  In Wang-Landau type algorithms, Monte-Carlo updates are performed with
respect to the density of states, which is iteratively refined during
simulations. The partition function and thermodynamic observables are then
obtained by standard integration. In this work, our recently introduced method
in this class (the LLR approach) is analysed and further developed. Our
approach is a histogram free method particularly suited for systems with
continuous degrees of freedom giving rise to a continuum density of states, as
it is commonly found in Lattice Gauge Theories and in some Statistical
Mechanics systems. We show that the method possesses an exponential error
suppression that allows us to estimate the density of states over several
orders of magnitude with nearly-constant {\it relative} precision. We explain
how ergodicity issues can be avoided and how expectation values of arbitrary
observables can be obtained within this framework. We then demonstrate the
method using Compact U(1) Lattice Gauge Theory. A thorough study of the
algorithm parameter dependence of the results is performed and compared with
the analytically expected behaviour. We obtain high precision values for the
critical coupling for the phase transition and for the peak value of the
specific heat for lattice sizes ranging from $8^4$ to $20^4$. Our results
perfectly agree with the reference values reported in the literature, which
covers lattice sizes up to $18^4$. Robust results for the $20^4$ volume are
obtained for the first time. This latter investigation, which, due to strong
metastabilities developed at the pseudo-critical coupling, so far has been out
of reach even on supercomputers with importance sampling approaches, has been
performed to high accuracy with modest computational resources. Other
situations where the method is expected to be superior to importance sampling
techniques are pointed out.
