A proximal iteratively regularized Gauss-Newton method for nonlinear inverse problems

In this paper we discuss the construction, convergence analysis, and implementation of a proximal iteratively regularized Gauss-Newton method for the solution of nonlinear inverse problems with a specific regularization that linearly combines the L 2 {L^{2}} -norm and L 1 {L^{1}} -norm penalties. This regularization combines two very powerful features: the advantages of L 1 {L^{1}} -norm based penalty which impose less smoothing on the reconstruction parameter, and the general L 2 {L^{2}} -norm stabilizing term which can lead to smaller errors in some cases. However, non-linearity and non-smoothness of the problem make it challenging to find an efficient numerical solution. By using the proximal mapping, we derive a generalization of the iteratively regularized Gauss-Newton algorithm to handle such nonsmooth objective functions. Analysis on local convergence is carried out in the presence of observation noise. Parameter identification in numerical simulations of partial differential equations demonstrates the efficiency of the proposed method.