Massively parallel multicanonical simulations

Abstract

Generalized-ensemble Monte Carlo simulations such as the multicanonical method and similar techniques are among the most efficient approaches for simulations of systems undergoing discontinuous phase transitions or with rugged freeenergy landscapes. As Markov chain methods, they are inherently serial computationally. It was demonstrated recently, however, that a combination of independent simulations that communicate weight updates at variable intervals allows for the efficient utilization of parallel computational resources for multicanonical simulations. Implementing this approach for the many-thread architecture provided by current generations of graphics processing units (GPUs), we show how it can be efficiently employed with of the order of 10 4 parallel walkers and beyond, thus constituting a versatile tool for Monte Carlo simulations in the era of massively parallel computing. We provide the fully documented source code for the approach applied to the paradigmatic example of the two-dimensional Ising model as starting point and reference for practitioners in the field. Solution method: The code uses a parallel variant of the multicanonical method employing many parallel walkers that accumulate a common histogram. The resulting histogram is used to determine the weight function for the next iteration. Once the iteration has converged, simulations visit all possible energies with the same probability. Restrictions: The system size and size of the population of replicas are limited depending on the memory of the GPU device used. Code repository at https://github.com/CQT-Leipzig/ cudamuca. Running time: Depends on system size (approx. 40 s for L=32 on a Tesla K20m)

Estimating the density of states of frustrated spin systems

“…If, after some time, the histogram H Ê ( ) of all possible energies is found to be 'sufficiently flat' (typically interpreted as no histogram bin having less than 80% of the average number of entries [8], but see also [50] for a related discussion), the modification factor is reduced as f f  , and the histogram H Ê ( ) is reset to an empty state. The algorithm stops if f is 'sufficiently small', for example f f exp 10…”

GPU accelerated population annealing algorithm

“…If only enough such thread groups are available, the compute cores will be kept constantly busy and hence the memory latencies are hidden away. Good GPU performance thus requires to break the work into many threads, optimal performance is often only reached for thread numbers in excess of ten times the number of available physical cores [30].…”

Monte Carlo Methods for Massively Parallel Computers

“…For the total times per spin-flip, including the time spent on histogram and weight updates, we arrive at peak performances of 0.22 ns and 0.16 ns for the Tesla K20m and GTX Titan Black cards, respectively, which corresponds to a 15-21 times speedup as compared to the performance of an MPI code on a full dual-CPU node with a total of 12 cores (24 hyper-threads) with Intel Xeon E5-2640 CPUs. 119 This optimal performance is found for fully loading the GPUs with threads, i.e., for the maximum occupancy, corresponding to 30 720 threads for the Titan Black and 26 624 threads for K20m. The total speedup of the parallel implementation also depends on the effect of the parallel calculation on the number of required iterations until divergence, which is found to be slowly decreasing with p, 119 at least if a number of equilibration updates in between iterations ensures that the walkers are thermalized with respect to the updated weights before collecting statistics for the next iteration.…”

“…119 This optimal performance is found for fully loading the GPUs with threads, i.e., for the maximum occupancy, corresponding to 30 720 threads for the Titan Black and 26 624 threads for K20m. The total speedup of the parallel implementation also depends on the effect of the parallel calculation on the number of required iterations until divergence, which is found to be slowly decreasing with p, 119 at least if a number of equilibration updates in between iterations ensures that the walkers are thermalized with respect to the updated weights before collecting statistics for the next iteration. The total speedup in the time-to-solution for the parallel multicanonical code on GPU as a function of p is shown in Fig.…”

“…A GPU implementation of this method was recently introduced in Ref. [119]. It parallelizes only over independent walkers, thus resulting in particularly simple and easily adaptable code.…”