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Hadamard Product: Solving Thousands of Small Problems in CUDA or Squeezing Water out of a Rock

Outline. Review ProblemSerial AlgorithmParallel AlgorithmResultsConclusion. The Problem. Let C be a sparse N x N matrixLet X be a dense N x L matrixLet N >> LTask:Compute Z = C ? (X*XT) where ? is the Hadamard, or element product. The Problem Current Implementation. For N = 1282, L = 256

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Hadamard Product: Solving Thousands of Small Problems in CUDA or Squeezing Water out of a Rock

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    1. Hadamard Product: Solving Thousands of Small Problems in CUDA or Squeezing Water out of a Rock

    2. Outline Review Problem Serial Algorithm Parallel Algorithm Results Conclusion

    3. The Problem Let C be a sparse N x N matrix Let X be a dense N x L matrix Let N >> L Task: Compute Z = C ? (X*XT) where ? is the Hadamard, or element product

    4. The Problem Current Implementation For N = 1282, L = 256 Initialization: 2.5s Measurement Update: Compute mean: 37.4s Compute Zi and HiZi : 1825.5s Compute Ki : 218.3s Update the ensemble: 729.8s Time Update: Add mean: 34.4s Update the ensemble (Fi = I): 215.3s

    5. Serial Algorithm

    6. Parallel Decomposition One block computes multiple inner products One block computes one inner product Multiple blocks compute one inner product

    7. General Parallel Structure

    8. Best Parallel Kernel

    9. Best Parallel Kernel

    10. Best Parallel Kernel

    11. Varying the Problem Three Variables Stencil Changing stencil varies number of inner products (Work Units) Ensemble Size Changing ensemble size varies length of inner products (amount of work done per Work Unit) Reconstruction Size Only varies the number of Work Units Keep this fixed

    12. CPU Performance

    13. CPU Performance

    14. GPU Memory Overhead

    15. GPU Performance

    16. GPU Performance

    17. GPU Performance

    18. GPU Performance

    19. Useful New Knowledge Generally float2 > float > float4 Using texture, generally the same In one case though, float4 wins 33% occupancy often better than higher occupancy Warp level synchronicity Extra address calculations can actually help Be careful with loop unrolling (preserve coalescing) Unroll by block size, not by unroll factor

    20. Conclusion 17 X speedup over current CPU implementation 30 minutes down to a few minutes A system like the g80 can really help embarrassingly parallel problems Significant speedup even though computation to memory load ratio is nearly 1

    21. Future Work and Ideas CudaArray + 2D Texture Fetch Fewer address calculations But is this a good thing? C matrix into constant memory Maybe for faster reads Not much reuse so cache use limited Try to fold more computation into kernels

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