An efficient and robust electrical resistivity tomographic inversion algorithm based on the Levenberg-Marquardt method is considered to obtain quantities like grain size, source discrimination and particle size distribution. The corresponding model in two-dimensions is based on the Maxwell equations and leads to a partial differential equation with mixed Dirichlet-Neumann boundary conditions. The forward problem is solved numerically with the finite-difference method. However, the inverse problem at hand is a classic nonlinear and ill-posed parameter estimation problem. Linearizing it and applying the Tikhonov regularization method yields an iterative scheme, the Levenberg-Marquardt method. Several systems of equations of large size have to be solved efficiently in each iteration step which is accomplished by the conjugate gradient method without setting up the corresponding matrix. Instead fast matrix-vector multiplications are performed directly. Therefore, the derivative and its adjoint for the parameter-to-solution map are needed. Numerical results demonstrate the performance of our method as well as the possibility to reconstruct some of the desired parameters.