  The electrical impedance tomography (EIT) problem of estimating the unknown
conductivity distribution inside a domain from boundary current or voltage
measurements requires the solution of a nonlinear inverse problem. Sparsity
promoting hierarchical Bayesian models have been shown to be very effective in
the recovery of almost piecewise constant solutions in linear inverse problems.
We demonstrate that by exploiting linear algebraic considerations it is
possible to organize the calculation for the Bayesian solution of the nonlinear
EIT inverse problem via finite element methods with sparsity promoting priors
in a computationally efficient manner. The proposed approach uses the Iterative
Alternating Sequential (IAS) algorithm for the solution of the linearized
problems. Within the IAS algorithm, a substantial reduction in computational
complexity is attained by exploiting the low dimensionality of the data space
and an adjoint formulation of the Tikhonov regularized solution that
constitutes part of the iterative updating scheme. Numerical tests illustrate
the computational efficiency of the proposed algorithm. The paper sheds light
also on the convexity properties of the objective function of the maximum a
posteriori (MAP) estimation problem.
