-\chapterauthor{Pierre Fortin}{Laboratoire d'Informatique de Paris 6, University Paris 6}
-\chapterauthor{Rachid Habel}{T\'el\'ecom SudParis}
-\chapterauthor{Fabienne J\'ez\'equel}{Laboratoire d'Informatique de Paris 6, University Paris 6}
-\chapterauthor{Jean-Luc Lamotte}{Laboratoire d'Informatique de Paris 6, University Paris 6}
+\chapterauthor{Rachid Habel}{T\'el\'ecom SudParis, France}
+\chapterauthor{Pierre Fortin, Fabienne J\'ez\'equel, and Jean-Luc Lamotte}{Laboratoire d'Informatique de Paris 6, Université Pierre et Marie Curie, France}
+
+%\chapterauthor{Fabienne J\'ez\'equel}{Laboratoire d'Informatique de Paris 6, University Paris 6}
+%\chapterauthor{Jean-Luc Lamotte}{Laboratoire d'Informatique de Paris 6, University Paris 6}
\chapterauthor{Stan Scott}{School of Electronics, Electrical Engineering \& Computer Science,
-The Queen's University of Belfast}
+The Queen's University of Belfast, United Kingdom}
%\newcommand{\fixme}[1]{{\bf #1}}
-\chapter[Numerical validation and performance optimization on GPUs in atomic physics]{Numerical validation and performance optimization on GPUs of an application in atomic physics}
+\chapter[Numerical validation and GPU performance in atomic physics]{Numerical validation and performance optimization on GPUs of an application in atomic physics}
\label{chapter15}
\section{Introduction}\label{ch15:intro}
performance on
GPUs, applications should be coarse-grained and have a high arithmetic
intensity
-($i.e.$ the ratio of arithmetic operations to memory operations).
+(i.e., the ratio of arithmetic operations to memory operations).
Another important aspect of GPU programming is that floating-point
operations are preferably performed in single precision\index{precision!single precision}, if the
validity of results is not impacted by that format.
The GPU compute power for floating-point operations is indeed greater in
single precision\index{precision!single precision} than in double precision\index{precision!double precision}.
The peak performance ratio between single precision\index{precision!single precision} and double
-precision varies for example for NVIDIA GPUs from $12$ for the first Tesla
+precision varies, for example, for NVIDIA GPUs from $12$ for the first Tesla
GPUs (C1060),
-to $2$ for the Fermi GPUs (C2050 and C2070)
+to $2$ for the Fermi GPUs (C2050 and C2070),
and to $3$ for the latest Kepler architecture (K20/K20X).
As far as AMD GPUs are concerned, the latest AMD GPU (Tahiti HD 7970)
presents a ratio of $4$.
Moreover, GPU internal memory accesses and CPU-GPU data transfers are
-faster in single precision\index{precision!single precision} than in double precision\index{precision!double precision},
+faster in single precision\index{precision!single precision} than in double precision\index{precision!double precision}
because of the different format lengths.
This chapter describes the deployment on GPUs of PROP, a program of the
2DRMP~\cite{FARM_2DRMP,2DRMP} suite which models electron collisions
with H-like atoms and ions at intermediate energies. 2DRMP operates successfully on serial
-computers, high performance clusters and supercomputers. The primary
+computers, high performance clusters, and supercomputers. The primary
purpose of the PROP program is to propagate a global
-R-matrix~\cite{Burke_1987}, $\Re$, in the two-electron configuration
+$R$-matrix~\cite{Burke_1987}, $\Re$, in the two-electron configuration
space.
The propagation needs to be performed for all collision energies,
-for instance hundreds of energies,
+for instance, hundreds of energies,
which are independent.
Propagation equations are dominated by matrix multiplications involving sub-matrices of $\Re$.
However, the matrix multiplications are not
In a preliminary investigation PROP was selected by GENCI\footnote{GENCI: Grand Equipement National
de Calcul Intensif, \url{www.genci.fr}} and
-CAPS\footnote{CAPS is a software company providing products and solutions
- for manycore application programming and deployment,
- \url{www.caps-entreprise.com}},
-following their first call for projects in 2009-2010
+CAPS,\footnote{CAPS is a software company providing products and solutions
+ for many-core application programming and deployment,
+ \url{www.caps-entreprise.com}}
+following their first call for projects in 2009--2010
aimed at
deploying applications on hybrid systems based on GPUs.
