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3 Commits
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c6866453d7
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@ -1,20 +1,16 @@
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# 1
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# DZ: AMD EPYC
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# 2
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1 266. 0.1015625
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# 4
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2 133.202792882919 0.203125
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# 8
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4 68.3963158130646 0.40625
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#16
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8 35.3168480396271 0.8125
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#32
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16 17.4276471138000 1.625
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#64
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24 11.5433599948883
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28 10.3698871135712
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# TZ : AMD EPYC 7402 24-Core Processor
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32 9.43897294998169 3.25 12.1150507926941
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#1
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40 8.34011387825012
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#2
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48 7.40271902084351
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#4
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56 6.72331714630127
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#6
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64 6.40971302986145 6.5
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#12
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#24
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#32 123.718727827072
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#48 121.613038063049
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# DZ: ARM Q80
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# DZ: ARM Q80
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189
Data/scaling.plt
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Data/scaling.plt
@ -1,180 +1,13 @@
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#!/usr/bin/gnuplot -persist
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#!/usr/bin/env gnuplot
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#
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#
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set grid
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# G N U P L O T
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set key bottom
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# Version 5.4 patchlevel 2 last modified 2021-06-01
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set format y "%.1f"
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#
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# Copyright (C) 1986-1993, 1998, 2004, 2007-2021
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# Thomas Williams, Colin Kelley and many others
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#
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# gnuplot home: http://www.gnuplot.info
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# faq, bugs, etc: type "help FAQ"
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# immediate help: type "help" (plot window: hit 'h')
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# set terminal qt 0 font "Sans,9"
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# set output
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unset clip points
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set clip one
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unset clip two
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unset clip radial
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set errorbars front 1.000000
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set border 31 front lt black linewidth 1.000 dashtype solid
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set zdata
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set x2data
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set boxwidth
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set boxdepth 0
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set style fill empty border
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set style rectangle back fc bgnd fillstyle solid 1.00 border lt -1
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set style circle radius graph 0.02
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set style ellipse size graph 0.05, 0.03 angle 0 units xy
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set dummy x, y
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set format x "% h"
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set format y "% h"
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set format x2 "% h"
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set format y2 "% h"
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set format z "% h"
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set format cb "% h"
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set format r "% h"
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set ttics format "% h"
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set timefmt "%d/%m/%y,%H:%M"
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set angles radians
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set tics back
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set grid nopolar
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set grid xtics nomxtics ytics nomytics noztics nomztics nortics nomrtics \
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nox2tics nomx2tics noy2tics nomy2tics nocbtics nomcbtics
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set grid layerdefault lt 0 linecolor 0 linewidth 0.500, lt 0 linecolor 0 linewidth 0.500
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unset raxis
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set theta counterclockwise right
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set style parallel front lt black linewidth 2.000 dashtype solid
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set key notitle
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set key fixed left top vertical Right noreverse enhanced autotitle nobox
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set key noinvert samplen 4 spacing 1 width 0 height 0
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set key maxcolumns 0 maxrows 0
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set key noopaque
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unset label
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unset arrow
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unset style line
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unset style arrow
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set style histogram clustered gap 2 title textcolor lt -1
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unset object
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unset walls
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set style textbox transparent margins 1.0, 1.0 border lt -1 linewidth 1.0
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set offsets 0, 0, 0, 0
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set pointsize 1
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set pointintervalbox 1
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set encoding default
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unset polar
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unset parametric
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unset spiderplot
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unset decimalsign
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unset micro
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unset minussign
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set view 60, 30, 1, 1
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set view azimuth 0
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set rgbmax 255
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set samples 100, 100
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set isosamples 10, 10
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set surface
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unset contour
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set cntrlabel format '%8.3g' font '' start 5 interval 20
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set mapping cartesian
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set datafile separator whitespace
