09.10.2018::

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:: Stuttgart, Germany

Data Reduction using Singular Value Decomposition (SVD) Algorithm

28th Workshop on Sustained Simulation Performance

Jing Zhang

jing.zhang@hlrs.de

High Performance Computing Center Stuttgart (HLRS), University of Stuttgart, Germany

09.10.2018::

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Overview

• Motivation and aim

• SVD algorithm

• Results by applying SVD

– 2D steady-state test case (1)

– 3D transient test cases (2 - 5)

• Summary and future work

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Motivation and aim

• Steady increase in computing power

– to use more complex models

– to simulate a lot of fluid-flow problems

– to perform simulations with large amount of CFD data

• I/O is bottleneck

– subsystem slow compared to other parts of computing system

• Aim of this study

– to reduce amount of CFD data transferred from memory to disk

– by using data-reduction algorithm SVD

– to minimize impact of I/O bottleneck on computing performance

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SVD algorithm

• 𝐴 = 𝑈 × 𝛴 × 𝑉𝑇 (if 𝑚 > 𝑛)

• 𝐴 = 𝑖=1𝑛 𝑢𝑖𝜎𝑖𝑣𝑖

𝑇 ≈ 𝑖=1𝑟 𝑢𝑖𝜎𝑖𝑣𝑖

𝑇 = 𝐴𝑟 (𝑖𝑓 𝜎𝑟 ≫ 𝜎𝑟+1)

• SVD subroutine PDGESVD in ScaLAPACK library

428th Workshop on Sustained Simulation Performance

𝐴 𝑈𝛴 𝑉𝑇

= x x

𝐴 ≈

𝑉𝑟𝑇

x x𝑈𝑟

𝛴𝑟

𝐴𝑟=

𝑚

𝑛 𝑛 𝑛 𝑛

𝑛

𝑚

𝑛 𝑟 𝑟 𝑛 𝑛

𝑚

𝑟

Singular values: 𝜎1 ≥ 𝜎2 ≥ ⋯ ≥ 𝜎𝑛 ≥ 0

0

0

𝜎1

𝜎n

𝜎2…( )

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Parameters for data-reduction algorithm SVD

• Compression ratio: 𝐶𝑅 =𝑚×𝑛

𝑚×𝑟+𝑟+𝑟×𝑛=

𝑚×𝑛

𝑟× 𝑚+1+𝑛

• Requirement of compression: r <𝑚×𝑛

𝑚+1+𝑛

• Mean Squared Error: 𝑀𝑆𝐸 = 𝑖=1𝑚 𝑗=1

𝑛 𝐴(𝑖,𝑗)−𝐴𝑟(𝑖,𝑗)2

𝑚×𝑛

• Peak Signal to Noise Ratio: 𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔10𝑚𝑎𝑥 𝐴 𝑖,𝑗 2

𝑀𝑆𝐸

– Good quality for SVD algorithm: 𝑃𝑆𝑁𝑅 ≥ 35 dB

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2D steady-state test case 1 (𝐴76×76)

• Original matrix is square and symmetric

• Original and reconstructed matrix are similar

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Original Reconstructed (rank=2)

1

0.5

0

-0.5

-1

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2D steady-state test case 1 (𝐴76×76)

• Limit on rank number 37

• Variation of CR, MSE and PSNR according to rank value up to 37

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2D steady-state test case 1 (𝐴76×76)

• Maximal absolute value from original matrix: 1.00

• Maximal absolute difference between original and reconstructed matrix (rank=2): 4×10-15

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Difference

2.8 × 10−15

1.5 × 10−15

2.5 × 10−16

−1.0 × 10−15

−2.3 × 10−15

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3D transient test cases

• Only one dataset Rhou2 (i.e. Rho × u2) is analysed

• Transformed data from 3D transient test cases into 2D dataset

• Data-transformation process

– Convert many HDF5 files at every time step into many single binary files using HDF5 utility h5dump

– Merge all binary files into one binary file including all data

– Reconvert the merged binary file into one 2D HDF5 file for applying SVD using HDF5 utility h5import

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𝑚: no. grid points

𝑛: no. time steps

𝐴

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3D transient test case 2 (m=363, n=8)

