**Keywords:**

We propose and analyse a regularization method for parameter identification

problems modeled by ill-posed nonlinear operator equations, where the

parameter to be identified is a piecewise constant function taking known

values.

Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise

constant level set approach is used to represent the unknown parameter, and

a corresponding Tikhonov functional is defined on an appropriated space of

level set functions. Additionally, a suitable constraint is enforced, resulting that

minimizers of our Tikhonov functional belong to the set of piecewise constant

level set functions. In other words, the original parameter identification

problem is rewritten in the form of a constrained optimization problem, which

is solved using an augmented Lagrangian method.

We prove the existence of zero duality gaps and the existence of generalized

Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al’s

work (2013 Inverse Problems 29 015003), proving convergence and stability

of the proposed parameter identification method.

A primal-dual algorithm is proposed to compute approximate solutions

of the original inverse problem, and its convergence is proved. Numerical

examples are presented: this algorithm is applied to a 2D diffuse optical

tomography problem.