First CAPS
with the matrix size, since the
CPU-GPU transfer overhead becomes less significant and since CPUs are
still more efficient for fine computation grains.
-Then, using HMPP\index{HMPP}\footnote{
-HMPP or {\em CAPS compiler}, see: \url{www.caps-entreprise.com/hmpp.html}},
+Then, using HMPP,\index{HMPP}\footnote{
+HMPP (Hybrid Multicore Parallel Programming) or {CAPS compiler}, see: \url{www.caps-entreprise.com/hmpp.html}}
a commercial
hybrid and parallel compiler, CAPS
-developed a version of PROP, in
+developed a version of PROP in
which matrix multiplications are performed on
the GPU or the CPU, depending on the matrix size.
Unfortunately this partial GPU implementation of PROP does not offer
GPUs to avoid the overhead generated by
data transfers
and we propose successive improvements
-(including a specific one to the Fermi architecture)
+(including one specific to the Fermi architecture)
in order to optimize the GPU code.
\section{2DRMP and the PROP program}
\label{s:2DRMP_PROP}
-\subsection{Principles of R-matrix propagation}
-2DRMP~\cite{FARM_2DRMP,2DRMP} is part of the CPC library\footnote{CPC:
+\subsection{Principles of $R$-matrix propagation}
+2DRMP~\cite{FARM_2DRMP,2DRMP} is part of the CPC library.\footnote{CPC:
Computer Physics Communications,
-\url{http://cpc.cs.qub.ac.uk/}}.
+\url{http://cpc.cs.qub.ac.uk/}}
It is a suite of seven
programs aimed at creating virtual experiments on high performance and grid
architectures to enable the study of electron scattering from H-like
\begin{figure}[h]
\begin{center}
\includegraphics*[width=0.8\linewidth]{Chapters/chapter15/figures/Domain.pdf}
-\caption{\label{domain} Propagation of the R-matrix from domain D to domain D'.}
+\caption{\label{domain} Propagation of the $R$-matrix from domain $D$ to domain $D'$.}
\end{center}
\end{figure}
We consider the general situation in
Fig.~\ref{domain} where we assume that we already know
the global $R$-matrix, $\Re^{I}$, associated with the boundary defined
-by edges 5, 2, 1 and 6
+by edges 5, 2, 1, and 6
in domain $D$ and we wish to
-evaluate the new global $R$-matrix, $\Re^{O}$, associated with edges 5, 3, 4 and 6
+evaluate the new global $R$-matrix, $\Re^{O}$, associated with edges 5, 3, 4, and 6
in domain $D'$ following propagation across subregion $d$.
Input edges are denoted by I (edges 1 and~2), output edges by O (edges 3 and 4) and
common edges by X (edges 5 and~6).
Because of symmetry, only the lower half of domains $D$ and $D'$ has to be considered.
The global $R$-matrices, $\Re^{I}$ in domain $D$ and $\Re^{O}$ in
-domain $D'$, can be written as:
+domain $D'$, can be written as
\begin{equation}
\Re^{I} = \left(\begin{array}{cc}
\Re_{II}^{I} & \Re_{IX}^{I}\\
-From the set of local $R$-matrices, $\mathbf{R}_{ij}$ ($i,j\in \{1,2,3,4\}$)
+From the set of local $R$-matrices, $\mathbf{R}_{ij}$ ($i,j\in \{1,2,3,4\}$),
associated
with subregion $d$, we can define
\begin{subequations}
\medskip
-While equations (\ref{eq1})-(\ref{eq4}) can be applied to the
+While equations (\ref{eq1})--(\ref{eq4}) can be applied to the
propagation across a general subregion two special situations should be
noted: propagation across a diagonal subregion and propagation across
a subregion bounded by the $r_{1}$-axis at the beginning of a new
In the case of a diagonal subregion, from symmetry considerations,
edge 2 is identical to edge 1 and edge 3 is identical to edge~4.