• Number of grid points: 𝑚 = 363 = 46656

• Number of time steps: 𝑛 = 8

• Non-dimensional simulation time: t = 0~20

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𝑛 = 1 𝑛 = 2 𝑛 = 3 𝑛 = 4

𝑛 = 5 𝑛 = 6 𝑛 = 7 𝑛 = 8

rhou20.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

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3D transient test case 2 (m=363, n=8)

• Limit on rank number 7

• Variation of CR, MSE and PSNR according to rank value (2, 4, 6)

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3D transient test case 2 (m=363, n=8)

• Comparison of original and reconstructed data at the last time step

• All reconstructed data show poor quality

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𝑛𝑡 = 8

Original r = 2 r = 4 r = 6

rhou20.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5

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3D transient test case 2 (m=363, n=8)

• Maximal absolute value of original data: 1.2

• Difference between original and reconstructed data at the last time step

• Three figures show similar results and difference is relative large

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r = 2 r = 4 r = 6

difference0.2

0.15

0.1

0.15

0

-0.05

-0.1

-0.15

-0.2

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3D transient test case 3 (m=363, n=99)

• Limit on rank number 98

• Variation of CR, MSE and PSNR according to rank value up to 98

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3D transient test case 3 (m=363, n=99)

• Comparison of original and reconstructed data at the last time step

• If the rank number is increased, reconstructed data approach the original data

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𝑛𝑡 = 8

Original r = 2 r = 9 r = 18

rhou20.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5

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3D transient test case 3 (m=363, n=99)

• Maximal absolute value from original data: 1.5

• Difference between original and reconstructed data at the last time step

• If rank number is increased, difference is reduced

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r = 2 r = 9 r = 18

difference0.2

0.15

0.1

0.15

0

-0.05

-0.1

-0.15

-0.2

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3D transient test case 4 (m=363, n=423)

• Limit on rank number 419

• Variation of CR, MSE and PSNR according to rank value up to 419

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3D transient test case 4 (m=363, n=423)

• Comparison of original and reconstructed data at the last time step

• With rank number of 18 reconstructed data are similar as original data

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𝑛𝑡 = 8

Original r = 2 r = 9 r = 18

rhou20.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5

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3D transient test case 4 (m=363, n=423)

• Maximal absolute value from original data: 1.5

• Difference between original and reconstructed data at the last time step

• Compression ratio almost 4x larger than for test case 3

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r = 2 r = 9 r = 18

difference0.2

0.15

0.1

0.15

0

-0.05

-0.1

-0.15

-0.2

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3D transient test case 5 (m=2603, n=8)

• Number of grid points: 𝑚 = 2603 = 17576000

• Number of time steps: 𝑛 = 8

• Non-dimensional simulation time: t = 0~20

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𝑛 = 1 𝑛 =2 𝑛 = 3 𝑛 = 4

𝑛 = 5 𝑛 = 6 𝑛 = 7 𝑛 = 8

rhou20.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

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3D transient test case 5 (m=2603, n=8)

• Limit on rank number 7

• Variation of CR, MSE and PSNR according to rank value (2 and 4)

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3D transient test case 5 (m=2603, n=8)

• Comparison of original and reconstructed data at the last time step

• Both reconstructed data show poor quality

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𝑛𝑡 = 8

Original r = 2 r = 4

rhou20.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5

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3D transient test case 5 (m=2603, n=8)

• Maximal absolute value from original data: 1.1

• Difference between original and reconstructed data at the last time step

• Difference is relative large

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r = 2 r = 4

difference0.2

0.15

0.1

0.15

0

-0.05

-0.1

-0.15

-0.2

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Summary and future work

• SVD has been implemented and parallel tested with 2D and 3D test cases

• Important parameters have been analysed for data-reduction algorithm SVD, e.g. CR, MSE and PSNR

• Results of applying SVD have been visualized and discussed

– 2D steady-state test case: square original matrix; with small rank no. CR is large; store less data; good quality for reconstructed data

– 3D transient test cases: same grid size and rank number; if number of time steps increased; CR increased; store less data; if PSNR value is equal to or just higher than 35 dB; good quality for results

• Future work

– SVD should be further improved with very large datasets

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:: Stuttgart, Germany

Thanks for your attention!

28th Workshop on Sustained Simulation Performance

Jing Zhang

jing.zhang@hlrs.de

High Performance Computing Center Stuttgart (HLRS), University of Stuttgart, Germany