In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which
allows the definition of a Tikhonov functional on a space of
level-set functions. We provide convergence analysis for the
Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization
method is presented.
Newton-type methods are used for the solution of the optimality systems,
which can be interpreted as stabilized versions of algorithms in a previous
work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited
for real life applications.