Level-set approaches of L2-type for recovering shape and contrast in ill-posed problems

Abstract

We investigate level-set-type approaches for solving ill-posed inverse problems, under the assumption that the solution is a piecewise constant function. Our goal is to identify the level sets as well as the level values of the unknown parameter function. Two distinct level-set frameworks are proposed for solving the inverse problem. Among both of them, the level-set function is assumed to be in L2. Corresponding Tikhonov regularization approaches are derived and analysed. Existence of minimizers for the Tikhonov functionals is proven. Moreover, convergence and stability results of the variational approaches are established, characterizing the Tikhonov approaches as regularization methods. © 2012 Copyright Taylor and Francis Group, LLC.