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Numerical Stability of Algorithms at Extreme Scale and Low Precisions
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- 1 Intro
- 2 TOP500: June 2022 Frontier at Oak Ridge AMD EPYC 64C 2GHz, AMD Radeon Instinct GPU. 8,730,112 cores
- 3 Growth of Problem Size in TOP500
- 4 Today's Floating-Point Arithmetics
- 5 Backward Error Analysis for LU Factorizatia
- 6 Low Precision in Deep Learning
- 7 The (Partial) Explanation
- 8 Blocked Inner Products: 2 Pieces
- 9 Block Summation
- 10 FABsum Error Bound
- 11 Random Uniform (0, 1), b = 128, fp32
- 12 Extended Precision Registers
- 13 Mixed Precision Block FMA
- 14 Block FMA Hardware
- 15 Error Analysis of Block FMAS Blanchard, H, Lopez, Mary, & Pranesh (2020). Analysis of algs for matrix mult c - AB based on block
- 16 NVIDIA V100
- 17 Probabilistic Error Analysis Rounding error bounds above are worst-case.
- 18 Statistical Effects
- 19 Standard Tool for Rounding Error Analysis Theorem If | sufori - 1 n and nu 1 then
- 20 Assumptions for Probabilistic Analysis
- 21 Probabilistic Analysis Theorem (Connolly, H & Mary, 2021)
- 22 Linear Systems
- 23 Real-Life Matrices
- 24 Probabilistic QR Error Bound Theorem (Connolly & H, 2022) Under Model Mand a technical assumption, for the
- 25 Stochastic Rounding Forsythe (1950). .... Croci et al. (2022).
- 26 Stagnation Harmonic sum 1/k in fp16.
- 27 Random Data
- 28 Putting It All Together
- 29 Conclusions
