Iterative Total Variation Schemes for Nonlinear Inverse Problems

In this paper we discuss the construction, analysis, and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg-Marquardt, and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in presence of data noise. Moreover we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations. AMS Subject classification: 65K10, 65J22, 65N21, 68U10, 47J30