GPU-based computation of the Monte Carlo simulation of classical spin systems

Transcription

1 Perspectives of GPU Computing in Physics and Astrophysics, Sapienza University of Rome, Rome, Italy, September 15-17, 2014 GPU-based computation of the Monte Carlo simulation of classical spin systems Yutaka Okabe a) and Yukihiro Komura a,b) a) Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo , Japan b) RIKEN Advanced Institute for Computational Science, Kobe , Japan

2 Tokyo Metropolitan University TMU is a public university in Tokyo, the host city for the Olympic and Paralympic Games in All of the campuses of TMU are located in Tokyo Prefecture, and it takes about 30 minutes by train from central Tokyo to the Minami-Osawa Campus (the main campus), which offers an excellent learning environment with lush greenery. Currently, a total of 9,300 students (about 7,000 undergraduate students and about 2,300 graduate students) are enrolled at TMU, and about 680 faculty members are employed. Many excellent researchers are employed at TMU, achieving significant growth in the fields of research. In the Times Higher Education World University Rankings , TMU ranks 6th among Japanese universities. TMU has four faculties with 28 divisions and six graduate schools with 30 departments, offering a wide range of education and research fields.

3 TSUBAME2.5 (Tokyo Institute of Technology)

4 TSUBAME2.5 (Tokyo Institute of Technology)

5 Abstract High performance computing accelerates advances in science. Recently the use of general purpose computing on graphics processing unit (GPU) is a hot topic in computer science. We present the GPU calculation of the Monte Carlo simulation of classical spin systems. We put emphasis on the Swendsen-Wang multi-cluster algorithm [1]. We study the Ising model, the Potts model, and the classical XY model on two- and three-dimensional lattices, and the CUDA code is put on the program library [2]. We also discuss the multiple GPU computing, which can treat very large systems [3]. As applications, the large-scale Monte Carlo study of the two-dimensional XY model [4], and other topics will be discussed. [1] Y. Komura and Y. Okabe, Comp. Phys. Commun. 183, 1155 (2012). [2] Y. Komura and Y. Okabe, Comp. Phys. Commun. 185, 1038 (2014). [3] Y. Komura and Y. Okabe, Comp. Phys. Commun. 184, 40 (2013). [4] Y. Komura and Y. Okabe, J. Phys. Soc. Jpn., 81, (2012).

6 Plan of talk 1. Introduction Monte Carlo simulation cluster-flip algorithm Graphics Processing Unit (GPU) 2. GPU calculation of cluster-flip algorithm 3. Multiple-GPU calculation of cluster-flip algorithm 4. Large-scale simulation of 2D XY model 5. Two-size probability changing cluster algorithm (preliminary)

7 1.1 Monte Carlo simulation The Monte Carlo simulation is now used as a standard method for studying many-body problems in physics. Metropolis method N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953) Metropolis, Nicolas Constantine ( ) canonical ensemble transition probability p = min[1,e -β E ] single spin flip

8 Problem of slow dynamics In performing simulations, however, we often encounter the problem of slow dynamics. critical slowing down near the critical point phase separation dynamics at low temperature, slow dynamics due to the randomness or frustration, low-temperature slow dynamics in quantum Monte Carlo simulation To conquer the problem of slow dynamics, or the long time-scale problem, is one of the most important subjects in simulational physics.

9 New Monte Carlo Methods in order to overcome the problems of slow dynamics 1. extended ensemble methods multi-canonical method Berg & Neuhaus (1991) simulated tempering Marinari & Parisi (1992) exchange Monte Carlo method Hukushima & Nemoto (1996) broad histogram Oliveira et al (1996) flat histogram J.-S. Wang (1998) 2. cluster algorithms multi-cluster Swendsen & J.-S. Wang (1987) single cluster Wolff (1989) loop algorithm Evertz et al (1993) quantum MC Kawashima & Gubernatis (1994) continuous imaginary time Beard &Wiese (1996) Wang-Landau algorithm F. Wang & Landau (2001) Probability-changing cluster algorithm Tomita & Okabe (2001)

10 1.2 Probability-changing cluster (PCC) algorithm Tuning the critical point automatically Since our ensemble is (asymptotically) canonical, we can use the finite-size scaling analysis for physical quantities Using the PCC algorithm, we can determine the critical point and critical exponents with much less numerical efforts Y. Tomita and Y. Okabe, PRL 86, 572 (2001)

11 Swendsen-Wang Cluster Algorithm Hamiltonian of the Ising model 1. Start from some spin configuration. 2. If neighboring spins are the same type, then put a bond between them with the probability p 3. Flip all the spins on any clusters to one of two states. 4. Go back to the process 2.

