1. Brookhaven Science Associates U.S. Department of Energy 1 Data Analysis and Visualization Numerical Simulations Using Programmable GPUs Stan Tomov September 5, 2003

2. Brookhaven Science Associates U.S. Department of Energy 1 Outline Motivation Literature review The graphics pipeline Programmable GPUs Block diagram of nVidia's GeForce FX Some probability based simulations - Monte Carlo simulations - Ising model - Percolation model Implementation Performance results and analysis Extensions and future work Conclusions

3. Brookhaven Science Associates U.S. Department of Energy 1 ● High flops count (nVidia has listed 200Gflops theoretical speed for NV30) ● Compatible price performance (0.1 cents per M flop) ● High rate of performance increase over time (doubling every 6 months) Motivation The GPUs have: Explore the possibility of extending GPUs' use to non-graphics applications Table 1. GPU vs CPU in rendering polygons. The GPU (Quadro2 Pro) is approximately 30 times faster than the CPU (Pentium III, 1 GHz) in rendering polygonal data of various sizes.

4. Brookhaven Science Associates U.S. Department of Energy 1 Literature review Cellular automata Reaction-diffusion simulation (Mark Harris, University of North Carolina) Matrix multiply (E. Larsen and D. McAllister, University of North Carolina) Lattice Boltzmann computation (Wei Li, Xiaoming Wei, and Arie Kaufman, Stony Brook) CG and multigrid (J. Bolz et al, Caltech, and N. Goodnight et al, University of Virginia) Convolution (University of Stuttgart) Using graphics hardware for non-graphics applications: Performance results: Significant speedup of GPU vs CPU are reported if the GPU performs low precision computations (30 to 60 times; depends on the configuration) The fact that the operations are low precision is often skipped which may be confusing: - NCSA, University of Illinois assembled a $50,000 supercomputer out of 70 PlayStation 2 consoles, which could theoretically deliver 0.5 trillion operations/second - also, currently $200 GPUs are capable of 1.2 trillion op/s GPU’s flops performance is comparable to the CPU’s

5. Brookhaven Science Associates U.S. Department of Energy 1 The graphics pipeline

6. Brookhaven Science Associates U.S. Department of Energy 1 Programmable GPUs (in particular NV30) Support floating point operations Vertex program - Replaces fixed-function pipeline for vertices - Manipulates single vertex data - Executes for every vertex Fragment program - Similar to vertex program but for pixels Programming in Cg: - High level language - Looks like C - Portable - Compiles Cg programs to assembly code

7. Brookhaven Science Associates U.S. Department of Energy 1 Block diagram of GeForce FX AGP 8x graphics bus bandwidth: 2.1GB/s Local memory bandwidth: 16 GB/s Chip officially clocked at 500 MHz Vertex processor: - execute vertex shaders or emulate fixed transfor- mations and lighting (T&L) ● Pixel processor : - execute pixel shaders or emulate fixed shaders - 2 int & 1 float ops or 2 texture accesses/clock circle ● Texture & color interpolators - interpolate texture coordinates and color values Hardware at Digit-Life.com, NVIDIA GeForce FX, or "Cinema show started", November 18, 2002. Performance (on processing 4D vectors): ● Vertex ops/sec - 1.5 Gops ● Pixel ops/sec - 8 Gops (int), or 4 Gops (float)

8. Brookhaven Science Associates U.S. Department of Energy 1 Monte Carlo simulations ● Used in variety of simulations in physics, finance, chemistry, etc. ● Based on probability statistics and use random numbers ● A classical example: compute area of a circle ● Computation of expected values: N can be very large : on a 1024 x 1024 lattice of particles, every particle modeled to have k states, N = ● Random number generation. We used linear congruential type generator: (1)

9. Brookhaven Science Associates U.S. Department of Energy 1 Ising model ● Simplified model for magnets (introduced by Wilhelm Lenz in 1920, further studied by his student Ernst Ising) ● Modeled on 2D lattice with a “spin” (corresponding to orientation of electrons) at every cell pointing up or down ● Uses temperature to couple 2 opposing physical principles - minimization of the system's energy - entropy maximization ● Want to compute - expected magnetization: - expected energy: ● Evolve the system into “higher probability” states and compute expected values as average over those states - evolving from state to state, based on certain probability decision, is related to so called Markov chains: W.Gilks, S.Richardson, and D.Spiegelhalter (Editors), Markov chain Monte Carlo in Practice, Chapman&Hall, 1996.