Accordingly, with only one input edge and one output edge equations
-(\ref{eqaa})-(\ref{eqdd}) become:
+(\ref{eqaa})--(\ref{eqdd}) become
\begin{subequations}
\begin{eqnarray}
\mathbf{r}_{II} = 2\mathbf{R}_{11}, \
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
- \multirow{2}{0.09\linewidth}{\centering Data set} &
+ \multirow{2}{0.09\linewidth}{\centering Data Set} &
\multirow{2}{0.15\linewidth}{\centering
- Local $R$-\\matrix size} &
+ Local $R$-\\Matrix Size} &
\multirow{2}{0.07\linewidth}{\centering Strips} &
\multirow{2}{0.09\linewidth}{\centering Sectors} &
- \multirow{2}{0.19\linewidth}{\centering Final global \\$R$-matrix size} &
- \multirow{2}{0.15\linewidth}{\centering Scattering\\energies} \\
+ \multirow{2}{0.19\linewidth}{\centering Final Global \\$R$-Matrix Size} &
+ \multirow{2}{0.15\linewidth}{\centering Scattering\\Energies}\\
& & & & & \\
\hline
- Small & 90x90 & 4 & 10 & 360x360 & 6\\
+ Small & $90\times90$ & 4 & 10 & $360\times360$ & 6\\
\hline
- Medium & 90x90 & 4 & 10 & 360x360 & 64\\
+ Medium & $90\times90$ & 4 & 10 & $360\times360$ & 64\\
\hline
- Large & 383x383 & 20 & 210 & 7660x7660 & 6\\
+ Large & $383\times383$ & 20 & 210 & $7660\times7660$ & 6\\
\hline
- Huge & 383x383 & 20 & 210 & 7660x7660 & 64\\ \hline
+ Huge & $383\times383$ & 20 & 210 & $7660\times7660$ & 64\\ \hline
\end{tabular}
-\caption{\label{data-sets}Characteristics of four data sets}
+\caption{\label{data-sets}Characteristics of four data sets.}
\end{center}
\end{table}
From $\Re^{I}$ and local $R$-matrices, compute $\Re^{O}$\;
$\Re^{O}$ becomes $\Re^{I}$ for the next sector\;
}
- Compute physical $R$-Matrix \;
+ Compute physical $R$-Matrix\;
}
%\end{algorithmic}
\end{algorithm}
In the PROP program, sectors are characterized into four types,
depending on the computation performed:
\begin{itemize}
-\item the starting sector (labeled 0 in Fig.~\ref{prop})
-\item the axis sectors (labeled 1, 3 and 6 in Fig.~\ref{prop})
-\item the diagonal sectors (labeled 2, 5 and 9 in Fig.~\ref{prop})
-\item the off-diagonal sectors (labeled 4, 7 and 8 in Fig.~\ref{prop}).
+\item the starting sector (labeled 0 in Fig.~\ref{prop});
+\item the axis sectors (labeled 1, 3, and 6 in Fig.~\ref{prop});
+\item the diagonal sectors (labeled 2, 5, and 9 in Fig.~\ref{prop});
+\item the off-diagonal sectors (labeled 4, 7, and 8 in Fig.~\ref{prop}).
\end{itemize}
-The serial version of PROP is implemented in Fortran~90 and uses
-for linear algebra operations BLAS\index{BLAS} and LAPACK\index{LAPACK} routines
+The serial version of PROP is implemented in Fortran~90 and
+for linear algebra operations uses BLAS\index{BLAS} and LAPACK\index{LAPACK} routines
which are fully optimized for x86 architecture.
This
program
In order to handle larger matrices, and thus obtain better GPU speedup, CAPS
recast equations (\ref{eq1}) to (\ref{eq4}) into one equation.
-The output $R$-matrix $\Re^{O}$ defined by equation~(\ref{eq:RI_RO}) is now computed as follows.
+The output $R$-matrix $\Re^{O}$ defined by equation~(\ref{eq:RI_RO}) is now computed as follows:
\begin{equation}\label{eq_CAPS_1}
\Re^{O} = \Re^{O^{\ \prime}} + U A^{-1} V,
\end{equation}
This reimplementation of PROP reduces the number of equations to be
solved and the number of matrix copies for evaluating each sector.
