Preprint D104/2013
On piecewise constant level-set (PCLS) methods for the identication of discontinuous parameters in ill-posed problems

Adriano De Cezaro | Antonio Leitao | Xue-Cheng Tai

**Keywords: **
Inverse Problems | Ill-posed equations | Level set methods

We investigate level-set type methods for solving ill-posed problems with discontinuous
(piecewise constant) coecients. The goal is to identify the level sets as well as the level values
of an unknown parameter function on a model described by a nonlinear ill-posed operator
equation. The PCLS approach is used here to parametrize the solution of a given operator
equation in terms of a L2 level-set function, i.e., the level-set function itself is assumed to be
a piecewise constant function.
Two distinct methods are proposed for computing stable solutions of the resulting ill-posed
problem: The rst one is based on Tikhonov regularization, while the second method is based
on the augmented Lagrangian approach with total variation penalization.
Classical regularization results [16] are derived for the Tikhonov method. On the other
hand, for the augmented Lagrangian method, we succeed in proving existence of (generalized)
Lagrangian multipliers in the sense of [35].
Numerical experiments are performed for a 2D inverse potential problem [22], demonstrating
the capabilities of both methods for solving this ill-posed problem in a stable way (complicated
inclusions are recovered without any a priori geometrical information on the unknown
parameter).