GPU Accelerated Monte Carlo Simulations and Time Series Analysis

Transcription

1 GPU Accelerated Monte Carlo Simulations and Time Series Analysis 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: Peter Virnau, Wolfgang Paul, and 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 24 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 24 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 1 0 β = Total processing time on CPU Total processing time on GPU!1! 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 2 1 0! 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 2 1 0! Block size n/2

15 Binder cumulant 3D k B T C = {4.53 J, 4.51 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 GPGPU / Time Series Analysis

17 GPGPU / Time Series Analysis

18 Random Walk

19 German Stock Index (Dax) PDF of returns Autocorrelation Pattern Conformity #$%&'()!!!!!"!"" Hurst Exponent./01('23,''4 "!!!!!"!""!&!"!5,!,!+!*!,'!&-!&'!- '!!!!"! " &' &-,' φ( p( t )) = u exp( v p2 )

20 GPU computing / Hurst exponent p(t + t) p(t) q 1/q t H q( t)

21 GPU computing / Hurst exponent Time [ms] Acceleration # ! Time on GPU for allocation Time on GPU for memory transfer Time on GPU for main function Time on GPU for post processing Time on GPU for final processing Total processing time on GPU Total processing time on CPU 1 0 "1 " Length parameter!

22 GPU computing / Hurst exponent p(t) [percent of par value] t [ 5 units of time tick] FGBL JUN Hurst exponent H(!t) Random walk FGBL (CPU) FGBL (GPU) Relative error " [%] Time lag!t [units of time tick] Time lag!t [units of time tick] TP, P. Virnau, W. Paul, and J. J. Schneider, Preprint submitted (2009)

23 Fluctuation Patterns The aim is to compare the current reference pattern of time interval length t with all previous patterns in the time series. p(t) [units of points] (a)! t= "(t) (b)

24 Fluctuation Patterns 1.0 True range adapted modified time series p(t) p p t l ( t, ) (t) = p h ( t, ) p l ( t, ) p t (t) ~ "t p t^ (t) ~ "t p (t!#) t^!# 0.5 ~ p "t (t^! 1) t^ 0.0 p t (t) [0; 1] t [ t ; ) ^ t!"t ^ t! 1 t^

25 Fluctuation Patterns Mean-square quality between current and comparison sequence with Q t (τ) = Q t (τ) [0, 1] t θ=1 ( p t ( θ) p t τ ( τ θ) 1.0 t ) 2 ~ "t p t^ (t) ~ "t p (t!#) t^!# 0.5 ~ p "t (t^! 1) t^ 0.0 ω t (τ t + )= ( p t ^ t!"t ) ( 1+ t + ) p t ( 1) ( p t ^ t! 1 t^ In order to quantify the value of reference and comparison pattern relative to the reference point, one can define..., ) τ ( τ 1+ t + ) p t ( 1)

26 Fluctuation Patterns Observable for pattern conformity:, ξ χ ( t + t ) = T t + = t τ =τ ( sgn ( exp ω t χq t ) (τ t, + ) ) (τ) Limitation: τ = { ˆτ if ˆτ t 0 t else Normalized pattern conformity:, Ξ χ ( t + t ) = Definition: T t + = t sgn (x) = 1 for x>0 0 for x =0 1 for x<0 ( ξ χ t + t ) ( ) sgn ω t (τ t, + ) ( ) exp (τ) τ =τ, χq t

27 Pattern Conformity / Trivial Cases Straight Line Random Walk (a) (b) " " !t +!t !t +!t 0

28 Pattern Conformity / FDAX (a) (b) " !t +!t Complex correlations for financial market time series especially for large pattern lengths. 0 " * γ = !t +!t Ξ =Ξ FDAX χ=0 Ξ ACRW χ=0 Q t (τ) =Q p, t (τ) 0

29 Inclusion of volumes and ITWT (c) T. Preis et al., Europhys. Lett. 82, (2008) (d) " * " * !t +!t !t + Same structure high values of the pattern conformity Q t (τ) =Q p, t (τ)+q v, t (τ) Q t!t (τ) =Q p, t 0 (τ)+q ι, t (τ)

30 GPU computing / Pattern Conformity (a) " (b) " !t (c) !t " # !t!t !t +!t Time [ms] "1 "2 Acceleration # ! 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 Scan interval parameter!

31 Final remarks TP, PV, WP, and JJS, GPU Accelerated Monte Carlo Simulation of 2D and 3D Ising Model, J. Comp. Phys. 228, (2009) TP, PV, WP, and JJS, Accelerated Fluctuation Analysis by Graphic Cards and Complex Pattern Formation in Econophysics, Preprint submitted (2009) Source code available:

32 Thank you!