A primal-dual semi-smooth Newton method for nonlinear L data fitting problems

This work is concerned with L 1 data fitting for nonlinear inverse problems. This formulation is advantageous if the data is corrupted by impulsive noise. However, the problem is not differentiable and lacks local uniqueness, which makes its efficient solution challenging. By considering a regularized primal-dual formulation of this problem, local uniqueness can be shown under a second order sufficient condition and a semi-smooth Newton method becomes applicable. In particular, its super-linear convergence is proved for the discretized optimality system. The convergence of the regularized formulation as the regularization and discretization parameters go to zero is shown. Additionally, approximation properties of the minimizers to nonlinear functionals with L 1 data fitting are analyzed and a strategy for selecting the regularization parameter based on a balancing principle is suggested. The efficiency is illustrated through the model problem of recovering the potential in an elliptic boundary value problem from distributed observational data, for which one-and two-dimensional numerical examples are presented.