  In this paper we discuss a level set approach for the identification of an
unknown boundary in a computational domain. The problem takes the form of a
Bernoulli problem where only the Dirichlet datum is known on the boundary that
is to be identified, but additional information on the Neumann condition is
available on the known part of the boundary. The approach uses a classical
constrained optimization problem, where a cost functional is minimized with
respect to the unknown boundary, the position of which is defined implicitly by
a level set function. To solve the optimization problem a steepest descent
algorithm using shape derivatives is applied. In each iteration the cut finite
element method is used to obtain high accuracy approximations of the pde-model
constraint for a given level set configuration without re-meshing. We consider
three different shape derivatives. First the classical one, derived using the
continuous optimization problem (optimize then discretize). Then the functional
is first discretized using the CutFEM method and the shape derivative is
evaluated on the finite element functional (discretize then optimize). Finally
we consider a third approach, also using a discretized functional. In this case
we do not perturb the domain, but consider a so-called boundary value
correction method, where a small correction to the boundary position may be
included in the weak boundary condition. Using this correction the shape
derivative may be obtained by perturbing a distance parameter in the discrete
variational formulation. The theoretical discussion is illustrated with a
series of numerical examples showing that all three approaches produce similar
result on the proposed Bernoulli problem.
