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AIP conference proceedings

Optimality of Wang-Landau Sampling and Performance Limitations of Flat Histogram Methods

Simon Trebst

Published Jan 1, 2003• 0 citations
Publisher

Abstract

We determine the optimal scaling of local‐update flat‐histogram methods (such as multicanonical, broad histograms, tempering, or Wang‐Landau sampling) with system size by using a perfect flat‐histogram scheme based on the exact density of states of 2D Ising models. We find that the Wang‐Landau algorithm shows the same scaling as the perfect scheme and is thus optimal. However, even for the perfect scheme the scaling with the number of spins N is slower than the minimal N2 of an unbiased random walk in energy space for both, local and N‐fold way updates. While it still follows a power law N2Lz for the ferromagnetic and fully frustrated Ising model we find exponential scaling of the tunneling times in the case of the ±J Ising spin glass. The tunneling times of the ±J Ising spin glass are found to follow a fat‐tailed Fréchet extremal value distribution.

DOI
10.1063/1.1632165

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We determine the optimal scaling of local‐update flat‐histogram methods (such as multicanonical, broad histograms, tempering, or Wang‐Landau sampling) with system size by using a perfect flat‐histogram scheme based on the exact density of states of 2D Ising models. We find that the Wang‐Landau algorithm shows the same scaling as the perfect scheme and is thus optimal. However, even for the perfect scheme the scaling with the number of spins N is slower than the minimal N2 of an unbiased random walk in energy space for both, local and N‐fold way updates. While it still follows a power law N2Lz for the ferromagnetic and fully frustrated Ising model we find exponential scaling of the tunneling times in the case of the ±J Ising spin glass. The tunneling times of the ±J Ising spin glass are found to follow a fat‐tailed Fréchet extremal value distribution.
In this work, we present a comparative study of the accuracy provided by the Wang-Landau sampling and the Broad Histogram method to estimate de density of states of the two dimensional Ising ferromagnet. The microcanonical averages used to describe the thermodynamic behaviour and to use the Broad Histogram method were obtained using the single spin-flip Wang-Landau sampling, attempting to convergence issues and accuracy improvements. We compare the results provided by both techniques with the exact ones for thermodynamic properties and critical exponents. Our results, within the Wang-Landau sampling, reveal that the Broad Histogram approach provides a better description of the density of states for all cases analysed.
We present a Monte Carlo method that efficiently computes the density of states for spin models having any number of interaction per spin. By combining a random walk in the energy space with collective updates controlled by the microcanonical temperature, our method yields dynamic exponents close to their ideal random-walk values, reduced equilibrium times, and very low statistical error in the density of states. The method can host any density of states estimation scheme, including the Wang-Landau algorithm and the transition matrix method. Our approach proves remarkably powerful in the numerical study of models governed by long-range interactions, where it is shown to reduce the algorithm complexity to that of a short-range model with the same number of spins. We apply the method to the -state Potts chains with power-law decaying interactions in their first-order regime; we find that conventional local-update algorithms are outperformed already for sizes above a few hundred spins. By considering chains containing up to spins, which we simulated in fairly reasonable time, we obtain estimates of transition temperatures correct to five-figure accuracy. Finally, we propose several efficient schemes aimed at estimating the microcanonical temperature.
At second-order phase transitions the critical energy range covered by a canonical Monte Carlo simulation close to the critical temperature is often smaller than the energy range needed for reliable reweighting analyses of certain observables. Such an extended energy range can be covered by performing a Wang–Landau recursion for the spectral density followed by a multicanonical simulation with fixed weights. But in the conventional approach based on local update rules one loses the advantage due to non-local cluster algorithms which are well known to drastically reduce critical slowing down. We develop a cluster version of the Wang–Landau recursion together with a subsequent multibondic simulation and show for 2D and 3D Ising models that the efficiency of the conventional Wang–Landau multicanonical approach can be improved by power laws in the lattice size. In our simulations real gains in CPU time reach two orders of magnitude.
•Goal: determine the density of states (g[E]): the number of spin configurations producing an energy, E. • In an unbiased random walk in energy space, realized by randomly flipping spins, the histogram of the energy distribution (h[E]) converges to the density of states (g[E]), but too slowly to be practical. •When the random walk is biased inversely to g[E] so that h[E] becomes flat, g[E] is constructed to within an error ∝ ln[f ] if at each energy visited, g[E] is redefined as g[E]× f . If f is incrementally reduced to = 1, g[E] becomes exact. •Random walk acceptance bias: P [E1 → E2] = min g[E1] g[E2] , 1 
For a second-order phase transition the critical energy range of interest is larger than the energy range covered by a canonical Monte Carlo simulation at the critical temperature. Such an extended energy range can be covered by performing a Wang-Landau recursion for the spectral density followed by a multicanonical simulation with fixed weights. But in the conventional approach one loses the advantage due to cluster algorithms. A cluster version of the Wang-Landau recursion together with a subsequent multibondic simulation improves for 2D and 3D Ising models the efficiency of the conventional Wang-Landau or multicanonical approach by power laws in the lattice size. In our simulations real gains in CPU time reach 2 orders of magnitude.