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PaSTR TRI: E : Err rror-Bou Bounded Los Lossy Comp Compression - - PowerPoint PPT Presentation

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PaSTR TRI: E : Err rror-Bou Bounded Los Lossy Comp Compression on for or Two-El Electron n Integr grals s in in Quan antu tum Chem emis istr try Ali Murat Gok (Northwestern University, USA) Sheng Di (Argonne National

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SLIDE 1

PaSTR TRI: E : Err rror-Bou Bounded Los Lossy Comp Compression

  • n for
  • r Two-El

Electron n Integr grals s in in Quan antu tum Chem emis istr try

Ali Murat Gok (Northwestern University, USA) Sheng Di (Argonne National Laboratory, USA) Yuri Alexeev (Argonne National Laboratory, USA) Dingwen Tao (The University of Alabama, USA) Vladimir Mironov (Lomonosov Moscow State University, Russia) Xin Liang (University of California, Riverside, USA ) Franck Cappello (Argonne National Laboratory, USA)

September 2018

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SLIDE 2

Sheng Di (ANL), Xin Liang (U. C. Riverside), Dingwen Tao (U. Alabama), Franck Cappello (Lead)

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SLIDE 3

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 4

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 5

Introduction

  • HPC applications work with extremely large data (Petabytes!)
  • Large data → System bottlenecks (Memory, Storage, Bandwidth)
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SLIDE 6

Introduction

  • HPC applications work with extremely large data (Petabytes!)
  • Large data → System bottlenecks (Memory, Storage, Bandwidth)
  • Electron Repulsion Integrals (ERIs):
  • Large data size: Petabytes
  • Costly computations: O(N4)
  • Data reuse: ~10-30 times
  • PaSTRI: Pattern Scaling for Two-Electron Repulsion Integrals
  • Calculate and compress once
  • Decompress whenever needed
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SLIDE 7

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 8

Electron Repulsion Integrals (ERIs)

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SLIDE 9

Electron Repulsion Integrals (ERIs)

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 10

Electron Repulsion Integrals (ERIs)

  • ERIs are a part of solving the Schrödinger equation:

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 11

Electron Repulsion Integrals (ERIs)

  • ERIs are a part of solving the Schrödinger equation:

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

scale as O(N4)

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SLIDE 12

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 13

ERI Data Representation

  • (ij|kl) representation examples: (dd|dd), (dp|ff), (ps|df), …
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SLIDE 14

ERI Data Representation

  • (ij|kl) representation examples: (dd|dd), (dp|ff), (ps|df), …

(dd|dd) block 0,0,0,0 1234E-6 0,0,0,1 2345E-7 0,0,0,5 3456E-6 0,0,1,0 4567E-8 5,5,5,5 6789E-5

6*6*6*6 = 1296 pts … … … …

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 15

ERI Data Representation

  • (ij|kl) representation examples: (dd|dd), (dp|ff), (ps|df), …

(dd|dd) block 0,0,0,0 1234E-6 0,0,0,1 2345E-7 0,0,0,5 3456E-6 0,0,1,0 4567E-8 5,5,5,5 6789E-5

6*6*6*6 = 1296 pts … … … …

(dp|ff) block 0,0,0,0 1234E-6 0,0,0,1 2345E-7 0,0,0,9 3456E-6 0,0,1,0 4567E-8 5,2,9,9 6789E-5

6*3*10*10 = 1800 pts … … … …

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 16

ERI Data Representation

  • (ij|kl) representation examples: (dd|dd), (dp|ff), (ps|df), …

(ff|ff) block 0,0,0,0 1234E-6 0,0,0,1 2345E-7 0,0,0,9 3456E-6 0,0,1,0 4567E-8 9,9,9,9 6789E-5