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set datafile nocolumnheaders
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unset hidden3d
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set cntrparam order 4
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set cntrparam linear
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set cntrparam levels 5
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set cntrparam levels auto
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set cntrparam firstlinetype 0 unsorted
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set cntrparam points 5
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set size ratio 0 1,1
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set origin 0,0
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set style data points
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set style function lines
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unset xzeroaxis
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unset yzeroaxis
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unset zzeroaxis
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unset x2zeroaxis
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unset y2zeroaxis
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set xyplane relative 0.5
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set tics scale 1, 0.5, 1, 1, 1
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set mxtics default
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set mytics default
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set mztics default
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set mx2tics default
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set my2tics default
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set mcbtics default
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set mrtics default
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set nomttics
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set xtics border in scale 1,0.5 mirror norotate autojustify
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set xtics norangelimit autofreq
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set ytics border in scale 1,0.5 mirror norotate autojustify
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set ytics norangelimit autofreq
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set ztics border in scale 1,0.5 nomirror norotate autojustify
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set ztics norangelimit autofreq
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unset x2tics
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unset y2tics
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set cbtics border in scale 1,0.5 mirror norotate autojustify
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set cbtics norangelimit autofreq
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set rtics axis in scale 1,0.5 nomirror norotate autojustify
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set rtics norangelimit autofreq
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unset ttics
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set title ""
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set title font "" textcolor lt -1 norotate
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set timestamp bottom
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set timestamp ""
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set timestamp font "" textcolor lt -1 norotate
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set trange [ * : * ] noreverse nowriteback
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set urange [ * : * ] noreverse nowriteback
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set vrange [ * : * ] noreverse nowriteback
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set xlabel "Number of cores"
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set xlabel "Number of cores"
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set xlabel font "" textcolor lt -1 norotate
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set x2label ""
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set x2label font "" textcolor lt -1 norotate
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set xrange [ * : * ] noreverse writeback
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set x2range [ * : * ] noreverse writeback
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set ylabel "Speedup"
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set ylabel "Speedup"
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set ylabel font "" textcolor lt -1 rotate
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set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
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set y2label ""
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set output 'scaling.pdf'
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set y2label font "" textcolor lt -1 rotate
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set pointsize 0.5
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set yrange [ * : * ] noreverse writeback
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plot 'scaling.dat' i 1 u 1:(740.99828964984044/$2) w lp title "ARM Q80", \
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set y2range [ * : * ] noreverse writeback
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'scaling.dat' i 0 u 1:(266./$2) w lp title "AMD EPYC", \
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set zlabel ""
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x title "Ideal"
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set zlabel font "" textcolor lt -1 norotate
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set zrange [ * : * ] noreverse writeback
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set cblabel ""
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set cblabel font "" textcolor lt -1 rotate
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set cbrange [ * : * ] noreverse writeback
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set rlabel ""
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set rlabel font "" textcolor lt -1 norotate
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set rrange [ * : * ] noreverse writeback
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unset logscale
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unset jitter
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set zero 1e-08
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set lmargin -1
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set bmargin -1
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set rmargin -1
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set tmargin -1
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set locale "en_AU.UTF-8"
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set pm3d explicit at s
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set pm3d scansautomatic
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set pm3d interpolate 1,1 flush begin noftriangles noborder corners2color mean
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set pm3d clip z
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set pm3d nolighting
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||||||
set palette positive nops_allcF maxcolors 0 gamma 1.5 color model RGB
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set palette rgbformulae 7, 5, 15
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set colorbox default
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set colorbox vertical origin screen 0.9, 0.2 size screen 0.05, 0.6 front noinvert bdefault
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set style boxplot candles range 1.50 outliers pt 7 separation 1 labels auto unsorted
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set loadpath
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set fontpath
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set psdir
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set fit brief errorvariables nocovariancevariables errorscaling prescale nowrap v5
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GNUTERM = "qt"
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I = {0.0, 1.0}
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VoxelDistance = 0.0
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## Last datafile plotted: "scaling.dat"
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plot 'scaling.dat' u 1:(740.99828964984044/$2) w lp notitle, x title "Ideal"
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# EOF
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BIN
Manuscript/benzene_qz.pdf
Normal file
BIN
Manuscript/benzene_qz.pdf
Normal file
Binary file not shown.
BIN
Manuscript/benzene_tz.pdf
Normal file
BIN
Manuscript/benzene_tz.pdf
Normal file
Binary file not shown.
