  The von Neumann lattice refers to a discrete basis of Gaussians located on a
lattice in phase space. It provides an attractive approach for solving quantum
mechanical problems, allowing the pruning of tensor-product basis sets using
phase space considerations. In a series of recent articles Shimshovitz et al.
[Phys. Rev. Lett. 109 7 (2012)], Takemoto et al. [Journal of Chemical Physics
137 1 (2012)] Machnes et al. [Journal of Chemical Physics, accepted (2016)]),
we have introduced two key new elements into the method: a formalism for
converging the basis and for efficient pruning by use of the biorthogonal
basis. In this paper we review the key components of the theory and then
present new, efficient and parallelizable iterative algorithms for solving the
time-independent and time-dependent Schr\"odinger equations. The algorithms
dynamically determine the active reduced basis iteratively without resorting to
classical analogs. These algorithmic developments, combined with the previous
formal developments, allow quantum dynamics to be performed directly and
economically in phase space. We provide two illustrative examples: double-well
tunneling and double ionization of helium.
