  Detecting inhomogeneities in the electrical conductivity is a special case of
the inverse problem in electrical impedance tomography, that leads to fast
direct reconstruction methods. One such method can, under reasonable
assumptions, exactly characterize the inhomogeneities based on monotonicity
properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr\'echet
derivative (linear). We give a comparison of the non-linear and linear approach
in the presence of measurement noise, and show numerically that the two methods
give essentially the same reconstruction in the unit disk domain. For a fair
comparison, exact matrix characterizations are used when probing the
monotonicity relations to avoid errors from numerical solution to PDEs and
numerical integration. Using a special factorization of the
Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear
method in the unit disk geometry.
