On multiple level-set regularization methods for inverse problems

We analyze a multiple level-set method for solving inverse problems with piecewise constant solutions. This method corresponds to an iterated Tikhonov method for a particular Tikhonov functional G(alpha) based on TV-H-1 penalization. We define generalized minimizers for our Tikhonov functional and establish an existence result. Moreover, we prove convergence and stability results of the proposed Tikhonov method. A multiple level-set algorithm is derived from the first-order optimality conditions for the Tikhonov functional G(alpha), similarly as the iterated Tikhonov method. The proposed multiple level-set method is tested on an inverse potential problem. Numerical experiments show that the method is able to recover multiple objects as well as multiple contrast levels.