Nonconvergence of the Wang-Landau algorithms with multiple random walkers

Abstract

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/ √ m. However, if the number of walkers increases above a certain critical value m > mx, the error reaches a constant value (i.e. it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value mx, since it does not reduces the error in the calculation. Therefore, the number of walkers does not guarantee convergence.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

“…The parallelization becomes more cumbersome for Wang–Landau simulations, but introducing a clever energy-window distribution of walkers ensures good performance [48]. This is similarly possible for the 1/t algorithm [49].…”

Implications of Occupational Disorder on Ion Mobility in Li4Ti5O12 Battery Materials

“…We explore the configuration space of Li 4 Ti 5 O 12 through canonical Metropolis Monte Carlo (MC) sampling and estimate the relative configurational density of states (DOS) in the relevant total energy range via Wang–Landau sampling . Both samplings are based on a density-functional theory (DFT) validated core/shell force field, , as implemented in LAMMPS, that we further describe in the Supporting Information (SI).…”

Large-deviations of the SIR model around the epidemic threshold

“…However the saturation of the final error becomes a problem for the original WL algorithm ( [62], see also Refs. [63][64][65]). The algorithm introduced by Belardinelli and Pereyra [63] is used to circumvent the problem, since it was shown [65], that error saturation does not become a problem for this alternative algorithm.…”

“…[63][64][65]). The algorithm introduced by Belardinelli and Pereyra [63] is used to circumvent the problem, since it was shown [65], that error saturation does not become a problem for this alternative algorithm. The main difference between this algorithm and the original WL is in how the factor f is updated during the simulation, for details see the citations.…”