Improving the efficiency of extended ensemble simulations: The accelerated weight histogram method

We propose a method for efficient simulations in extended ensembles, useful, e.g., for the study of problems with complex energy landscapes and for free energy calculations. The main difficulty in such simulations is the estimation of the a priori unknown weight parameters needed to produce flat histograms. The method combines several complementary techniques, namely, a Gibbs sampler for the parameter moves, a reweighting procedure to optimize data use, and a Bayesian update allowing for systematic refinement of the free energy estimate. In a certain limit the scheme reduces to the $1/t$ algorithm of B. E. Belardinelli and V. D. Pereyra [Phys. Rev. E 75, 046701 (2007)]. The performance of the method is studied on the two-dimensional Ising model, where comparison with the exact free energy is possible, and on an Ising spin glass.

Figure 1

(a) Mean absolute deviation of estimated and exact free energy differences $\delta F$ of the $64\times 64$ Ising model as a function of number of samples, $N$. From top to bottom: results from Wang-Landau iterations, the $1/t$ method, and the AWH method. Also included is the average behavior of the AWH method and a curve showing a $1/\sqrt{N}$ dependence. Inset: Difference between estimated and exact free energy. The error bars (the shaded area) represent one standard deviation. (b) As in (a), but for an $8\times 8\times 8$ Ising spin glass.