  We numerically study the superfluid to Mott insulator transition for bosonic
atoms in a one dimensional lattice by exploiting a recently developed
simulation method for strongly correlated systems. We demonstrate this methods
accuracy and applicability to Bose-Hubbard model calculations by comparison
with exact results for small systems. By utilizing the efficient scaling of
this algorithm we then concentrate on systems of comparable size to those
studied in experiments and in the presence of a magnetic trap. We investigate
spatial correlations and fluctuations of the ground state as well as the nature
and speed at which the superfluid component is built up when dynamically
melting a Mott insulating state by ramping down the lattice potential. This is
performed for slow ramping, where we find that the superfluid builds up on a
time scale consistent with single-atom hopping and for rapid ramping where the
buildup is much faster than can be explained by this simple mechanism. Our
calculations are in remarkable agreement with the experimental results obtained
by Greiner et al. [Nature (London) 415, 39 (2002)].