For instance, for an off-diagonal sector,
-copies fall from 22 to 5, matrix multiplications from 4 to~1 and calls
+copies fall from 22 to 5, matrix multiplications from 4 to~1, and calls
to a linear equation solver from 2 to 1.
To implement this version, CAPS
used HMPP\index{HMPP}, a
commercial hybrid and parallel compiler,
-based on compiler directives like the new OpenACC\index{OpenACC} standard\footnote{See: \url{www.openacc-standard.org}}.
+based on compiler directives such as the new OpenACC\index{OpenACC} standard.\footnote{See: \url{www.openacc-standard.org}}
If the matrices are large enough (the limit sizes are experimental parameters),
they are multiplied on the GPU, otherwise on the CPU.
CAPS
- used the MKL BLAS\index{BLAS} implementation on an Intel Xeon
+ used Intel's MKL (Math Kernel Library) BLAS\index{BLAS} implementation on an Intel Xeon
x5560 quad core CPU (2.8 GHz)
and the CUBLAS\index{CUBLAS} library (CUDA 2.2) on one Tesla C1060 GPU.
On the large data set (see Table~\ref{data-sets}), CAPS
obtained a speedup of 1.15 for the GPU
-version over the CPU one (with multi-threaded MKL calls on the four
+version over the CPU one (with multithreaded MKL calls on the four
CPU cores). This limited gain in performance is mainly
due to the use of double precision\index{precision!double precision} computation
and to the small or medium sizes of most matrices.
For these matrices, the computation gain on
-the GPU is indeed
+the GPU is indeed
strongly affected by the overhead
generated by transferring these matrices from
the CPU memory to the GPU memory to perform each matrix multiplication and then
transferring the result back to the CPU memory.
-Our goal is to speedup PROP more significantly by porting the whole
+Our goal is to speed up PROP more significantly by porting the whole
code to the GPU and therefore avoiding
the
intermediate data transfers between
the host (CPU) and the GPU. We will also study the
stability of PROP in single precision\index{precision!single precision} because
-single precision\index{precision!single precision} computation is faster on the GPU
+single-precision\index{precision!single precision} computation is faster on the GPU
and CPU-GPU data transfers are twice as fast as those performed in
double precision\index{precision!double precision}.
\end{comment}
-Floating-point input data, computation and output data of PROP are
-originally in double precision\index{precision!double precision} format.
+Floating-point input data, computation, and output data of PROP are
+originally in double-precision\index{precision!double precision} format.
PROP produces a standard $R$-matrix H-file \cite{FARM_2DRMP}
- and a collection of Rmat00X files (where X
-ranges from 0 to the number of scattering energies - 1)
-holding the physical R-matrix for each
+ and a collection of {Rmat}00X files (where X
+ranges from 0 to the number of scattering energies $-$ 1)
+holding the physical $R$-matrix for each
energy.
-The H-file and the Rmat00X files are binary input files of the FARM program \cite{FARM_2DRMP}
+The H-file and the {Rmat}00X files are binary input files of the FARM program \cite{FARM_2DRMP}
(last program of the 2DRMP suite).
-Their text equivalent are the prop.out
+Their text equivalents are the prop.out
and the prop00X.out files.
To study the validity of PROP results in single precision\index{precision!single precision},
first,
\begin{figure}[h]
\begin{center}
\includegraphics*[width=0.9\linewidth]{Chapters/chapter15/figures/error.pdf}
-\caption{\label{fig:sp-distrib} Error distribution for medium case in single precision\index{precision!single precision}}
+\caption{\label{fig:sp-distrib} Error distribution for medium case in single precision.\index{precision!single precision}}
\end{center}
\end{figure}
given in Fig.~\ref{fig:sp-distrib}.
We focus on the largest errors.
\begin{itemize}
-\item Errors greater than 100: the only impacted value is of order 1.E-6
+\item Errors greater than $10^{2}$: %100:
+the only impacted value is of order $10^{-6}$ %1.E-6
and is negligible compared to the other ones
in the same prop00X.out file.
