A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions

Abstract

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. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663-83) demonstrating the effectiveness of this primal-dual algorithm.