  We present a study of the numerical solution of the two dimensional
electrical impedance tomography problem, with noisy measurements of the
Dirichlet to Neumann map. The inversion uses parametrizations of the
conductivity on optimal grids. The grids are optimal in the sense that finite
volume discretizations on them give spectrally accurate approximations of the
Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of
special resistor networks, that are uniquely recoverable from the measurements.
Inversion on optimal grids has been proposed and analyzed recently, but the
study of noise effects on the inversion has not been carried out. In this paper
we present a numerical study of both the linearized and the nonlinear inverse
problem. We take three different parametrizations of the unknown conductivity,
with the same number of degrees of freedom. We obtain that the parametrization
induced by the inversion on optimal grids is the most efficient of the three,
because it gives the smallest standard deviation of the maximum a posteriori
estimates of the conductivity, uniformly in the domain. For the nonlinear
problem we compute the mean and variance of the maximum a posteriori estimates
of the conductivity, on optimal grids. For small noise, we obtain that the
estimates are unbiased and their variance is very close to the optimal one,
given by the Cramer-Rao bound. For larger noise we use regularization and
quantify the trade-off between reducing the variance and introducing bias in
the solution. Both the full and partial measurement setups are considered.
