  Monte Carlo simulations of the 4-dimensional compact U(1) lattice gauge
theory in the neighborhood of the transition point are made difficult by the
suppression of tunneling between the phases, which becomes very strong as soon
as the volume of the lattice grows to any appreciable size. This problem can be
avoided by making the monopole coupling a dynamical variable. In this manner
one can circumvent the tunneling barrier by effectively riding on top of the
peaks in the energy distribution which meet for sufficiently large monopole
coupling. Here we present an efficient method for determining the parameters
needed for this procedure, which can thus be implemented at low computational
cost also on large lattices. This is particularly important for a reliable
determination of the transition point. We demonstrate the working of our method
on a 16^4 lattice. We obtain an equidistribution of configurations across the
phase transition even for such a relatively large lattice size.
