  The Logarithmic Linear Relaxation (LLR) algorithm is an efficient method for
computing densities of states for systems with a continuous spectrum. A key
feature of this method is exponential error reduction, which allows us to
evaluate the density of states of a system over hundreds of thousands of orders
of magnitude with a fixed level of relative accuracy. As a consequence of
exponential error reduction, the LLR method provides a robust alternative to
traditional Monte Carlo calculations in cases in which states suppressed by the
Boltzmann weight play nevertheless a relevant role, e.g., as transition regions
between dominant configuration sets. After reviewing the algorithm, we will
show an application in U(1) Lattice Gauge Theory that has enabled us to obtain
the most accurate estimate of the critical coupling with modest computational
resources, defeating exponential tunneling times between metastable vacua. As a
further showcase, we will then present an application of the LLR method to the
decorrelation of the topological charge in SU(3) Lattice Gauge Theory near the
continuum limit. Finally, we will review in general applications of the LLR
algorithm to systems affected by a strong sign problem and discuss the case of
the Bose gas at finite chemical potential.
