An algorithm to calculate the density of states, based on the well-known Wang-Landau method, is introduced. Independent random walks are performed in different restricted ranges of energy, and the resultant density of states is modified by a function of time, F(t) proportional to t-1, for large time. As a consequence, the calculated density of state, gm(E,t) , approaches asymptotically the exact value g(ex)(E) as proportional to t-1/2, avoiding the saturation of the error. It is also shown that the growth of the interface of the energy histogram belongs to the random deposition universality class.
In this work we present a theoretical analysis of the convergence of the Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced years ago to calculate the density of states in statistical models. We study the dynamical behavior of the error in the calculation of the density of states. We conclude that the source of the saturation of the error is due to the decreasing variations of the refinement parameter. To overcome this limitation, we present an analytical treatment in which the refinement parameter is scaled down as a power law instead of exponentially. An extension of the analysis to the N-fold way variation of the method is also discussed.
In this Brief Report, the convergence of the 1t and Wang-Landau algorithms in the calculation of multidimensional numerical integrals is analyzed. Both simulation methods are applied to a wide variety of integrals without restrictions in one, two, and higher dimensions. The efficiency and accuracy of both algorithms are determined by the dynamical behavior of the errors between the exact and the calculated values of the integral. It is observed that the time dependence of the error calculated with the 1t algorithm varies as N;{-12} [with N the number of Monte Carlo (MC) trials], in quantitative agreement with the simple sampling Monte Carlo method. In contrast, the error calculated with the Wang-Landau algorithm saturates in time, evidencing the nonconvergence of this method. The sources of error for both methods are also determined.
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