Applying the conventional Output Least Squares method to the nonlinear inverse problem of Electrical Impedance Tomography involves the inversion of a sequence of dense nonstructured matrix problems. Given the dimensionality of currently studied 3D problems, such an approach becomes computationally prohibitive. Starting from a modified problem formulation which retains the underlying sparsity structure, we derived a fast iterative solution method based on imbedding the original displacement rank concept in a conjugate gradient scheme.