BIN
Manuscript/scaling.pdf
Normal file
BIN
Manuscript/scaling.pdf
Normal file
Binary file not shown.
@ -260,3 +260,16 @@ i@article{watson_2016,
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publisher = {North-Holland},
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publisher = {North-Holland},
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doi = {10.1016/0009-2614(91)87003-T}
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doi = {10.1016/0009-2614(91)87003-T}
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}
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}
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@article{garniron_2017,
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author = {Garniron, Yann and Scemama, Anthony and Loos, Pierre-Fran{\c{c}}ois and Caffarel, Michel},
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title = {{Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory}},
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journal = {J. Chem. Phys.},
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volume = {147},
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number = {3},
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year = {2017},
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month = jul,
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issn = {0021-9606},
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publisher = {AIP Publishing},
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doi = {10.1063/1.4992127}
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}
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@ -121,13 +121,20 @@ that were previously computationally prohibitive.
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\section{Introduction}
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\section{Introduction}
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\label{sec:introduction}
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\label{sec:introduction}
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Coupled cluster (CC) theory is a powerful quantum mechanical approach widely used in computational chemistry and physics to describe the electronic structure of atoms, molecules, and materials.
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Coupled cluster (CC) theory is an accurate quantum mechanical approach widely used in computational chemistry and physics to describe the electronic structure of atoms, molecules, and materials.
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It offers a systematic and rigorous framework for accurate predictions of molecular properties and reactions by accounting for electron correlation effects beyond the mean-field approximation.
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It offers a systematic and rigorous framework for accurate predictions of molecular properties and reactions by accounting for electron correlation effects beyond the mean-field approximation.
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Among the various variants of the CC method, the Coupled Cluster Singles and Doubles with perturbative Triples method, CCSD(T), stands as the gold standard of quantum chemistry.
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CC theory starts with a parametrized wave function, typically referred to as the CC wave function, which is expressed as an exponential series of excitation operators acting on a reference:
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\begin{equation}
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\ket{\Psi_{\text{CC}}} = e^{\hat{T}} \ket{\Phi}
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\end{equation}
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where $|\ket{\Phi}$ is the reference determinant, and $\hat{T}$ is the cluster operator representing single, double, triple, and higher excitations from the reference wave function.
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Coupled Cluster with Singles and Doubles (CCSD) includes single and double excitations and represents the most commonly used variant of CC theory due to its favorable balance between accuracy and computational cost.
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Coupled Cluster with Singles, Doubles, and perturbative Triples (CCSD(T)) incorporates a perturbative correction to the CCSD energy to account for some higher-order correlation effects, and stands as the gold standard of quantum chemistry.
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CCSD(T) has demonstrated exceptional accuracy and reliability, making it one of the preferred choices for benchmark calculations and highly accurate predictions.
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CCSD(T) has demonstrated exceptional accuracy and reliability, making it one of the preferred choices for benchmark calculations and highly accurate predictions.
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It has found successful applications in a diverse range of areas, including spectroscopy,\cite{villa_2011,watson_2016,vilarrubias_2020} reaction kinetics,\cite{dontgen_2015,castaneda_2012} and materials design,\cite{zhang_2019} and has played a pivotal role in advancing our understanding of complex chemical phenomena.
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It has found successful applications in a diverse range of areas, including spectroscopy,\cite{villa_2011,watson_2016,vilarrubias_2020} reaction kinetics,\cite{dontgen_2015,castaneda_2012} and materials design,\cite{zhang_2019} and has played a pivotal role in advancing our understanding of complex chemical phenomena.
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In the context of CC theory, perturbative triples represent an important contribution to the accuracy of electronic structure calculations.\cite{stanton_1997}
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In the context of CC theory, the perturbative triples correction represents an important contribution to the accuracy of electronic structure calculations.\cite{stanton_1997}
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However, the computational cost associated with the calculation of this correction can be prohibitively high, especially for large systems.
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However, the computational cost associated with the calculation of this correction can be prohibitively high, especially for large systems.
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The inclusion of the perturbative triples in the CCSD(T) method leads to a computational scaling of $\order{N^7}$, where $N$ is proportional to the number of molecular orbitals.