10*10*10*10 = 10000 pts … … … …

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 17

ERI Data Representation

  • (ij|kl) representation examples: (dd|dd), (dp|ff), (ps|df), …

Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

4D 0,0,0,0 0,0,0,1 0,0,0,9 0,0,1,0 9,9,9,9

… 1D Index

1D 1 9 10 9999 4D 0,0,0,0 0,0,0,1 0,0,0,6 0,0,1,0 5,5,5,5 1D 1 6 7 1295

(ff|ff) (dd|dd) … … … … … … …

4D 0,0,0,0 0,0,1,0 1,0,1,0 1,0,2,0 9,5,2,0 1D 1 19 20 179

(fd|ps) … … … …

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SLIDE 18

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 19

Patterns in ERIs

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

(dd|dd)

215

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SLIDE 20

Patterns in ERIs

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215]

(dd|dd)

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SLIDE 21

Patterns in ERIs

Data Ranges [0:35] , [36:71]

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215]

4E-7

  • 4E-7

(dd|dd)

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SLIDE 22

Patterns in ERIs

Data Ranges [0:35] , [36:71] Data Ranges [0:35] , [36:71]

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215]

4E-7

  • 4E-7

1E-7

  • 1E-7

4E-7

  • 4E-7

(dd|dd)

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SLIDE 23

Patterns in ERIs

Data Ranges [0:35] , [36:71] Data Ranges [0:35] , [36:71] |Deviation| and |Compr. Error|

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215] 1E-12 1E-10 1E-8 1E-6 1E-4 1E-2 1E+0

4E-7

  • 4E-7

1E-7

  • 1E-7

4E-7

  • 4E-7

(dd|dd)

Reasonable Absolute Error Bound: 10-10

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SLIDE 24

Patterns in ERIs

Data Ranges [0:35] , [36:71] Data Ranges [0:35] , [36:71] |Deviation| and |Compr. Error|

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215] 1E-12 1E-10 1E-8 1E-6 1E-4 1E-2 1E+0

4E-7

  • 4E-7

1E-7

  • 1E-7

4E-7

  • 4E-7

(dd|dd) → (6*6 | 6*6) → (36 | 36) → (1296) # of SBs Period (SB Size) Block Size Orbital # of BFs

s 1 p 3 d 6 f 10 g 15 … …

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SLIDE 25

Patterns in ERIs

Data Ranges [0:35] , [36:71] Data Ranges [0:35] , [36:71] |Deviation| and |Compr. Error|

  • 4E-07
  • 2E-07

0E+00 2E-07 4E-07

Original Data, Range: [0:215]

Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block Sub-Block [0:35] [36:71] [72:107] [108:143] [144:179] [179:215] 1E-12 1E-10 1E-8 1E-6 1E-4 1E-2 1E+0

4E-7

  • 4E-7

1E-7

  • 1E-7

4E-7

  • 4E-7

(dd|dd) → (6*6 | 6*6) → (36 | 36) → (1296) # of SBs Period (SB Size) Block Size Original Data: Full Block: 1296 (64-bit) Compressed: Pattern: 36 (<64-bit) Scale: 36 (<64-bit) Error Correction: ? bits

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SLIDE 26

Why are there patterns in ERIs?

  • ERI values are calculated in ordered loops
  • ERI values depend on both the shape and the distance of electron

clouds

  • For distant clouds, the shape loses its importance, distance dominates
  • Most of the electron clouds are distant from each other
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SLIDE 27

Generating Pattern and Scaling Coefficients

FR (Ratio of Firsts) ER (Ratio of Extremums) AR (Ratio of Averages)

|Sub-Block|

AAR (Ratio of Abs. Averages) IS (Interval Scaling)

a b a a a a b b b b |Pattern| Sub-Block Pattern Sub-Block Pattern Sub-Block Pattern Sub-Block Pattern

Scaling coefficient = a / b (Note: |b| ≥ |a|)

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SLIDE 28

Generating Pattern and Scaling Coefficients

FR (Ratio of Firsts) ER (Ratio of Extremums) AR (Ratio of Averages)

|Sub-Block|

AAR (Ratio of Abs. Averages) IS (Interval Scaling) Requires Sign Correction

a b a a a a b b b b |Pattern| Sub-Block Pattern Sub-Block Pattern Sub-Block Pattern Sub-Block Pattern

Best Compression, Fast

Scaling coefficient = a / b (Note: |b| ≥ |a|)

“a” or “b” can be close to zero !