12 GPU-based computation for Metropolis method Metropolis method (2D Ising model) Ising spin Spin flip with the probability exp(-β E) Sublattice decomposition of checker-board type Preis et. al. (2009) proposed the parallel computation for the Ising model on GPU. q=8 Acceleration 60 times for 2D and 35 times for 3D (comparison with a current core calculation) T. Preis, P. Virnau, W. Paul, J.J. Schneider, J. Comp. Phys. 228 (2009)

13 2. GPU calculation of cluster-flip algorithm Y. Komura and Y. Okabe, Comp. Phys. Commun. 183, 1155 (2012) There are 3 steps in the Swendsen-Wang multi-cluster flip algorithm (1) Bond generation (2) Cluster labeling (3) spin flip In the cluster labeling process for CPU calculation, Hoshen-Kopelman (HK) algorithm [1] is often used. However, the HK algorithm (sequential calculation) cannot be used in GPU calculation. [1] J. Hoshen et. al., Phys. Rev. B 14,3438 (1976)

14 GPU calculation of cluster-flip algorithm HK algorithm is sequential The number of the previous cluster was determined in the HK algorithm Initially the cluster number is set by the site number. Then, the cluster number is updated by comparing with neighboring sites. GPU calculation becomes possible. Label Equivalence algorithm [1,2] Comparison with neighboring sites Update of cluster number loop [1] K.A. Hawick, et al., Parallel Computing 36, 655 (2010). [2] O. Kalentev,et. Al., Parallel Distrib. Comput. 71, 615 (2011).

15 GPU calculation of cluster-flip algorithm GPU (GTX580) vs. 1CPU (Intel Xeon W3680) Average computational time per a spin flip at Tc for the 2D Ising model. The time for only a spin update and that including the measurement of energy and magnetization are given. The acceleration of computational speed as 13 times was realized. The efficiency is better for larger lattices (parallelization).

16 GPU calculation of cluster-flip algorithm Program library GPU computing of Swendsen-Wang multi-cluster spin ip algorithm: 2D and 3D Ising, Potts, and XY models Y. Komura and Y. Okabe, Comp. Phys. Commun. 185, 1038 (2014). Program: Models 1. 2D Ising model 2. 2D q-state Potts model 3. 2D q-state clock model 4. 3D Ising model 5. 3D q-state Potts model 6. 3D q-state clock model Cluster-labeling algorithms 1. the work by Hawick et al. 2. the work by Kalentev et al. System size: 2D: nx*nx maximum size 4096 or so, depending on memory size 3D: nx*nx*nx maximum size 256 or so, depending on memory size

17 GPU calculation of cluster-flip algorithm Program library ver.2 (in preparation) What is new? 1. new cluster labeling algorithm due to Wende and Steinke F. Wende and T. Steinke, Proceedings of the 2013 ACM/IEEE conference on Supercomputing (SC'13), New York, NY, USA, Nov 2013, Article No classical Heisenberg model (2D, 3D) 3. all double precisions 4. minor refinements

18 3. Multiple-GPU calculation of cluster-flip algorithm Y. Komura and Y. Okabe, Comp. Phys. Commun. 184, 40 (2013) The memory size within a single GPU is limited, for example, 3G bytes. For large-size calculation, the use of multiple GPU is necessary. The memory structure is distributed memory type. We need data communication between GPU through MPI of CPU. single GPU calculation multiple-gpu calculation simple extension. parallel calculation with shared memories parallel calculation with distributed memories

19 Multiple-GPU calculation of cluster-flip algorithm We should take care of the distributed memory structure for multiple- GPU calculation. Multi-cluster method for distributed memory type (CPU) Master-slave method Method to make a table for global label reference Relaxation method (communication of information of boundaries; repeat the label update)

20 Multiple-GPU calculation of cluster-flip algorithm (a) The 2x2 super-lattice structure for 4 GPUs. (b) The information on a single GPU. It has the arrays to preserve the data of surrounding boundary layers and to transfer the data of boundary layers. The same structure was used in the multiple-gpu calculation for Metropolis-type Monte Carlo simulation [Block et al. (2010)]. In this case, all the process is the copy of information of boundary layers.

21 Multiple-GPU calculation of cluster-flip algorithm Two-step cluster labeling method is employed in the multiple GPU calculation. example: calculation for 4 GPUs (1) cluster labeling within each GPU (Label Equivalence algorithm ) (2) inter-gpu cluster labeling (relaxation method) Data transfer Some clusters are merged within a single GPU. We repeat this process with data communication until the update of cluster number ends.