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The inclusion of the perturbative triples in the CCSD(T) method leads to a computational scaling of $\order{N^7}$, where $N$ is proportional to the number of molecular orbitals.
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This scaling can rapidly become impractical, posing significant challenges in terms of computational resources and time requirements.
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This scaling can rapidly become impractical, posing significant challenges in terms of computational resources and time requirements.
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@ -196,13 +203,62 @@ In the algorithm proposed by Rendell\cite{rendell_1991}, for each given triplet
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\subsection{Stochastic formulation}
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\subsection{Stochastic formulation}
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\subsection{Test code}
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We propose an algorithm influenced by the semi-stochastic approach originally developed for computing the Epstein-Nesbet second-order perturbation correction to the energy. \cite{garniron_2017}
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\label{subsec:test_code}
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% Include the test code here, if applicable.
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The perturbative triples correction is expressed as a sum of corrections, each indexed solely by virtual orbitals:
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\begin{equation}
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E_{(T)} = \sum_{abc} E^{abc} \text{, where }
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E^{abc} = \sum_{ijk} E_{ijk}^{abc}.
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\end{equation}
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Monte Carlo sampling is employed by selecting samples $E_{abc}$.
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The principal advantage of this formulation is that the number of triplet combinations $(a,b,c)$, given by $N_v^3$, is sufficiently small to allow for all contributions $E_{abc}$ to be stored in memory.
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The first time a triplet $(a,b,c)$ is drawn, its corresponding value $E_{abc}$ is computed and then stored.
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Subsequent drawings of the same triplet retrieve the value from memory. We refer to this technique as \emph{memoization}.
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Thus, the computational expense of calculating the sample, which scales as $N_\text{o}^3 \times N_\text{v}$, is incurred only once, with all subsequent accesses being computationally trivial.
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Consequently, employing a sufficient number of Monte Carlo samples to ensure that each contribution is selected at least once results in a total computational cost that is only negligibly higher than that of an exact computation.
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To reduce the variance of the samples, the samples are drawn using the
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probability
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\begin{equation}
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P(a,b,c) = \frac{1}{\mathcal{N}} \frac{1}{\bar{\epsilon}_{ijk} - \epsilon_a - \epsilon_b - \epsilon_c}
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\end{equation}
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where $\mathcal{N}$ normalizes the sum such that $\sum P(a,b,c) = 1$. Here, $\bar{\epsilon}_{ijk}$ represents the average energy of the occupied orbitals, calculated as follows:
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\begin{equation}
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\bar{\epsilon}_{ijk} = \frac{3}{N_\text{o}} \sum_{i=1}^{N_\text{o}} \epsilon_i.
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\end{equation}
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The perturbative contribution is then computed by
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\begin{equation}
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E_{(T)} = \mathcal{N} \sum_{abc} P(a,b,c) \, E^{abc} \,
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(\bar{\epsilon}_{ijk} - \epsilon_a - \epsilon_b - \epsilon_c).
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\end{equation}
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This approach effectively reduces the statistical error bars by approximately a factor of two for the same computational expense due to two primary reasons: i) the estimator exhibits reduced fluctuations, ii) some triplet combinations are more likely to be selected than others, enhancing the efficiency of memoization.
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We employ the inverse transform sampling technique to select samples, where an array of pairs $\qty(P(a,b,c), (a,b,c))$ is stored.
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To further reduce the variance of the samples, this array is sorted in descending order based on $P(a,b,c)$ and subsequently partitioned into buckets, $B$.
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Each bucket is designed such that the sum $\sum_{(a,b,c) \in B} P(a,b,c)$ within it is as uniform
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as possible across all buckets.
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As each bucket is equiprobable, samples are defined as combinations of triplets, with one triplet drawn from each bucket.
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Should the values of $E_{abc}$ be skewed, this advanced refinement significantly diminishes the variance.
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The total perturbative contribution is computed as the aggregate of contributions from various buckets:
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\begin{equation}
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E_{(T)} = \sum_B E_B = \sum_B\sum_{(a,b,c) \in B} E_{abc}.
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\end{equation}
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Once every triplet within a bucket $B$ has been drawn at least once, the contribution $E_B$ can be determined.