-\item Errors between 1 and 100: the values corresponding to the
- largest errors are of order 1.E-3 and are negligible compared to
- the majority of the other values which range between 1.E-2 and
- 1.E-1.
+\item Errors between 1 and $10^{2}$: %100:
+the values corresponding to the
+ largest errors are of order $10^{-3}$ %1.E-3
+and are negligible compared to
+ the majority of the other values which range between $10^{-2}$ %1.E-2
+and $10^{-1}$.
-\item Errors between 1.E-2 and 1: the largest errors ($\ge$ 6\%)
- impact values the order of magnitude of which is at most 1.E-1.
+\item Errors between $10^{-2}$ %1.E-2
+and 1: the largest errors ($\ge$ 6\%)
+ impact values the order of magnitude of which is at most $10^{-1}$. %1.E-1.
These values are negligible.
Relative errors of approximately 5\% impact values the order of
- magnitude of which is at most 1.E2.
+ magnitude of which is at most $10^{2}$. %1.E2.
For instance, the value 164 produced by the reference version of
- PROP becomes 172 in the single precision\index{precision!single precision} version.
-
+ PROP becomes 172 in the single precision\index{precision!single precision} version.
\end{itemize}
To study the impact of the single precision\index{precision!single precision} version of PROP on the
Table~\ref{sp-farm} shows that all cross-section files are impacted by
errors. Indeed in the {2p4d} file, four relative errors are
greater than one and the maximum relative error is 1.60.
-However the largest errors impact negligible values. For example, the maximum
-error (1.60) impacts a reference value which is 4.5E-4. The largest
+However, the largest errors impact negligible values. For example, the maximum
+error (1.60) impacts a reference value which is $4.5\ 10^{-4}$. %4.5E-4.
+The largest
values are impacted by low errors. For instance, the maximum value
-(1.16) is impacted by a relative error of the order 1.E-3.
+(1.16) is impacted by a relative error of the order $10^{-3}$. %1.E-3.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c||c|c|} \hline
- file & largest relative error & file & largest relative error\\ \hline
+ File & Largest Relative Error & File & Largest Relative Error\\ \hline
{1s1s} & 0.02& {1s3p} & 0.11 \\ \hline
{1s2s} & 0.06 & {1s3d} & 0.22 \\ \hline
{1s2p} & 0.08 & {1s4s} & 0.20 \\ \hline
{1s3s} & 0.17 &2p4d & 1.60 \\ \hline
\end{tabular}
-\caption{\label{sp-farm}Impact on FARM of the single precision\index{precision!single precision} version of PROP}
+\caption{\label{sp-farm}Impact on FARM of the single precision\index{precision!single precision} version of PROP.}
\end{center}
\end{table}
To examine in more detail the impact of PROP on FARM,
-cross sections above the ionization threshold (1 Ryd)
+cross-sections above the ionization threshold (1 Ryd)
are compared in single and
double precision\index{precision!double precision} for
-transitions amongst the 1s, \dots 4s, 2p, \dots 4p, 3d, 4d target states.
+transitions among the 1s, \dots 4s, 2p, \dots 4p, 3d, 4d target states.
This comparison is carried out by generating 45 plots. In all the
- plots, results in single and double precision\index{precision!double precision} match except for few
+ plots, results in single and double precision\index{precision!double precision} match except for a few
scattering energies which are very close to pseudo-state thresholds.
-For example Fig.~\ref{1s2p} and \ref{1s4d} present the scattering energies corresponding to the
+For example, Fig.~\ref{1s2p} and \ref{1s4d} present the scattering energies corresponding to the
{1s2p} and {1s4d} cross-sections computed in single and double precision\index{precision!double precision}. For some cross-sections,
-increasing a threshold parameter from 1.E-4 to 1.E-3 in the FARM
-program
-results in energies close to threshold being avoided
- and therefore
+increasing a threshold parameter from $10^{-4}$ %1.E-4
+to $10^{-3}$ %1.E-3
+in the FARM
+program
+results in energies close to threshold being avoided,
+ and therefore,
the cross-sections in double and single precision\index{precision!single precision} match more
accurately.
This is the case for instance for cross-section 1s2p (see Fig.~\ref{1s2p3}).
