  We study the tunneling decay of a Bose-Einstein condensate out of tilted
optical lattices within the mean-field approximation. We introduce a novel
method to calculate also excited resonance eigenstates of the Gross-Pitaevskii
equation, based on a grid relaxation procedure with complex absorbing
potentials. This algorithm works efficiently in a wide range of parameters
where established methods fail. It allows us to study the effects of the
nonlinearity in detail in the regime of resonant tunneling, where the decay
rate is enhanced by resonant coupling to excited unstable states.
