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Scaling

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Anthony Scemama 2024-04-16 14:45:18 +02:00
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commit 792ab5ceb2
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Data/scaling.plt
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@ -1,180 +1,13 @@
#!/usr/bin/gnuplot -persist
#
#
# G N U P L O T
# Version 5.4 patchlevel 2 last modified 2021-06-01
#
# Copyright (C) 1986-1993, 1998, 2004, 2007-2021
# Thomas Williams, Colin Kelley and many others
#
# gnuplot home: http://www.gnuplot.info
# faq, bugs, etc: type "help FAQ"
# immediate help: type "help" (plot window: hit 'h')
# set terminal qt 0 font "Sans,9"
# set output
unset clip points
set clip one
unset clip two
unset clip radial
set errorbars front 1.000000
set border 31 front lt black linewidth 1.000 dashtype solid
set zdata
set ydata
set xdata
set y2data
set x2data
set boxwidth
set boxdepth 0
set style fill empty border
set style rectangle back fc bgnd fillstyle solid 1.00 border lt -1
set style circle radius graph 0.02
set style ellipse size graph 0.05, 0.03 angle 0 units xy
set dummy x, y
set format x "% h"
set format y "% h"
set format x2 "% h"
set format y2 "% h"
set format z "% h"
set format cb "% h"
set format r "% h"
set ttics format "% h"
set timefmt "%d/%m/%y,%H:%M"
set angles radians
set tics back
set grid nopolar
set grid xtics nomxtics ytics nomytics noztics nomztics nortics nomrtics \
nox2tics nomx2tics noy2tics nomy2tics nocbtics nomcbtics
set grid layerdefault lt 0 linecolor 0 linewidth 0.500, lt 0 linecolor 0 linewidth 0.500
unset raxis
set theta counterclockwise right
set style parallel front lt black linewidth 2.000 dashtype solid
set key notitle
set key fixed left top vertical Right noreverse enhanced autotitle nobox
set key noinvert samplen 4 spacing 1 width 0 height 0
set key maxcolumns 0 maxrows 0
set key noopaque
unset label
unset arrow
unset style line
unset style arrow
set style histogram clustered gap 2 title textcolor lt -1
unset object
unset walls
set style textbox transparent margins 1.0, 1.0 border lt -1 linewidth 1.0
set offsets 0, 0, 0, 0
set pointsize 1
set pointintervalbox 1
set encoding default
unset polar
unset parametric
unset spiderplot
unset decimalsign
unset micro
unset minussign
set view 60, 30, 1, 1
set view azimuth 0
set rgbmax 255
set samples 100, 100
set isosamples 10, 10
set surface
unset contour
set cntrlabel format '%8.3g' font '' start 5 interval 20
set mapping cartesian
set datafile separator whitespace
set datafile nocolumnheaders
unset hidden3d
set cntrparam order 4
set cntrparam linear
set cntrparam levels 5
set cntrparam levels auto
set cntrparam firstlinetype 0 unsorted
set cntrparam points 5
set size ratio 0 1,1
set origin 0,0
set style data points
set style function lines
unset xzeroaxis
unset yzeroaxis
unset zzeroaxis
unset x2zeroaxis
unset y2zeroaxis
set xyplane relative 0.5
set tics scale 1, 0.5, 1, 1, 1
set mxtics default
set mytics default
set mztics default
set mx2tics default
set my2tics default
set mcbtics default
set mrtics default
set nomttics
set xtics border in scale 1,0.5 mirror norotate autojustify
set xtics norangelimit autofreq
set ytics border in scale 1,0.5 mirror norotate autojustify
set ytics norangelimit autofreq
set ztics border in scale 1,0.5 nomirror norotate autojustify
set ztics norangelimit autofreq
unset x2tics
unset y2tics
set cbtics border in scale 1,0.5 mirror norotate autojustify
set cbtics norangelimit autofreq
set rtics axis in scale 1,0.5 nomirror norotate autojustify
set rtics norangelimit autofreq
unset ttics
set title ""
set title font "" textcolor lt -1 norotate
set timestamp bottom
set timestamp ""
set timestamp font "" textcolor lt -1 norotate
set trange [ * : * ] noreverse nowriteback
set urange [ * : * ] noreverse nowriteback
set vrange [ * : * ] noreverse nowriteback
#!/usr/bin/env gnuplot
set grid
set key bottom
set format y "%.1f"
set xlabel "Number of cores"
set xlabel font "" textcolor lt -1 norotate
set x2label ""
set x2label font "" textcolor lt -1 norotate
set xrange [ * : * ] noreverse writeback
set x2range [ * : * ] noreverse writeback
set ylabel "Speedup"
set ylabel font "" textcolor lt -1 rotate
set y2label ""
set y2label font "" textcolor lt -1 rotate
set yrange [ * : * ] noreverse writeback
