Das Ising-Modell auf Grafikkarten
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1 Das Ising-Modell auf Grafikkarten Institute of Physics, Johannes Gutenberg-University of Mainz Center for Polymer Studies, Department of Physics, Boston University Artemis Capital Asset Management GmbH With thanks to: Benjamin Block, Peter Virnau, Wolfgang Paul, Johannes J. Schneider
2 GPGPU computing GT200 single precision G80 Realistic illustrations Driving force: computer game industry NV GHz Harpertown Source: NVIDIA CUDA programming guide
3 GPU device architecture GeForce GTX 280: Global memory 1024 MB Number of multiprocessors 30 Number of cores 240 Constant memory 64 kb Shared memory 16 kb Clock rate 1.30 GHz
4 GPU device / Reference system GeForce GTX 280: Global memory 1024 MB Number of multiprocessors 30 Number of cores 240 Constant memory 64 kb Reference CPU: Intel Core 2 Quad 2.66 GHz Shared memory 16 kb Clock rate 1.30 GHz Cache size 4096 KB
5 C code with extensions global void gpu_function(int n, float* a, float* b) { } //Determine array element int i = threadidx.x + blockidx.x * blockdim.x; if(i<n) b[i] += a[i] * a[i]; Block 0 Block 1 Block 2... host void cpu_function() { int n = 128 * 128; int n_blocks = 128; int n_threads = 128; gpu_function<<<n_blocks,n_threads>>>(n, a, b); // Global barrier between GPU functions Thread 0 Block 1 Thread 1 Thread 2... } gpu_function<<<n_blocks/2,n_threads*2>>>(n, a, b);
6 Linear congruential RNGs x i+1,j =(a x i,j + c) mod m x 0,j+1 = (16807 x 0,j ) mod m a = c = bit architecture provided by the GPU x i,j [ 2 31 ;2 31 1] y i,j = abs ( x i,j /2 31) abs (x i,j )
7 Computation times Random numbers Time [ms] Acceleration " s Time on GPU for allocation Time on GPU for memory transfer Time on GPU for main function Total processing time on GPU Total processing time on CPU Speedup factor β = Total processing time on CPU Total processing time on GPU 10!1 10! Block number s
8 Ising model H = J i,j S i S j H i S i nearest neighbors Spin update: Metropolis criterion W a b = exp( H /k B T ) if H > 0 W a b =1 H 0 if
9 2D Ising: GPU implementation Noninteracting domains where Monte Carlo moves are performed in parallel Checkerboard algoritm
10 Computation times 2D Time [ms] Acceleration " n/2 Time on GPU for allocation Time on GPU for memory transfer Time on GPU for main function Total processing time on GPU Total processing time on CPU Speedup factor ! Block size n/2
11 Binder cumulant 2D kbtc = J "M (T )4 # U4 (T ) = 1 3"M (T )2 # U4 0.5 Critical temperature n/2 = 16 n/2 = 32 n/2 = 64 n/2 = 128 n/2 = kbt [J]
12 Ising model 3D H = J i,j S i S j H i S i nearest neighbors
13 3D Ising: GPU implementation Noninteracting domains where Monte Carlo moves are performed in parallel
14 Computation times 3D Time [ms] Acceleration " n/2 Time on GPU for allocation Time on GPU for memory transfer Time on GPU for main function Total processing time on GPU Total processing time on CPU Speedup factor ! Block size n/2
15 T. Preis, P. Virnau, W. Paul, and J. J. Schneider, J. Comp. Phys. 228, (2009) Binder cumulant 3D k B T C = { J, J} Critical temperature U U 4 (T )=1 M(T )4 3 M(T ) n/2 = 16 n/2 = 32 n/2 = 64 n/2 = k B T [J]
16 B. Block, P. Virnau, and T. Preis, manuscript in preparation Multi-GPU version 2D Communicated by MPI
17 B. Block, P. Virnau, and T. Preis, manuscript in preparation Multi-GPU Computation times Time [ms] Acceleration! n Time on 1 CPU Time on single GPU Time on 4 GPUs Time on 2 GPUs in 2 machines Speedup: n
18 Final remarks, Peter Virnau, Wolfgang Paul, and Johannes J. Schneider, GPU Accelerated Monte Carlo Simulation of 2D and 3D Ising Model, J. Comp. Phys. 228, (2009), Peter Virnau, Wolfgang Paul, and Johannes J. Schneider, Accelerated fluctuation analysis by graphic cards and complex pattern formation in financial markets, New J. Phys. 11, (2009) Source code available:
19 Thank you!
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