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Journal of Statistical Mechanics: Theory and Experiment Volume 2006 March 2006 Create an alert RSS this journal

Helmut G Katzgraber et al J. Stat. Mech. (2006) P03018 doi:10.1088/1742-5468/2006/03/P03018

Feedback-optimized parallel tempering Monte Carlo

Helmut G Katzgraber1, Simon Trebst1,2,3, David A Huse4 and Matthias Troyer1

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Figure 1

Figure 1. Sketch of the random walk that a given replica performs in temperature space in the course of the simulation. Ideally, the replica will wander up (τup) and down (τdown) in the simulated temperature range [Tmin, Tmax]. The goal of the feedback method is to maximize the number of round-trips each replica performs in this temperature range, and thereby minimize the average round-trip time τrt = τup + τdown.



Figure 2

Figure 2. Fraction f(T) of replicas diffusing from the lowest to the highest temperature as a function of the temperature index for the ferromagnetic Ising model. For the initial temperature set based on a geometric progression (filled squares), the fraction shows a sharp drop between two temperature points. A similar behaviour is found for a temperature set where the acceptance rates are constant  ~ 40% independent of temperature (temperature set with `flat' acceptance rates, open squares). In contrast, for the optimized temperature set (triangles) the fraction constantly decreases. The inset shows the fraction as a function of temperature T. The dashed line in the inset represents the critical temperature of the two-dimensional Ising model, Tc ≈ 2.269.



Figure 3

Figure 3. Local diffusivity D(T) of the random walk a replica performs in temperature space for the ferromagnetic Ising model as a function of temperature T after the feedback optimization for several system sizes L. Notice the logarithmic vertical scale. The vertical dashed line represents Tc ≈ 2.269.



Figure 4

Figure 4. Temperature sets for the ferromagnetic Ising model for different feedback steps. Starting from a geometric progression temperature set (step 0), we apply a feedback loop until the temperature set converges. While the geometric progression places many temperatures at low temperatures, the density of temperatures after the feedback optimization is highest at the bottleneck of the simulation around the critical temperature (marked by a vertical dashed line). Rapid convergence of the optimized temperature set is found after 3-4 feedback steps and a total of Nsw ≈ 1.6 × 107 swap moves in our parallel tempering simulations.



Figure 5

Figure 5. Acceptance probabilities A(T) as a function of temperature T for the ferromagnetic Ising model. The inset shows the acceptance rates as a function of temperature in the optimized case for varying system sizes L and a fixed number of temperature points. The vertical dashed line marks the critical temperature.



Figure 6

Figure 6. Average round-trip times \overline {\tau }_{\mathrm {rt}} before the optimization divided by the average round-trip times after the feedback optimization (\overline {\tau }_{\mathrm {rt}}^{\mathrm {opt}} ) as a function of system size. The data for the filled squares are for a system starting from a geometric progression and represent the speedup obtained by the feedback method. The open symbols correspond to a temperature set which initially has `flat' acceptance probabilities. The dashed lines are guides to the eye.



Figure 7

Figure 7. Fraction f(T) of replicas diffusing from the lowest to the highest temperature for the fully frustrated Ising model. Displayed are data for an initial `flat' temperature set with M = 21 temperature points that yields temperature-independent acceptance probabilities for swap moves (open squares). In addition, data for a geometric progression (M = 21) are also shown (filled squares). After the optimization, the change in the fraction is independent of the temperature index (triangles). The inset shows the fractions as a function of temperature T. Data for Nsw = 2 × 106.



Figure 8

Figure 8. Local diffusivity D(T) of a random walk in temperature space for the fully frustrated Ising model as a function of temperature T after the feedback optimization for several system sizes L. Notice the logarithmic vertical scale.



Figure 9

Figure 9. Energy per spin e(T)=(1/N)[\langle {\mathcal H}\rangle]_{\mathrm {av}} as a function of temperature T for the fully frustrated Ising model for several system sizes. The data show that already for T \lesssim 0.5 the energy is independent of temperature, thus signalling that the system has reached the ground state. The inset zooms into the temperature range around T = 0.



Figure 10

Figure 10. Temperature sets for the fully frustrated Ising model for different feedback steps. Starting from a temperature set where the acceptance probabilities are independent of temperature with M = 21 temperature points (open symbols) and a temperature set obtained by a geometric progression also with M = 21 temperature points (filled symbols), we apply repeated feedback steps until the temperature sets converge to the optimal distributions. Also shown are data for an initial temperature set with M = 21 equidistant temperature points (stars). Independent of the initial temperature set, an optimal temperature distribution is found after 3-4 iterations and  ~ 7.5 × 106 swaps. After the successful feedback, temperature points accumulate near the transition to the ground state slightly above zero temperature.



Figure 11

Figure 11. Acceptance probabilities A(T) for replica swap moves as a function of temperature T for the fully frustrated Ising model. While the acceptance probabilities for a geometric progression temperature set show a pronounced dip close to T = 0, the optimized ensemble shows a peak close to zero temperature where the system enters the ground-state manifold. The inset shows, for a fixed number of temperatures, the acceptance rates as a function of temperature for different system sizes L. As for the Ising model, the `mean' value away from the bottlenecks can be tuned by increasing the number of temperatures. This illustrates that in order to obtain higher acceptance rates away from the bottlenecks of the simulation, the number of temperatures has to be increased with increasing L.



Figure 12

Figure 12. Average round-trip times \overline {\tau }_{\mathrm {rt}} before the optimization divided by the average round-trip times after the feedback optimization (\overline {\tau }_{\mathrm {rt}}^{\mathrm {opt}} ) as a function of system size for the fully frustrated Ising model. The data for the filled squares are for a system starting from a geometric progression temperature set and represent the speedup obtained by the feedback method. In addition, we show data for a temperature set with `flat' temperature-independent acceptance rates (open squares). The dashed lines are guides to the eye.



Figure 13

Figure 13. Average round-trip times \overline {\tau }_{\mathrm {rt}} as a function of the number of temperatures M for the fully frustrated Ising model with L = 20 after the feedback optimization. The data show that the round-trip times only depend moderately on the number of temperatures M, provided that there is sufficient overlap of the energy distributions. For a small number of replicas, this is no longer the case and the round-trip times increase drastically, e.g., for M \lesssim 12 in this plot. The inset shows the CPU time which is the product of the average round-trip time and the number of temperatures M. The data show a more pronounced minimum.



Figure 14

Figure 14. Distribution of average round-trip times for 5000 different samples of the 3D Edwards-Anderson Ising spin glass with Gaussian disorder and fixed system size L = 4 in the temperature range from Tmin = 0.10 to Tmax = 2.0. The data follow a fat-tailed Fréchet distribution (solid line) with a shape parameter ξ = 035(1). The inset shows the shape parameter ξ as a function of system size L. Already for L \gtrsim 5 , the shape parameter becomes \xi \gtrsim 1/2 , indicating that the variance of the distribution is no longer properly defined. The simulations have been performed using a fixed temperature set with M = 27 temperature points distributed such that, on average, replica swap moves have a nearly flat acceptance rate.




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