set y2range [ * : * ] noreverse writeback
set zlabel ""
set zlabel font "" textcolor lt -1 norotate
set zrange [ * : * ] noreverse writeback
set cblabel ""
set cblabel font "" textcolor lt -1 rotate
set cbrange [ * : * ] noreverse writeback
set rlabel ""
set rlabel font "" textcolor lt -1 norotate
set rrange [ * : * ] noreverse writeback
unset logscale
unset jitter
set zero 1e-08
set lmargin -1
set bmargin -1
set rmargin -1
set tmargin -1
set locale "en_AU.UTF-8"
set pm3d explicit at s
set pm3d scansautomatic
set pm3d interpolate 1,1 flush begin noftriangles noborder corners2color mean
set pm3d clip z
set pm3d nolighting
set palette positive nops_allcF maxcolors 0 gamma 1.5 color model RGB
set palette rgbformulae 7, 5, 15
set colorbox default
set colorbox vertical origin screen 0.9, 0.2 size screen 0.05, 0.6 front noinvert bdefault
set style boxplot candles range 1.50 outliers pt 7 separation 1 labels auto unsorted
set loadpath
set fontpath
set psdir
set fit brief errorvariables nocovariancevariables errorscaling prescale nowrap v5
GNUTERM = "qt"
I = {0.0, 1.0}
VoxelDistance = 0.0
## Last datafile plotted: "scaling.dat"
plot 'scaling.dat' u 1:(740.99828964984044/$2) w lp notitle, x title "Ideal"
# EOF
set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
set output 'scaling.pdf'
set pointsize 0.5
plot 'scaling.dat' i 1 u 1:(740.99828964984044/$2) w lp title "ARM Q80", \
'scaling.dat' i 0 u 1:(266./$2) w lp title "AMD EPYC", \
x title "Ideal"

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40
Manuscript/stochastic_triples.tex
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@ -243,6 +243,20 @@ As each bucket is equiprobable, samples are defined as combinations of triplets,
Should the values of $E_{abc}$ be skewed, this advanced refinement significantly diminishes the variance.
The total perturbative contribution is computed as the aggregate of contributions from various buckets:
\begin{equation}
E_{(T)} = \sum_B E_B = \sum_B\sum_{(a,b,c) \in B} E_{abc}.
\end{equation}
Once every triplet within a bucket $B$ has been drawn at least once, the contribution $E_B$ can be determined.
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:
\begin{equation}
E_{(T)} = \sum_{B \in \mathcal{D}} E_B + \frac{1}{|\mathcal{S}|} \sum_{B \in \mathcal{S}}
\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}.
\end{equation}
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.
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.
%=================================================================%
\subsection{Implementation Details}
\label{sec:implementation}
@ -410,15 +424,12 @@ 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.
\subsection{Parallel efficiency}
\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}.
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.
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.
@ -429,7 +440,7 @@ For this, we utilized the ArmPL library for BLAS operations.
CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
\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
ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
\end{tabular}
@ -442,7 +453,7 @@ Peak performance is determined by calculating the maximum achievable Flops/s on
\begin{equation}
P = N_{\text{cores}} \times N_{\text{FMA}} \times 2 \times V \times F
\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.
@ -453,7 +464,22 @@ I = \frac{2\, {N_\text{o}}^3\, N_\text{v}}{8\, \qty({N_\text{o}}^3 + {N_\text{o}
\end{equation}
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 3.33 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}
Figure~\ref{fig:speedup} shows the parallel speedups obtained with the ARM and AMD servers for the benzene molecule in the triple-zeta basis set.
Three distinct regimes appear.
The first one, up to 24 cores is close to the ideal regime
The second one, between 24 and 64 cores is decent and enables an acceleration of $40 \times$ with 64 cores. Then, beyond 64 cores, the parallel efficiency drops quickly.
These behaviors can be explained by the arithmetic intensity and the bandwidth of these machines.
On the ARM server, we have seen that the critical arithmetic intensity to leverage peak performance was 8.8 flops/byte. However, if the number of cores decreases, the bandwidth per core increases and so does the efficiency.