  We investigate the effects of phase noise and particle loss on the dynamics
of a Bose-Einstein condensate in an optical lattice. Starting from the
many-body master equation, we discuss the applicability of generalized
mean-field approximations in the presence of dissipation as well as methods to
simulate quantum effects beyond mean-field by including higher-order
correlation functions. It is shown that localized particle dissipation leads to
surprising dynamics, as it can suppress decay and restore the coherence of a
Bose-Einstein condensate. These effects can be applied to engineer coherent
structures such as stable discrete breathers and dark solitons.
