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Työväentutkimus Vuosikirja

Measuring the efficiency of the Worm algorithm and the Metropolis algorithm in producing statistically independent observations

Lauri A. Lehtimäki

Published Jan 1, 2018• 0 citations

Abstract

In lattice field theories the partition function is often a very high dimensional integral which can’t be written in a closed form. In order to study the behaviour of the system one needs an efficient algorithm to approximate the high dimensional integral. In this thesis two Markov chain Monte Carlo algorithms the Metropolis algorithm and the Worm algorithm are used to simulate the three dimensional xy-model near its critical point. As the critical point is approached from below in terms of the inverse coupling the correlation length of the system diverges which may lead to the critical slowing down. An efficient algorithm is such that it suffers from the critical slowing down as little as possible which corresponds to a small dynamical critical exponent value. The efficiency of the algorithms is compared by measuring the integrated autocorrelation time of energy for different lattice sizes. The dynamical critical exponent, which controls the computational cost of the simulation, is calculated as a linear least squares fit to the integrated autocorrelation time data. From the simulated data the measured value of the dynamical critical exponent for the Metropolis algorithm is 1.85 and for the Worm algorithm the value is 0.37.

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In lattice field theories the partition function is often a very high dimensional integral which can’t be written in a closed form. In order to study the behaviour of the system one needs an efficient algorithm to approximate the high dimensional integral. In this thesis two Markov chain Monte Carlo algorithms the Metropolis algorithm and the Worm algorithm are used to simulate the three dimensional xy-model near its critical point. As the critical point is approached from below in terms of the inverse coupling the correlation length of the system diverges which may lead to the critical slowing down. An efficient algorithm is such that it suffers from the critical slowing down as little as possible which corresponds to a small dynamical critical exponent value. The efficiency of the algorithms is compared by measuring the integrated autocorrelation time of energy for different lattice sizes. The dynamical critical exponent, which controls the computational cost of the simulation, is calculated as a linear least squares fit to the integrated autocorrelation time data. From the simulated data the measured value of the dynamical critical exponent for the Metropolis algorithm is 1.85 and for the Worm algorithm the value is 0.37.

We show that high-temperature expansions provide a basis for the novel approach to efficient Monte Carlo simulations. "Worm" algorithms utilize the idea of updating closed-path configurations (produced by high-temperature expansions) through the motion of end points of a disconnected path. An amazing result is that local, Metropolis-type schemes using this approach appear to have dynamical critical exponents close to zero (i.e., their efficiency is comparable to the best cluster methods) as proved by finite-size scaling of the autocorrelation time for various universality classes.

The worm algorithm is a versatile technique in the Markov chain Monte Carlo method for both classical and quantum systems. The algorithm substantially alleviates critical slowing down and reduces the dynamic critical exponents of various classical systems. It is crucial to improve the algorithm and push the boundary of the Monte Carlo method for physical systems. We here propose a directed worm algorithm that significantly improves computational efficiency. We use the geometric allocation approach to optimize the worm scattering process: worm backscattering is averted, and forward scattering is favored. Our approach successfully enhances the diffusivity of the worm head (kink), which is evident in the probability distribution of the relative position of the two kinks. Performance improvement is demonstrated for the Ising model at the critical temperature by measurement of exponential autocorrelation times and asymptotic variances. The present worm update is approximately 25 times as efficient as the conventional worm update for the simple cubic lattice model. Surprisingly, our algorithm is even more efficient than the Wolff cluster algorithm, which is one of the best update algorithms. We estimate the dynamic critical exponent of the simple cubic lattice Ising model to be z≈0.27 in the worm update. The worm and the Wolff algorithms produce different exponents of the integrated autocorrelation time of the magnetic susceptibility estimator but the same exponent of the asymptotic variance. We also discuss how to quantify the computational efficiency of the Markov chain Monte Carlo method. Our approach can be applied to a wide range of physical systems, such as the |ϕ|^{4} model, the Potts model, the O(n) loop model, and lattice QCD.

I simulated the two-dimensional Ising model at its exact critical temperature with two Markov chain Monte Carlo methods, the single-spin Metropolis algorithm and the Wolff cluster algorithm, on periodic L×L lattices from L = 8 to L = 64. For each method I measured the integrated autocorrelation time of the energy and of the absolute magnetization, using four independent runs per lattice size. Both methods reproduce the exact finite-lattice energy within statistical error. For Metropolis, the autocorrelation time of |M| grows as L^z with z = 2.11(3) for L ≥ 16 and 2.16(5) for L ≥ 24, in line with the accepted value z ≈ 2.17 for local dynamics. For Wolff, measured in sweep-equivalent units, I get z = 0.29(1) from the energy and 0.15(1) from |M| for L ≥ 24, within the range of published values. In CPU time per statistically independent sample of |M|, Wolff was 5 times faster at L = 8 and 275 times faster at L = 64, and the gap grows roughly as L^2. All code and raw results are available, and the full study runs in under an hour on two CPU cores.

In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.

The critical properties of the three-dimensional XY model are investigated through measurements of the two-point correlation functions via a worm-algorithm Monte Carlo calculation, which automatically crunches the correlation functions while tracing worms, in large systems. This method has the advantage of directly measuring the order parameter, as well as the correlation length, the susceptibility, and their associated critical exponents. From the finite-size scaling behavior, we find 2 β/ν = 1.033(5) and γ/ν = 1.967(5), confirming the hyperscaling relations.