However for other cross-sections (such as 1s4d) some problematic energies remain even if the
threshold parameter in the FARM
-program is increased to 1.E-3 (see Fig.~\ref{1s4d3}). A higher
+program is increased to $10^{-3}$ %1.E-3
+ (see Fig.~\ref{1s4d3}). A higher
threshold parameter would be required for such cross-sections.
\begin{figure}[t]
\centering
-\subfigure[threshold = 1.E-4]{
+\subfigure[threshold = $10^{-4}$]{ %1.E-4]{
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s2p.pdf}
\label{1s2p}
}
-\subfigure[threshold = 1.E-3]{
+\subfigure[threshold = $10^{-3}$]{ %1.E-3]{
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s2p3.pdf}
\label{1s2p3}
}
\label{fig:1s2p_10sectors}
-\caption{1s2p cross-section, 10 sectors}
+\caption{1s2p cross-section, 10 sectors.}
\end{figure}
\begin{figure}[t]
\centering
-\subfigure[threshold = 1.E-4]{
+\subfigure[threshold = $10^{-4}$]{ %1.E-4]{
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s4d.pdf}
\label{1s4d}
}
-\subfigure[threshold = 1.E-3]{
+\subfigure[threshold = $10^{-3}$]{ %1.E-3]{
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s4d3.pdf}
\label{1s4d3}
}
\label{fig:1s4d_10sectors}
-\caption{1s4d cross-section, 10 sectors}
+\caption{1s4d cross-section, 10 sectors.}
\end{figure}
As a conclusion, the medium case study shows that the execution of
be computed by the FARM program for some cross-sections.
Thanks to a suitable threshold parameter in the FARM program these problematic energies may possibly
be skipped.
-Instead of investigating deeper the choice of such a parameter for the medium case, we analyze the
-single precision\index{precision!single precision} computation in a more
+Instead of investigating more deeply the choice of such a parameter for the medium case, we analyze the
+single-precision\index{precision!single precision} computation in a more
realistic case in Sect.~\ref{huge}.
\begin{comment}
The conclusion of the medium case study is that running PROP in single
\begin{figure}[t]
\centering
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s2pHT.pdf}
-\caption{\label{1s2pHT}1s2p cross-section, threshold = 1.E-4, 210 sectors}
+\caption{\label{1s2pHT}1s2p cross-section, threshold = $10^{-4}$, 210 sectors.} %1.E-4, 210 sectors.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s2pHT.pdf}
-\caption{\label{1s4dHT}1s4d cross-section, threshold = 1.E-4, 210 sectors}
+\caption{\label{1s4dHT}1s4d cross-section, threshold = $10^{-4}$, 210 sectors.} %1.E-4, 210 sectors.}
\end{figure}
We study here the impact on FARM of the PROP program run in
single precision\index{precision!single precision} for the huge case (see Table~\ref{data-sets}).
-The cross-sections
+The cross-sections
corresponding to all
atomic target states 1s \dots 7i are explored, which
leads to
As expected, all the plots exhibit large differences between single and double
precision cross-sections.
For example Fig.~\ref{1s2pHT} and \ref{1s4dHT} present the 1s2p and 1s4d cross-sections computed in
-single and double precision\index{precision!double precision} for the huge case.
+single---and double---precision\index{precision!double precision} for the huge case.
We can conclude that PROP in single precision\index{precision!single precision} gives invalid results
for realistic simulation cases above the ionization threshold.
-Therefore the deployment of PROP on GPU, described in Sect.~\ref{gpu-implem},
+Therefore, the deployment of PROP on GPU, described in Sect.~\ref{gpu-implem},
has been carried out in double precision\index{precision!double precision}.
\section{Towards a complete deployment of PROP on GPUs}
\end{figure}
As mentioned in Algorithm~\ref{prop-algo}, evaluating a sector
-mainly consists in constructing local $R$-matrices and in computing
+mainly consists of constructing local $R$-matrices and computing
one output $R$-matrix, $\Re^{O}$. In this first step of the porting
process, referred to as GPU V1\label{gpuv1},
-we only consider the computation of $\Re^{O}$ on the GPU.
