  In this work, we develop an efficient high order discontinuous Galerkin (DG)
method for solving the Electrical Impedance Tomography (EIT). EIT is a highly
nonlinear ill-posed inverse problem where the interior conductivity of an
object is recovered from the surface measurements of voltage and current flux.
We first propose a new optimization problem based on the recovery of the
conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between
the predicted current and the measured current on the boundary. And we further
prove the existence of the minimizer. Numerically the optimization problem is
solved by a third order DG method with quadratic polynomials. Numerical results
for several two-dimensional problems with both single and multiple inclusions
are demonstrated to show the high {accuracy and efficiency} of the proposed
high order DG method. Analysis and computation for discontinuous conductivities
are also studied in this work.
