Preprint D98/2012
On piecewise constant level-set (PCLS) methods for the identication of discontinuous parameters in ill-posed problems

Adriano De Cezaro | Antonio Leitao

**Keywords: **
Inverse Problems; Discontinuous Parameters; Levelset 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 (according to this approach, the levelset
function itself is 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 method 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 the inverse potential problem on a 2D-domain,
testing the performance of both methods. Our tests demonstrate that the proposed reconstruction
methods are capable of solving the ill-posed problem in a stable way, being
able to recover complicated inclusions without any a priori geometrical information on the
unknown parameter.