+we consider only the computation of $\Re^{O}$ on the GPU.
We distinguish the following six steps, related to equations
-(\ref{eq_CAPS_1}), (\ref{eq_CAPS_2}) and (\ref{eq_CAPS_3}), and illustrated in
+(\ref{eq_CAPS_1}), (\ref{eq_CAPS_2}), and (\ref{eq_CAPS_3}), and illustrated in
Fig.~\ref{offdiagonal} for an off-diagonal sector.
\begin{description}
to temporary arrays ($A$, $U$, $V$) and to $\Re^{O}$.
These copies, along with possible scalings or transpositions, are
implemented as CUDA kernels which can be applied to two
- matrices of any size and starting at any offset.
+ matrices of any size starting at any offset.
Memory accesses are coalesced\index{coalesced memory accesses} \cite{CUDA_ProgGuide} in order to
provide the best performance for such memory-bound kernels.
\item[Step 2] (``Local copies''):~data are copied from
\item[Step 3] (``Linear system solving''):~matrix $A$ is factorized
using the MAGMA DGETRF\index{MAGMA functions!DGETRF}
routine and the result is stored in-place.
-\item[Step 4] (``Linear system solving'' cont.):~the matrix system
+\item[Step 4] (``Linear system solving,'' cont.):~the matrix system
of linear equations $AW$ = $V$ is solved using the MAGMA DGETRS\index{MAGMA functions!DGETRS}
routine. The solution is stored in matrix $V$.
\item[Step 5] (``Output matrix product''):~matrix $U$
\subsection{Constructing the local $R$-matrices on GPU}
-\begin{figure}[t]
+\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{Chapters/chapter15/figures/amplitudes_nb.pdf}
\caption{\label{amplitudes} Constructing the local $R$-matrix R34
amplitude array associated with edge~3.}
\end{figure}
-Local $R$-matrices are constructed using two three dimensional arrays,
-$i$ and $j$. Each three dimensional array contains four
+Local $R$-matrices are constructed using two three-dimensional arrays,
+$i$ and $j$. Each three-dimensional array contains four
matrices corresponding to the surface amplitudes associated with the
four edges of a sector. Those matrices are named {\em amplitude arrays}.
$j$ amplitude arrays are read from data files and $i$ amplitude arrays
the GPU (via CUBLAS\index{CUBLAS} routines).
-The involved matrices having medium sizes (either $3066 \times 383$ or
-$5997 \times 383$),
+The involved matrices have medium sizes (either $3066 \times 383$ or
+$5997 \times 383$), and
performing these matrix multiplications
on the GPU is expected to be faster than on the CPU.
However, this implies a greater communication volume
each sector by transferring only this 1D array of scaling factors.
Moreover, scaling $j$ amplitude arrays is expected to be faster on the
GPU than on the CPU, thanks to the massively parallel architecture of
-the GPU and thanks to its higher internal memory bandwidth.
+the GPU and its higher internal memory bandwidth.
\subsection{Using double-buffering\index{double-buffering} to overlap I/O and computation}
The I/O times are roughly constant among all strips.
The evaluation time is equivalent to the I/O
time for the first sectors. But this evaluation time grows
-linearly with the strip number, and rapidly exceeds the I/O
+linearly with the strip number and rapidly exceeds the I/O
time.
It is thus interesting to use a double-buffering\index{double-buffering} technique to overlap the
\subsection{Matrix padding\index{padding}}
-The CUBLAS DGEMM
-performance and the MAGMA DGETRF\index{MAGMA functions!DGETRF}/DGETRS\index{MAGMA functions!DGETRS}
-performance is reduced when the sizes (or
+The MAGMA DGETRF\index{MAGMA functions!DGETRF}/DGETRS\index{MAGMA functions!DGETRS}
+performance and the CUBLAS DGEMM performance
+are reduced when the sizes (or
the leading dimensions) of the matrix are not multiples of the inner blocking size \cite{NTD10a}.
This inner blocking size can be 32 or 64, depending on the computation
and on the underlying
so that their sizes are
multiples of 64.
