dorsal/arxiv
View SchemaEntropic effects in large-scale Monte Carlo simulations
| Authors | Cristian Predescu |
|---|---|
| Categories | |
| ArXiv ID | physics/0703225 |
| URL | https://arxiv.org/abs/physics/0703225 |
| DOI | 10.1103/PhysRevE.76.016704 |
Abstract
The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Renyi divergence of order 1/2. We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for non-vanishing acceptance probabilities in the limit of large dimensions. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.
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"abstract": "The efficiency of Monte Carlo samplers is dictated not only by energetic\neffects, such as large barriers, but also by entropic effects that are due to\nthe sheer volume that is sampled. The latter effects appear in the form of an\nentropic mismatch or divergence between the direct and reverse trial moves. We\nprovide lower and upper bounds for the average acceptance probability in terms\nof the Renyi divergence of order 1/2. We show that the asymptotic finitude of\nthe entropic divergence is the necessary and sufficient condition for\nnon-vanishing acceptance probabilities in the limit of large dimensions.\nFurthermore, we demonstrate that the upper bound is reasonably tight by showing\nthat the exponent is asymptotically exact for systems made up of a large number\nof independent and identically distributed subsystems. For the last statement,\nwe provide an alternative proof that relies on the reformulation of the\nacceptance probability as a large deviation problem. The reformulation also\nleads to a class of low-variance estimators for strongly asymmetric\ndistributions. We show that the entropy divergence causes a decay in the\naverage displacements with the number of dimensions n that are simultaneously\nupdated. For systems that have a well-defined thermodynamic limit, the decay is\ndemonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart\nMonte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is\nvirtually as efficient as the Markov chain implementation of the Gibbs sampler,\nwhich is normally utilized for Lennard-Jones clusters. An application of the\nentropic inequalities to the parallel tempering method demonstrates that the\nnumber of replicas increases as the square root of the heat capacity of the\nsystem.",
"arxiv_id": "physics/0703225",
"authors": [
"Cristian Predescu"
],
"categories": [
"physics.comp-ph",
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"doi": "10.1103/PhysRevE.76.016704",
"title": "Entropic effects in large-scale Monte Carlo simulations",
"url": "https://arxiv.org/abs/physics/0703225"
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