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
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  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 . Numerical experiments are performed for a 2D inverse potential problem , 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).