On the regularization of an augmented-Lagrangian approach for the identification of discontinuous parameters in inverse problems

AGNELLI J.P.; LEITAO A.; DE CESARO A.; FERNANDEZ FERREYRA D.

Buenos Aires

Congreso; Primer Encuentro Conjunto de la Real Sociedad Matemática Española y la Unión Matemática Argentina; 2017

We investigate a level-set type method for solving inverse problems withdiscontinuous coefficients. The goal is to identify the level sets of an unknown parameter function on a model described by a nonlinear ill-posed operator equation.Imposing a suitable constraint, the level-set function itself is forced to be a piecewise constant function and hence the original problem is recast as an ill-posed operator equation whose solution satisfies the required restriction.On the other hand, to cope with the ill-posedness we consider a Tikhonov-type functional. In this way, the inverse problem can be stated and treated as an optimization problem with constraints.  To solve the constrained optimization problem we proposed an augmented Lagrangian approach. Following the ideas presented in~\cite{CLT13} it is possible to show the existence of a zero duality gap and the existence of generalized Lagrangian multipliers. Moreover, here we complete the analysis initiated in~\cite{CLT13}proving that method is convergent and stable with respect to the noise level in the data. In other words, we show that the suggested approach is a regularization method.Additionally, we present a convergent primal-dual algorithm to compute approximated solutions of the corresponding inverse problem.We complete this contribution with some numerical examples applied to the diffuse optical tomography problem that show the feasibility of the proposed approach.