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.