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At this juncture, there is no longer a necessity to evaluate \(E_B\) stochastically, and the buckets can be categorized into stochastic ($\mathcal{S}$) and deterministic ($\mathcal{D}$) groups:
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\begin{equation}
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E_{(T)} = \sum_{B \in \mathcal{D}} E_B + \frac{1}{|\mathcal{S}|} \sum_{B \in \mathcal{S}}
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\left \langle E^B_{abc} \times \frac{- \epsilon_a - \epsilon_b - \epsilon_c}{\mathcal{N}} \right \rangle_{P(a,b,c), (a,b,c) \in B}.
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\end{equation}
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Not all buckets are of equal size; the number of triplets per bucket decreases with the bucket's index. Consequently, the initial buckets transition into the deterministic set first, gradually reducing the stochastic contribution. When every triplet has been drawn, the exact value of $E_{(T)}$ is obtained, devoid of statistical error.
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To accelerate the completion of the buckets, each Monte Carlo iteration triggers the computation of the first non-computed triplet. This ensures that after $N$ drawings, the
|
||||||
|
exact contribution from each bucket can be obtained.
|
||||||
|
|
||||||
%=================================================================%
|
%=================================================================%
|
||||||
\section{Implementation Details}
|
\subsection{Implementation Details}
|
||||||
\label{sec:implementation}
|
\label{sec:implementation}
|
||||||
|
|
||||||
The algorithm presented in Algorithm~\cite{alg:stoch} was implemented in the \textsc{Quantum Package} software.
|
The algorithm presented in Algorithm~\cite{alg:stoch} was implemented in the \textsc{Quantum Package} software.
|
||||||
@ -368,18 +424,15 @@ The vibrational frequency and equilibrium distance estimated using this data, $\
|
|||||||
Figure \ref{fig:cucl} illustrates the potential energy surface of \ce{CuCl}, displaying both the exact CCSD(T) energies and those estimated via the semi-stochastic method.
|
Figure \ref{fig:cucl} illustrates the potential energy surface of \ce{CuCl}, displaying both the exact CCSD(T) energies and those estimated via the semi-stochastic method.
|
||||||
|
|
||||||
|
|
||||||
\subsection{Parallel efficiency}
|
|
||||||
|
|
||||||
|
|
||||||
\subsection{Performance analysis}
|
\subsection{Performance analysis}
|
||||||
|
|
||||||
The primary bottleneck of our proposed algorithm lies in the generation of the sub-tensor $W^{abc}$ for each $(a,b,c)$ triplet, as discussed in Section~\ref{sec:theory}.
|
The primary bottleneck of our proposed algorithm lies in the generation of the sub-tensor $W^{abc}$ for each $(a,b,c)$ triplet, as discussed in Section~\ref{sec:theory}.
|
||||||
However, we have outlined a strategy to reframe this operation into BLAS matrix multiplications,\cite{form_w_abc} offering the potential for significantly enhanced efficiency.
|
However, we have outlined a strategy to reframe this operation into BLAS matrix multiplications,\cite{form_w_abc} offering the potential for significantly enhanced efficiency.
|
||||||
|
|
||||||
We evaluated the efficiency of our implementation using the Likwid\cite{treibig_2010} performance analysis tool on two distinct x86 platforms: an AMD EPYC 7513 dual-socket server equipped with 64 cores at \SI{2.6}{\giga\hertz}, and an Intel Xeon Gold 6130 dual-socket server with 32 cores at \SI{2.1}{\giga\hertz}.
|
We evaluated the efficiency of our implementation using the Likwid\cite{treibig_2010} performance analysis tool on two distinct x86 platforms: an AMD \textsc{Epyc} 7513 dual-socket server equipped with 64 cores at \SI{2.6}{\giga\hertz}, and an Intel Xeon Gold 6130 dual-socket server with 32 cores at \SI{2.1}{\giga\hertz}.
|
||||||
We linked our code with the Intel MKL library for BLAS operations.
|
We linked our code with the Intel MKL library for BLAS operations.
|
||||||
Additionally, we executed the code on an ARM Q80 server featuring 80 cores at \SI{2.8}{\giga\hertz}, and although performance counters were unavailable, we approximated the Flop/s rate by comparing the total execution time with that measured on the AMD CPU.
