Accuracy and convergence of the Wang-Landau sampling algorithm

We present estimations of the accuracy and convergence of the Wang-Landau algorithm. Both accuracy and the related length of the Monte Carlo run depend on the modification parameter $f$ and the density of states. The analytical solution obtained for the two-level system was checked numerically on the two-dimensional Ising model. Although the two-level system is a very simple one, it appears that the proposed solution describes the generic features of the Wang-Landau algorithm. The estimations should prove useful in Monte Carlo calculations of protein folding, first-order transitions, and other systems with a rough energy landscape.

Figure 1

Dashed lines with scatters: the dispersion of the error of the WL algorithm for the two-level system (${\Delta }_0=1$, $q∕p=e$), calculated by averaging over 10 000 stochastic trajectories, vs the number of MC steps. Solid lines: the related analytical calculations of accuracy and convergence using Eqs. (19, 21). The related values of $⟨\Delta ⟩$ averaged over 10 000 trajectories are $⟨\Delta ⟩\cong 0.002$ for $f=1+2^{-10}$ and $⟨\Delta ⟩\cong 0.001$ for $f=1+2^{-12}$.

Figure 2

The ratio ${\sigma }_m\left(\Delta \right)∕\sqrt{\frac{g_m}{g_k}{\epsilon }_i}$, $g_m⩾g_k$, for the 2D Ising model calculated with (a) $\mid m-k\mid =1$ and (b) $\mid m-k\mid =2$. ${\sigma }_m\left(\Delta \right)$ was averaged over 1600 WL trajectories for the two different sizes of the system: (◻) $-10\times 10$ and (엯) $-16\times 16$.

Figure 3

Accuracy and convergence of WL algorithm: the 2D Ising model of finite size $16\times 16$. Solid lines: the dispersion of the error, calculated by averaging over 1600 stochastic trajectories, vs the histogram number: (a) $E=-2$ and (b) $E=-1$ ($E$ denotes energy per spin). Dashed lines: the related analytical calculations of convergence using Eq. (25): (a) ${\gamma }_m=256$, $a=2$ and (b) ${\gamma }_m\cong 25$, $a=2$.

Figure 4

The effect of the system size on the statistical error of the WL algorithm $(f=1+2^{-14})$. $E$ denotes the energy per spin in the 2D Ising model.