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SLIDE 29

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 30
  • Calculate period (based on the last two BFs)
  • Determine Pattern (P), then quantize P to PQ
  • Calculate Scaling coefficients (S), then quantize S to SQ
  • # of elements in PQ and SQ depend on block type (s, p, d, f, g,…)
  • Calculate Error Correction (EC), then quantize EC to ECQ
  • EC = Original data - PQ * P_binsize * SQ * S_binsize
  • # of elements in ECQ depends on deviation (atoms are distant or not)
  • Decide encoding mode
  • Sparse or Non-sparse
  • Encode PQ, SQ, and ECQ and write to output file

PaSTRI Compression

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SLIDE 31

PaSTRI Decompression

  • Read encoding mode, error bound
  • Calculate period
  • Read PQ and reconstruct Pattern
  • Read SQ and reconstruct Scaling coefficients
  • Read ECQ and reconstruct Error Correction
  • Reconstruct data values:
  • Decompressed Data = Pattern_DQ * Scale_DQ + ErrorCorrection_DQ

Scaled Pattern

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SLIDE 32

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 33

Optimizations

Error Correction

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SLIDE 34

Optimizations

Error Correction How many bits per element for Pattern (PQ), Scaling coefficients (SQ) and Error Correction (ECQ)?

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SLIDE 35

Optimizations

  • EB: 10-10
  • Ranges:
  • PQ: [BlockMin, BlockMax], e.g., [-10-6; 10-6] à 10 bits
  • SQ: [-1,1] HUGE RANGE! à 33 bits
  • ECQ: Smaller range than PQ, e.g., [-10-8; 10-8] à 7 bits
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SLIDE 36

Optimizations

  • EB: 10-10
  • Ranges:
  • PQ: [BlockMin, BlockMax], e.g., [-10-6; 10-6] à 10 bits
  • SQ: [-1,1] HUGE RANGE! à 33 bits
  • ECQ: Smaller range than PQ, e.g., [-10-8; 10-8] à 7 bits
  • Cannot enforce 2*EB quantization bin size on everyone!
  • Cannot find exact optimal solution. Why?
  • This is a nonlinear optimization problem (or even harder).

Too costly to solve!

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SLIDE 37

Practical solution

  • PQ: 2*EB quantization bin size
  • SQ: Same number of bits as PQ
  • ECQ: 2*EB quantization bin size
  • Uses a special encoding tree
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SLIDE 38

ECQ Characteristics

1 bit : 0 2 bits: 1,-1 3 bits: {2, 3, -2, -3} 4 bits: {4, 5, 6, 7,

  • 4, -5, -6, -7}

5 bits: {8, 9, … , 14, 15,

  • 8, -9, …, -14, -15}

ECb,max : Max bits needed for ECQ

…

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SLIDE 39

ECQ Encoding Trees

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SLIDE 40

ECQ Encoding Trees

Why better than Huffman?

1) NO require generating a dictionary 2) NO require storing a dictionary 3) Single occurrences HURT Huffman compression ratio

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SLIDE 41

Outline

  • Introduction
  • Background
  • Electron Repulsion Integrals (ERIs)
  • ERI Data Representation
  • Patterns in ERIs
  • PaSTRI Compression
  • Optimizations of Quantization & Encoding
  • Experimental Evaluation
  • Conclusion
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SLIDE 42

Evaluation

Benzene Glutamine tri-Alanine

  • Bebop supercomputer at Argonne
  • 64 nodes (two Intel Xeon E5-2695 v4 and

128 GB memory) with 2048 cores

  • General Parallel File Systems (GPFS)
  • Experimental Data
  • Tri-Alanine, Benzene, Glutamine molecules
  • (dd|dd) and (ff|ff) BF configurations
  • s and p are too small
  • g and above are not common
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SLIDE 43

