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Register a new userError estimation in the histogram Monte Carlo method
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M. E. J. Newman
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We examine the sources of error in the histogram reweighting method for Monte Carlo data analysis. We demonstrate that, in addition to the standard statistical error which has been studied elsewhere, there are two other sources of error, one arising through correlations in the reweighted samples, and one arising from the finite range of energies sampled by a simulation of finite length. We demonstrate that while the former correction is usually negligible by comparison with statistical fluctuations, the latter may not be, and give criteria for judging the range of validity of histogram extrapolations based on the size of this latter correction.
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The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Greens function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version Flat Histogram Diagrammatic Monte Carlo method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Greens function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem
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 tau scales with the number of spins N=L^d as tau 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 tau propto N^2 up to logarithmic corrections for the d=1 and d=2 Ising model.
We discuss the detailed balance condition for hybrid Monte Carlo method
While the Quasi-Monte Carlo method of numerical integration achieves smaller integration error than standard Monte Carlo, its use in particle physics phenomenology has been hindered by the abscence of a reliable way to estimate that error. The standard Monte Carlo error estimator relies on the assumption that the points are generated independently of each other and, therefore, fails to account for the error improvement advertised by the Quasi-Monte Carlo method. We advocate the construction of an estimator of stochastic nature, based on the ensemble of pointsets with a particular discrepancy value. We investigate the consequences of this choice and give some first empirical results on the suggested estimators.
We present a method for Monte Carlo sampling on systems with discrete variables (focusing in the Ising case), introducing a prior on the candidate moves in a Metropolis-Hastings scheme which can significantly reduce the rejection rate, called the reduced-rejection-rate (RRR) method. The method employs same probability distribution for the choice of the moves as rejection-free schemes such as the method proposed by Bortz, Kalos and Lebowitz (BKL) [Bortz et al. J.Comput.Phys. 1975]; however, it uses it as a prior in an otherwise standard Metropolis scheme: it is thus not fully rejection-free, but in a wide range of scenarios it is nearly so. This allows to extend the method to cases for which rejection-free schemes become inefficient, in particular when the graph connectivity is not sparse, but the energy can nevertheless be expressed as a sum of two components, one of which is computed on a sparse graph and dominates the measure. As examples of such instances, we demonstrate that the method yields excellent results when performing Monte Carlo simulations of quantum spin models in presence of a transverse field in the Suzuki-Trotter formalism, and when exploring the so-called robust ensemble which was recently introduced in Baldassi et al. [PNAS 2016]. Our code for the Ising case is publicly available [https://github.com/carlobaldassi/RRRMC.jl], and extensible to user-defined models: it provides efficient implementations of standard Metropolis, the RRR method, the BKL method (extended to the case of continuous energy specra), and the waiting time method [Dall and Sibani Comput.Phys.Commun. 2001].
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