|
Additionally, we executed the code on an ARM Q80 server featuring 80 cores at \SI{2.8}{\giga\hertz}, and although performance counters were unavailable, we approximated the Flop/s rate by comparing the total execution time with that measured on the AMD CPU.
|
||||||
For this, we utilized the ArmPL library for BLAS operations.
|
For this, we utilized the \textsc{ArmPL} library for BLAS operations.
|
||||||
|
|
||||||
\begin{table*}
|
\begin{table*}
|
||||||
\begin{ruledtabular}
|
\begin{ruledtabular}
|
||||||
@ -387,7 +440,7 @@ For this, we utilized the ArmPL library for BLAS operations.
|
|||||||
CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
|
CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
|
||||||
& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
|
& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
|
||||||
\hline
|
\hline
|
||||||
EPYC 7513 & 64 & 4 & 2.6 & 409.6 & 2~662 & 1~576 \\
|
\textsc{EPYC} 7513 & 64 & 4 & 2.6 & 409.6 & 2~662 & 1~576 \\
|
||||||
Xeon Gold 6130 & 32 & 8 & 2.1 & 256.0 & 2~150 & 667 \\ % 239.891
|
Xeon Gold 6130 & 32 & 8 & 2.1 & 256.0 & 2~150 & 667 \\ % 239.891
|
||||||
ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
|
ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
@ -400,7 +453,7 @@ Peak performance is determined by calculating the maximum achievable Flops/s on
|
|||||||
\begin{equation}
|
\begin{equation}
|
||||||
P = N_{\text{cores}} \times N_{\text{FMA}} \times 2 \times V \times F
|
P = N_{\text{cores}} \times N_{\text{FMA}} \times 2 \times V \times F
|
||||||
\end{equation}
|
\end{equation}
|
||||||
where $F$ represents the frequency, $V$ the number of double precision elements in a vector register, $N_{\text{FMA}}$ denotes the number of vector FMA units per core (all considered CPUs possess two), and $N_{\text{cores}}$ reflects the number of cores. Notably, the Xeon and ARM CPUs both operate at approximately 30\% of peak performance, while the AMD EPYC CPU demonstrates twice the efficiency, achieving 60\% of the peak.
|
where $F$ represents the frequency, $V$ the number of double precision elements in a vector register, $N_{\text{FMA}}$ denotes the number of vector FMA units per core (all considered CPUs possess two), and $N_{\text{cores}}$ reflects the number of cores. Notably, the Xeon and ARM CPUs both operate at approximately 30\% of peak performance, while the AMD \textsc{Epyc} CPU demonstrates twice the efficiency, achieving 60\% of the peak.
|
||||||
|
|
||||||
|
|
||||||
The relatively modest performance, at around 30\% efficiency, is attributed to the small dimensions of the matrices involved.
|
The relatively modest performance, at around 30\% efficiency, is attributed to the small dimensions of the matrices involved.
|
||||||
@ -410,9 +463,29 @@ These multiplications exhibit an arithmetic intensity of
|
|||||||
I = \frac{2\, {N_\text{o}}^3\, N_\text{v}}{8\, \qty({N_\text{o}}^3 + {N_\text{o}}^2 N_\text{v} + {N_\text{o}} N_\text{v})}
|
I = \frac{2\, {N_\text{o}}^3\, N_\text{v}}{8\, \qty({N_\text{o}}^3 + {N_\text{o}}^2 N_\text{v} + {N_\text{o}} N_\text{v})}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
which can be approximated by $N_\text{o} / 4$ flops/byte as an upper bound, which is usually relatively low.
|
which can be approximated by $N_\text{o} / 4$ flops/byte as an upper bound, which is usually relatively low.
|
||||||
For instance, in the case of benzene with a triple-zeta basis set, the arithmetic intensity is calculated to be 3.52 flops/byte, falling short of the threshold required to attain peak performance on any of the CPUs.
|
For instance, in the case of benzene with a triple-zeta basis set, the arithmetic intensity is calculated to be 3.33 flops/byte, falling short of the threshold required to attain peak performance on any of the CPUs.
