Overcoming the critical slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles

Abstract

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time τscales with the number of spins N=L^d as τ\propto N^2L^z. The critical exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z>0 for finite d can be overcome by two complementary approaches - cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that τ\propto N^2 up to logarithmic corrections for the d=1 and d=2 Ising model.

References in corpus (11)

• An efficient, multiple range random walk algorithm to calculate the density of states
• Multicanonical Ensemble: A New Approach to Simulate First-order Phase Transitions
• Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram
• Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations
• Performance Limitations of Flat Histogram Methods and Optimality of Wang-Landau Sampling
• Wang-Landau sampling for quantum systems: algorithms to overcome tunneling problems and calculate the free energy
• Multibondic Cluster Algorithm for Monte Carlo Simulations of First-Order Phase Transitions
• Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0
• Combination of improved multibondic method and the Wang-Landau method
• Application of Multicanonical Multigrid Monte Carlo Method to the Two-Dimensional ${\varphi }^4$-Model: Autocorrelations and Interface Tension
• Multicanonical Multigrid Monte Carlo

Cited by in corpus (11)

• A generic, hierarchical framework for massively parallel Wang-Landau sampling
• Microcanonical Approach to the Simulation of First-Order Phase Transitions
• A new paradigm for petascale Monte Carlo simulation: Replica exchange Wang-Landau sampling
• Fast Flat-Histogram Method for Generalized Spin Models
• Optimized multicanonical simulations: a new proposal based on classical fluctuation theory
• Optimized Broad-Histogram Ensembles for the Simulation of Quantum Systems
• Exploring new frontiers in statistical physics with a new, parallel Wang-Landau framework
• Optimized broad-histogram simulations for strong first-order phase transitions: Droplet transitions in the large-Q Potts model
• Multiphase structure of finite-temperature phase diagram of the Blume-Capel model. Wang-Landau sampling method
• The Wang-Landau Algorithm as Stochastic Optimization and Its Acceleration
• On the nature of the phase transitions in two-dimensional type II superconductors