10. Brookhaven Science Associates U.S. Department of Energy 1 Ising model computational procedure ● Choose an absolute temperature of interest T (in Kelvin) ● Color lattice in a checkerboard manner ● Start consecutive black and white “sweeps” ● Change the spin at a site based on the procedure 1. Denote current state as S, the state with flipped spin as S' 2. Compute 3. If accept S' else generate and accept S' if, where P(S) is given by the Boltzmann probability distribution function

11. Brookhaven Science Associates U.S. Department of Energy 1 Percolation model ● First studied by Broadbent and Hemmercley in 1957 ● Used in studies of disordered medium (usually specified by a probability distribution) ● Applied in studies of various phenomena such as spread of diseases, flow in porous media, forest fire propagation, clustering, etc. ● Of particular interest are: - media modeling threshold after which there exists a “spanning cluster” - relations between different media models - time to reach steady state spanning cluster

12. Brookhaven Science Associates U.S. Department of Energy 1 Implementation Approaches: Pure OpenGL (simulations using the fixed-function pipeline) Shaders in assembly Shaders in Cg Dynamic texturing: Create a texture T (think of a 2D lattice) Loop: - Render an image using T (in an off-screen buffer) - Update T from the resulting image

13. Brookhaven Science Associates U.S. Department of Energy 1 Performance results and analysis 256x256512x512 traffic 0.00063 0.0024 +, -, *, / 0.00010 0.0003 cos, sin 0.00026 0.0010 log, exp 0.00045 0.0015 if, ? : 0.00016 0.0008  3.5 Gflops Time in s. (approximate) for different vector flops on the GPU:  48 B per node – speed limited by GPU’s memory speed (16 GB/s) Time in s. (approximate) including traffic for different vector flops on the CPU: 256x256512x5121024x1024 +, -, *, / 0.0011 0.0046 0.017 cos, sin 0.0540 0.06500.267 log, exp 0.0609 0.11000.426  32 B per node – speed limited by CPU’s memory speed (4.2 GB/s)  20 x faster then CPU but the operations are of low accuracy

14. Brookhaven Science Associates U.S. Department of Energy 1  2.64 Gflops, i.e. 15% GPU theoretical power utilization (too many ifs): - if (flag) { … } : exec. time = time to compute the block even if flag = 0 Performance compatible with visualization related sample shaders from nVidia Cg assembly - Performance is the same for using runtime Cg or the generated assembly code - The assembly code generated is not optimal: we found cases where the code could be optimized and performance increased Performance results and analysis GPU and CPU (2.8 GHz) performance on the Ising model Lattice size (not necessary power of 2) 64x64128x128256x256512x5121024x1024 GPU sec/frame 0.0006 0.00230.00810.033 0.14 CPU no opt. 0.0009 0.00240.00830.032 0.13 CPU with –O4 0.0008 0.00200.00690.026 0.10 GPU instr./sec 0.55 G 0.57 G0.66 G 0.63 G 0.61 G

15. Brookhaven Science Associates U.S. Department of Energy 1 Extensions and future work Code optimization (through optimization of Cg generated assembly) More applications: - QCD ? - Fluid flow ? Parallel algorithms (or just as a coprocessor) - domain decomposition type in cluster environment - Motivation: communication rates CPU GPU for lattices of different sizes in seconds 64x64128x128256x256512x512  speed Read bdr (glReadPixels) 0.000160.00020.00060.002414 MB/s Read all (glReadPixels) 0.000400.00150.00620.0250167 MB/s Write bdr (glDrawPixels) 0.000220.00030.00070.002414 MB/s Write all (glTexSubImage2D) 0.000200.00080.00320.0120350 MB/s Write bdr (glTexSubImage2D) 0.000500.00200.00710.02501.3 MB/s Not a bottleneck in cluster with 1Gbit network Other ideas?

16. Brookhaven Science Associates U.S. Department of Energy 1 Conclusions GPUs have higher rate of performance increase over time than CPUs - always appealing as “research for the future” In certain applications GPUs are 30 to 60 times faster than CPUs for low precision computations (depending on configuration) For certain floating point applications GPU’s and CPU’s performance is comparable - can be used as coprocessor GPUs are often constrained in memory, but Preliminary results show it is feasible to use GPUs in parallel Cg is a convenient tool (but cgc could be optimized) It is feasible to use GPUs for numerical simulations - we demonstrated it by implementing 2 models (with many applications), and - used the implementation in benchmarking NV30 and Cg