  We provide analysis of the convergence properties and applicability
extensions of flat-histogram algorithms, with a particular focus on the
Wang-Landau algorithms (exemplified by converging stochastic approximation
Monte Carlo (SAMC)) and multicanonical (MUCA) algorithms. Our investigation
reveals that the optimal decay rate of the modification factor in SAMC
algorithms is influenced by the number of energy bins rather than the width of
the energy range. Despite the frequent naming of these algorithms based on the
histogram flatness, our findings indicate that flatness demonstrates a limited
correlation with estimation accuracy. We explore the implications of
integrating the importance sampling technique with flat-histogram algorithms,
demonstrating that this combination yields comparable or better accuracy in
density of states estimations, almost independent of specific algorithmic
parameters within certain bounds. Furthermore, our research extends the
possibilities of the flat-histogram and importance sampling combination for
investigating a range of underlying system parameters simultaneously within a
single simulation. These system parameters could both originate from the
potential, e.g., various relative contributions of different energy terms or
characteristic interaction range, and characterize the accessible
configurations, e.g., through the size of the simulation box.
