Inverse doping profile problems are linked to inverse conductivity problems under the assumptions of zero space charge and low injection. Unipolar inverse conductivity problems are analyzed theoretically via three uniqueness proofs. Optimized numerical methods are developed for solving the unipolar direct conductivity problem with a piecewise constant conductivity coefficient. Finally, the unipolar inverse conductivity problem is solved for inclusions defined by as many as 9 entirely unknown parameters, or by as many as 120 parameters when an initial guess for each parameter is known with less than 10% error. Our free boundary identification algorithm produces a sequence of improved approximations in a way that provides both regularization and accelerated convergence towards the solution.