Scaling
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Data/scaling.plt
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Data/scaling.plt
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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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# G N U P L O T
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# Version 5.4 patchlevel 2 last modified 2021-06-01
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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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plot 'scaling.dat' i 1 u 1:(740.99828964984044/$2) w lp title "ARM Q80", \
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'scaling.dat' i 0 u 1:(266./$2) w lp title "AMD EPYC", \
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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
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Manuscript/benzene_qz.pdf
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Manuscript/benzene_tz.pdf
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Manuscript/benzene_tz.pdf
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Manuscript/scaling.pdf
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Manuscript/scaling.pdf
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@ -243,6 +243,20 @@ As each bucket is equiprobable, samples are defined as combinations of triplets,
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Should the values of $E_{abc}$ be skewed, this advanced refinement significantly diminishes the variance.
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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
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exact contribution from each bucket can be obtained.
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%=================================================================%
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%=================================================================%
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\subsection{Implementation Details}
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\subsection{Implementation Details}
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\label{sec:implementation}
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\label{sec:implementation}
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@ -410,15 +424,12 @@ The vibrational frequency and equilibrium distance estimated using this data, $\
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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.
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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.
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\subsection{Parallel efficiency}
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\subsection{Performance analysis}
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\subsection{Performance analysis}
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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}.
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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}.
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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.
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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.
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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}.
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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}.
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We linked our code with the Intel MKL library for BLAS operations.
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We linked our code with the Intel MKL library for BLAS operations.
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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.
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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.
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For this, we utilized the ArmPL library for BLAS operations.
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For this, we utilized the ArmPL library for BLAS operations.
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@ -429,7 +440,7 @@ For this, we utilized the ArmPL library for BLAS operations.
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CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
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CPU & $N_{\text{cores}}$ & $V$ & $F$ & Memory Bandwidth & Peak DP & Measured performance \\
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& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
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& & & (GHz) & (GB/s) & (GFlop/s) & (GFlop/s) \\
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\hline
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\hline
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EPYC 7513 & 64 & 4 & 2.6 & 409.6 & 2~662 & 1~576 \\
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\textsc{EPYC} 7513 & 64 & 4 & 2.6 & 409.6 & 2~662 & 1~576 \\
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Xeon Gold 6130 & 32 & 8 & 2.1 & 256.0 & 2~150 & 667 \\ % 239.891
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Xeon Gold 6130 & 32 & 8 & 2.1 & 256.0 & 2~150 & 667 \\ % 239.891
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ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
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ARM Q80 & 80 & 2 & 2.8 & 204.8 & 1~792 & 547 \\ % 292.492
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\end{tabular}
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\end{tabular}
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@ -442,7 +453,7 @@ Peak performance is determined by calculating the maximum achievable Flops/s on
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\begin{equation}
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\begin{equation}
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P = N_{\text{cores}} \times N_{\text{FMA}} \times 2 \times V \times F
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P = N_{\text{cores}} \times N_{\text{FMA}} \times 2 \times V \times F
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\end{equation}
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\end{equation}
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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.
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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.
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The relatively modest performance, at around 30\% efficiency, is attributed to the small dimensions of the matrices involved.
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The relatively modest performance, at around 30\% efficiency, is attributed to the small dimensions of the matrices involved.
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@ -453,7 +464,22 @@ I = \frac{2\, {N_\text{o}}^3\, N_\text{v}}{8\, \qty({N_\text{o}}^3 + {N_\text{o}
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\end{equation}
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\end{equation}
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which can be approximated by $N_\text{o} / 4$ flops/byte as an upper bound, which is usually relatively low.
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which can be approximated by $N_\text{o} / 4$ flops/byte as an upper bound, which is usually relatively low.
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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.
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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.
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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.
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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.
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\subsection{Parallel efficiency}
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\begin{figure}
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\includegraphics[width=\columnwidth]{scaling.pdf}
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\caption{\label{fig:speedup} Parallel speedup obtained with the ARM Q80 and AMD \textsc{Epyc} servers.}
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\end{figure}
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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.
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Three distinct regimes appear.
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The first one, up to 24 cores is close to the ideal regime
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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.
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These behaviors can be explained by the arithmetic intensity and the bandwidth of these machines.
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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.
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