22 Multiple-GPU calculation of cluster-flip algorithm Metropolis single-spin-flip algorithm Single CPU (shared memory) Multiple-CPU (distributed memory) Simple update of single spin Parallelization with MPI (data communication) copy of boundary layers Single GPU (shared memory) Parallelization for many threads within each GPU sublattice decomposition Multiple-GPU (distributed memory) Parallelization for many threads within each GPU sublattice decomposition Parallelization with MPI (data communication) copy of boundary layers B. Block, P. Virnau, T. Preis, Comp. Phys. Comm. 181 (2010)

23 Multiple-GPU calculation of cluster-flip algorithm Swendsen-Wang multi-cluster spin-flip algorithm Single CPU (shared memory) Multiple-CPU (distributed memory) Hoshen-Kopelman algorithm for data labeling Parallelization with MPI (data communication) Relaxation method Single GPU (shared memory) Parallelization for many threads within each GPU Label equivalence algorithm Multiple-GPU (distributed memory) Parallelization for many threads within each GPU Label equivalence algorithm Parallelization with MPI (data communication) Relaxation method

24 Multiple-GPU calculation of cluster-flip algorithm TSUBAME 2.0 system at Tokyo Institute of Technology GPU (Tesla M2050) Average computational time per a spin update at Tc for the 2D Ising model with multiple GPUs. The time for only a spin update and that including the measurement of energy and magnetization are given.

25 Multiple-GPU calculation of cluster-flip algorithm Plot of spin flips per nano second with fixing the sub-lattice size as 4096x4096 are plotted as a function of the number of GPUs. The bestfitted curve to get the power dependence is given by the dotted line A good weak scalability is obtained. Larger size of a single GPU is better in efficiency. Data communication becomes more important for 3D lattices.

26 4. Large-scale simulation of 2D XY model 2D XY model: (KT transition) The analysis of Monte Carlo data is difficult because of logarithmic corrections. The magnetic susceptibility scales as Theoretical predictions: η = 1/4, r = -1/16 = Several estimates of r by Monte Carlo simulations Janke (1997) : r = (1) Chandrasekharan et.al (2003) : r = (10) Hasenbusch (2005) : r = (7) The largest size of previous studies was L = We make simulations up to L = 65536, and compare with theoretical predictions. Y. Komura and Y. Okabe, J. Phys. Soc. Jpn., 81, (2012)

27 Large-scale simulation of 2D XY model The Kosterlitz renormalization-group equations lead to the universal jump of the helicity modulus, that is, from the value (2/π)/T KT to 0 at T KT in the thermodynamic limit. Helicity modulus We use the PCC algorithm; we change the temperature by checking whether the helicity modus is larger than (2/π)/T or not. β KT (L)

28 Large-scale simulation of 2D XY model KT form of the correlation length Simulation up to L=65536 FSS form of for β KT (L) Our estimate of β KT (L) is (6) Our estimate of -2r is 0.128

29 5. Two-size Probability-Changing Cluster Algorithm Moment ratio (Binder ratio) Ratio of moment ratio U(T,2L)/U(T,L) FSS Crossing of different sizes T c, U(T c ) is some value S. Caraccido, et al., Phys. Rev. Lett. 74, 2969 (1995). J. Salas and A. D. Sokal, J. Stat. Phys. 88, 567 (1997). Crossing of different sizes T c, U(T c,2l)/u(t c,l) is 1 PCC algorithm U(T) > U* U(T) < U* T T T T T + T Two-size PCC algorithm U(T,2L)/U(T,L) > 1 T T T U(T,2L)/U(T,L) < 1 T T + T T c (L) such that U(T c (L)) = U* T c, FSS T* such that U(T*,2L)/U(T*,L) = 1 T c no extrapolation Simulation of two systems of different sizes at the same temperature

30 Two-size Probability-Changing Cluster Algorithm Time evolution of β with the two-size PCC algorithm for the 2D Ising model. The set of system sizes are (32,64) and (256,512); that is, L = 32 and L = 256. Short time average 4000MCS β = Correlation ratio R(T)= <g(l/2)>/<g(l/4)> is used instead of moment ratio U(T).

31 Summary We present the GPU calculation for the Swendsen-Wang multi-cluster algorithm of classical spin systems. We also present the GPU calculation for the Swendsen-Wang multicluster algorithm of classical spin systems. We show the large-scale Monte Carlo simulation of the classical XY model using the multi-gpu-based algorithm. We propose the two-size Probability Changing Cluster algorithm. Computational physics Effective algorithm two-size PCC algorithm Sophisticated analysis clever FSS analysis High-performance computer powerful GPUs