Compression Ratios

5 10 15 20 25 30 35 SZ ZFP PaSTRI SZ ZFP PaSTRI SZ ZFP PaSTRI EB = 1E-11 EB = 1E-10 EB = 1E-9

Compression Ratios

alanine,(dd|dd) alanine,(ff|ff) benzene,(dd|dd) benzene,(ff|ff) glutamine,(dd|dd) glutamine,(ff|ff) Average

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SLIDE 44

Compression Rates

200 400 600 800 1000 SZ ZFP PaSTRI SZ ZFP PaSTRI SZ ZFP PaSTRI EB = 1E-11 EB = 1E-10 EB = 1E-9

  • Comp. Rates (MB/s)

alanine,(dd|dd) alanine,(ff|ff) benzene,(dd|dd) benzene,(ff|ff) glutamine,(dd|dd) glutamine,(ff|ff) Average

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SLIDE 45

Decompression Rates

200 400 600 800 1000 1200 1400 1600 SZ ZFP PaSTRI SZ ZFP PaSTRI SZ ZFP PaSTRI EB = 1E-11 EB = 1E-10 EB = 1E-9

  • Decomp. Rates(MB/s)

alanine,(dd|dd) alanine,(ff|ff) benzene,(dd|dd) benzene,(ff|ff) glutamine,(dd|dd) glutamine,(ff|ff) Average

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SLIDE 46

Rate-distortion

160 170 180 190 200 210 220 230 2 4 6 8 10 12 14

PSNR (dB) Bitrate SZ ZFP PaSTRI Alanine (dd|dd)

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SLIDE 47

Baseline Comparison

0.00 0.20 0.40 0.60 0.80 1.00 EB = EB = EB = EB = EB = EB = 1E-11 1E-10 1E-9 1E-11 1E-10 1E-9 Original PaSTRI infras. Original PaSTRI infras. (dd|dd) (ff|ff)

Normalized Time Calculate ERI Compress Decompress

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SLIDE 48

Parallel Performance (MPI)

5 10 15 20 25 30 D L D L D L D L D L D L D L D L D L D L D L D L SZ ZFP PaSTRI SZ ZFP PaSTRI SZ ZFP PaSTRI SZ ZFP PaSTRI 256 512 1024 2048

Elapsed Time (min) Number of Cores Compress Write to Disk Decompress Read Original Alanine (dd|dd)

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SLIDE 49

Conclusions

  • 16.8X compression ratio, over 2.3X better
  • 661 MB/s compression rate, over 2.1X better
  • 1.1 GB/s decompression rate, over 4.3X better
  • Significantly better rate-distortion curve
  • More than 73% energy saving compared to original GAMESS

infrastructure

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SLIDE 50

PaSTRI’s Contributions

  • Developed a model to understand ERI data
  • Data=Pattern*Scale + Deviation
  • Exploited the inherent pattern features in the ERIs
  • Proposed an effective lossy compression algorithm for ERIs
  • PaSTRI is implemented and released as open source
  • PaSTRI is evaluated by using original GAMESS data, and compared to
  • ther state-of-the-art lossy compressors
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SLIDE 51

Ackn knowledge

51

This research was supported by the Exascale Computing Project (17-SC-20-SC), a joint project of the U.S. Department of Energy’s Office of Science and National Nuclear Security Administration, responsible for delivering a capable exascale ecosystem, including software, applications, and hardware technology, to support the nation’s exascale computing imperative. The material was also supported by and supported by the National Science Foundation under Grant No. 1305624,

  • No. 1513201, and No. 1619253.
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SLIDE 52

PaSTRI was integrated into SZ already!

SZ is available at: https://github.com/disheng222/SZ Feel free to Download and Test! Contact: Sheng Di (sdi01@anl.gov) Franck Cappello (cappello@mcs.anl.gov)

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SLIDE 53

Thank you!

Any questions are welcome!

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SLIDE 54

Supporting Slides