing. In the interesting work of \textit{F. Schöpfer, A. K. Louis} and \textit{T. Schuster} [ibid. 22, No. 1, 311--329 (2006; Zbl 1088.65052)], the method was extended to the nonsmooth context, i.e., uniformly smooth and uniformly convex spaces. In this work, it is further extended to a nonsmooth but uniformly convex penalty, which encompasses useful \(L^1\) and total variation like penalty functions. The convergence of the method is established, and numerically verified on three examples, including photoacoustic tomography, parameter identification and Schlieren imaging.