|
||||||
By leveraging memory bandwidth and double precision throughput peak, we determined the critical arithmetic intensity necessary to achieve peak performance. On the Xeon and ARM CPUs, this critical value stands at approximately 8.4 and 8.8 flops/byte, respectively. Meanwhile, the EPYC CPU exhibits a value of 6.5 flops/byte, thanks to its superior memory bandwidth.
|
By leveraging memory bandwidth and double precision throughput peak, we determined the critical arithmetic intensity necessary to achieve peak performance. On the Xeon and ARM CPUs, this critical value stands at approximately 8.4 and 8.8 flops/byte, respectively. Meanwhile, the \textsc{EPYC} CPU exhibits a value of 6.5 flops/byte, thanks to its superior memory bandwidth.
|
||||||
|
|
||||||
|
\subsection{Parallel efficiency}
|
||||||
|
|
||||||
|
\begin{figure}
|
||||||
|
\includegraphics[width=\columnwidth]{scaling.pdf}
|
||||||
|
\caption{\label{fig:speedup} Parallel speedup obtained with the ARM Q80 and AMD \textsc{Epyc} servers.}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The parallel speedup performance of the ARM and AMD servers for computations involving the benzene molecule in a triple-zeta basis set is illustrated in Figure~\ref{fig:speedup}. The results delineate three distinct performance regimes:
|
||||||
|
\begin{itemize}
|
||||||
|
\item In the first regime, encompassing up to 24 cores, the performance closely approximates the ideal, with nearly linear speedup.
|
||||||
|
\item The second regime, spanning 24 to 64 cores, shows decent performance, achieving a 40-fold acceleration with 64 cores.
|
||||||
|
\item The third regime begins beyond 64 cores, where parallel efficiency rapidly deteriorates.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
This performance behavior can largely be attributed to the arithmetic intensity and the bandwidth characteristics of these servers.
|
||||||
|
On the ARM server, the peak performance is attained at an arithmetic intensity of 8.75~flops/byte.
|
||||||
|
Notably, with fewer cores, the bandwidth per core increases, thereby enhancing efficiency.
|
||||||
|
For the benzene molecule in the triple-zeta basis set, the critical arithmetic intensity is 3.33~flops/byte.
|
||||||
|
This intensity corresponds to a threshold of approximately 30 cores for the ARM server and 32 cores for the AMD server.
|
||||||
|
Beyond these thresholds, particularly after 64 cores on the ARM server, the heavy demand on memory bandwidth results in a rapid decline in speedup.
|
||||||
|
|
||||||
|
|
||||||
%%%
|
%%%
|
||||||
|
@ -59,7 +59,7 @@ fetched from memory. In this way, the $N_o^3 \times N_v$ cost of
|
|||||||
computing the sample is paid only once, and all other evaluations are
|
computing the sample is paid only once, and all other evaluations are
|
||||||
negligible.
|
negligible.
|
||||||
|
|
||||||
Hence, using an number of Monte Carlo samples large enough such that each
|
Hence, using a number of Monte Carlo samples large enough such that each
|
||||||
contribution has been drawn at least once will have a computational
|
contribution has been drawn at least once will have a computational
|
||||||
cost larger only by a negligible amount than the cost of making the
|
cost larger only by a negligible amount than the cost of making the
|
||||||
exact computation.
|
exact computation.
|
||||||
@ -80,7 +80,6 @@ contributions of the occupied orbitals:
|
|||||||
|
|
||||||
The perturbative contribution is computed as
|
The perturbative contribution is computed as
|
||||||
|
|
||||||
\[
|
|
||||||
\[
|
\[
|
||||||
E_{(T)} = \mathcal{N} \sum_{abc} P(a,b,c) \, E_{abc} \,
|
E_{(T)} = \mathcal{N} \sum_{abc} P(a,b,c) \, E_{abc} \,
|
||||||
(\epsilon_{\text{occ}} - \epsilon_a - \epsilon_b - \epsilon_c) \,
|
(\epsilon_{\text{occ}} - \epsilon_a - \epsilon_b - \epsilon_c) \,
|
||||||
|
Loading…
Reference in New Issue
Block a user