Yukihiro Komura

GPU-based Swendsen–Wang multi-cluster algorithm for the simulation of two-dimensional classical spin systems

Abstract We present the GPU calculation with the common unified device architecture (CUDA) for the Swendsen–Wang multi-cluster algorithm of two-dimensional classical spin systems. We adjust the two connected component labeling algorithms recently proposed with CUDA for the assignment of the cluster in the Swendsen–Wang algorithm. Starting with the q -state Potts model, we extend our implementation to the system of vector spins, the q -state clock model, with the idea of embedded cluster. We test the performance, and the calculation time on GTX580 is obtained as 2.51 nsec per a spin flip for the q = 2 Potts model (Ising model) and 2.42 nsec per a spin flip for the q = 6 clock model with the linear size L = 4096 at the critical temperature, respectively. The computational speed for the q = 2 Potts model on GTX580 is 12.4 times as fast as the calculation speed on a current CPU core. That for the q = 6 clock model on GTX580 is 35.6 times as fast as the calculation speed on a current CPU core.

Large-scale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs

We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L =65536, we investigate the Kosterlitz–Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β KT is 1.11996(6). We estimate the exponent to specify the multiplicative logarithmic correction, -2 r , and precisely reproduce the theoretical prediction -2 r =1/8.

GPU-based single-cluster algorithm for the simulation of the Ising model

We present the GPU calculation with the common unified device architecture (CUDA) for the Wolff single-cluster algorithm of the Ising model. Proposing an algorithm for a quasi-block synchronization, we realize the Wolff single-cluster Monte Carlo simulation with CUDA. We perform parallel computations for the newly added spins in the growing cluster. As a result, the GPU calculation speed for the two-dimensional Ising model at the critical temperature with the linear size L=4096 is 5.60 times as fast as the calculation speed on a current CPU core. For the three-dimensional Ising model with the linear size L=256, the GPU calculation speed is 7.90 times as fast as the CPU calculation speed. The idea of quasi-block synchronization can be used not only in the cluster algorithm but also in many fields where the synchronization of all threads is required.

GPU-based cluster-labeling algorithm without the use of conventional iteration: Application to the Swendsen–Wang multi-cluster spin flip algorithm

Abstract Cluster-labeling algorithms that use a single GPU can be roughly divided into direct and two-stage approaches. To date, both types use an iterative method to compare the labels of nearest-neighbor sites. In this paper, I present a GPU-based cluster-labeling algorithm that does not use conventional iteration. The proposed method is applicable to both direct algorithms and two-stage approaches. Under the proposed approach, only one comparison with the nearest-neighbor site is needed for a two-dimensional (2D) system, and just two comparisons are needed for three-dimensional (3D) systems. As an application of the new cluster-labeling algorithm, I consider the Swendsen–Wang (SW) multi-cluster spin flip algorithm. The performance of the proposed method is compared with that of other cluster-labeling algorithms for the SW multi-cluster spin flip problem using the 2D and 3D Ising models. As a result, the computation time of the new algorithm is shown to be 40% faster than that of the previous algorithm for the 2D Ising model, and 20% faster than that of the previous algorithm for the 3D Ising model at the critical temperature.

Difference of energy density of states in the Wang-Landau algorithm.

Paying attention to the difference of density of states, Δln g(E)≡ln g(E+ΔE)-lng(E), we study the convergence of the Wang-Landau method. We show that this quantity is a good estimator to discuss the errors of convergence and refer to the 1/t algorithm. We also examine the behavior of the first-order transition with this difference of density of states in connection with Maxwells equal area rule. A general procedure to judge the order of transition is given.

Large-scale calculation of ferromagnetic spin systems on the pyrochlore lattice

Abstract We perform the high-performance computation of the ferromagnetic Ising model on the pyrochlore lattice. We determine the critical temperature accurately based on the finite-size scaling of the Binder ratio. Comparing with the data on the simple cubic lattice, we argue the universal finite-size scaling. We also calculate the classical XY model and the classical Heisenberg model on the pyrochlore lattice.

High-Precision Monte Carlo Simulation of the Ising Models on the Penrose Lattice and the Dual Penrose Lattice

We study the Ising models on the Penrose lattice and the dual Penrose lattice by means of the high-precision Monte Carlo simulation. Simulating systems up to the total system size N = 20633239, we estimate the critical temperatures on those lattices with high accuracy. For high-speed calculation, we use the generalized method of the single-GPU-based computation for the Swendsen–Wang multi-cluster algorithm of Monte Carlo simulation. As a result, we estimate the critical temperature on the Penrose lattice as Tc/J = 2.39781 ± 0.00005 and that of the dual Penrose lattice as \(T_{\text{c}}^{*}/J = 2.14987 \pm 0.00005\). Moreover, we definitely confirm the duality relation between the critical temperatures on the dual pair of quasilattices with a high degree of accuracy, \(\sinh (2J/T_{\text{c}})\sinh (2J/T_{\text{c}}^{*}) = 1.00000 \pm 0.00004\).

Probing phase transition order of q-state Potts models using Wang-Landau algorithm

Phase transitions are ubiquitous phenomena, exemplified by the melting of ice and spontaneous magnetization of magnetic material. In general, a phase transition is associated with a symmetry breaking of a system; occurs due to the competition between coupling interaction and external fields such as thermal energy. If the phase transition occurs with no latent heat, the system experiences continuous transition, also known as second order phase transition. The ferromagnetic q-state Potts model with r extra invisible states, introduced by Tamura, Tanaka, and Kawashima [Prog. Theor. Phys. 124, 381 (2010)], is studied by using the Wang-Landau method. The density of states difference (DOSD), ln g(E + ΔE) − ln g(E), is used to investigate the order of the phase transition and examine the critical value of r changing the second to the first order transition.

Phase transition of a two-dimensional generalized XY model

We study a two-dimensional generalized XY model that depends on an integer q by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz–Thouless (KT) transition for all values of q, in contrast with the previous speculation that there may be two transitions, a regular KT transition and a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of q without assuming a specific universal form in terms of the KT transition temperature TKT.

Multi-GPU-based Swendsen–Wang multi-cluster algorithm with reduced data traffic

Abstract The computational performance of multi-GPU applications can be degraded by the data communication between each GPU. To realize high-speed computation with multiple GPUs, we should minimize the cost of this data communication. In this paper, I propose a multiple GPU computing method for the Swendsen–Wang (SW) multi-cluster algorithm that reduces the data traffic between each GPU. I realize this reduction in data traffic by adjusting the connection information between each GPU in advance. The code is implemented on the large-scale open science TSUBAME 2.5 supercomputer, and its performance is evaluated using a simulation of the three-dimensional Ising model at the critical temperature. The results show that the data communication between each GPU is reduced by 90%, and the number of communications between each GPU decreases by about half. Using 512 GPUs, the computation time is 0.005 ns per spin update at the critical temperature for a total system size of N = 4096 3 .