  A new approximate computational framework is proposed for computing the
non-equilibrium charge density in the context of the non-equilibrium Green's
function (NEGF) method for quantum mechanical transport problems. The framework
consists of a new formulation, called the X-formulation, for single-energy
density calculation based on the solution of sparse linear systems, and a
projection-based nonlinear model order reduction (MOR) approach to address the
large number of energy points required for large applied biases. The advantages
of the new methods are confirmed by numerical experiments.