This corresponds indeed to the optimal size for the matrix product on the
-Fermi architecture \cite{NTD10b}. And as far as linear system solving is
+Fermi architecture \cite{NTD10b}. And as far as linear system solving is
concerned, all the matrices have sizes which are multiples of 383: we
therefore use padding\index{padding} to obtain multiples of 384 (which are
again multiples of 64).
It can be noticed that this padding\index{padding} has to be performed dynamically
as the matrices increase in size during the propagation
(when possible the
- maximum required storage space is however allocated only once in the
+ maximum required storage space is, however, allocated only once in the
GPU memory).
-
-
\section{Performance results}
\subsection{PROP deployment on GPU}
-\begin{table*}[ht]
+\begin{table}[ht]
\begin{center}
\begin{tabular}{|c||c|c||}
\hline
- PROP version & \multicolumn{2}{c|}{Execution time} \\
+ PROP Version & \multicolumn{2}{c|}{Execution Time} \\
\hline
\hline
- CPU version: 1 core & \multicolumn{2}{c|}{201m32s} \\
+ CPU Version: 1 Core & \multicolumn{2}{c|}{201m32s} \\
\hline
- CPU version: 4 cores & \multicolumn{2}{c|}{113m28s} \\
+ CPU Version: 4 Cores & \multicolumn{2}{c|}{113m28s} \\
\hline \hline
-GPU version & C1060 & C2050 \\
+GPU Version & C1060 & C2050 \\
\hline\hline
GPU V1 (\S~\ref{gpuv1}) & 79m25s & 66m22s \\
\hline
GPU V5 (\S~\ref{gpuv5}) & 24m27s & 12m39s \\
\hline
\end{tabular}
-\caption{\label{table:time}
-Execution time of PROP on CPU and GPU}
\end{center}
-\end{table*}
+\caption{Execution time of PROP on CPU and GPU.}
+\label{table:time}
+\end{table}
-\begin{comment}
-\begin{table*}[ht]
-\begin{center}
-\begin{tabular}{|c||c|c||}
- \hline
- PROP version & \multicolumn{2}{c|}{Execution time} \\
- \hline \hline
-CPU version & 1 core & 4 cores \\\hline
-& {201m32s} & {113m28s} \\ \hline \hline
-GPU version & C1060 & C2050 \\
- \hline\hline
- GPU V1 (\ref{gpuv1}) & 79m25s & 66m22s \\
- \hline
- GPU V2 (\ref{gpuv2}) & 47m58s & 29m52s \\
- \hline
- GPU V3 (\ref{gpuv3}) & 41m28s & 23m46s \\
- \hline
- GPU V4 (\ref{gpuv4}) & 27m21s & 13m55s\\
- \hline
- GPU V5 (\ref{gpuv5}) & 24m27s & 12m39s \\
- \hline
-\end{tabular}
-\caption{\label{table:time}
-Execution time of the successive GPU versions}
-\end{center}
-\end{table*}
-\end{comment}
+
+%% \begin{table}[ht]
+%% \begin{center}
+%% \begin{tabular}{|c||c|c||}
+%% \hline
+%% PROP version & \multicolumn{2}{c|}{Execution time} \\
+%% \hline \hline
+%% CPU version & 1 core & 4 cores \\\hline
+%% & {201m32s} & {113m28s} \\ \hline \hline
+%% GPU version & C1060 & C2050 \\
+%% \hline\hline
+%% GPU V1 (\ref{gpuv1}) & 79m25s & 66m22s \\
+%% \hline
+%% GPU V2 (\ref{gpuv2}) & 47m58s & 29m52s \\
+%% \hline
+%% GPU V3 (\ref{gpuv3}) & 41m28s & 23m46s \\
+%% \hline
+%% GPU V4 (\ref{gpuv4}) & 27m21s & 13m55s\\
+%% \hline
+%% GPU V5 (\ref{gpuv5}) & 24m27s & 12m39s \\
+%% \hline
+%% \end{tabular}
+%% \end{center}
+%% \caption{Execution time of the successive GPU versions}
+%% \label{table:time}
+%% \end{table}
\begin{figure}[h]
\centering
of PROP on CPUs and GPUs,
each version solves the propagation equations in the
form~(\ref{eq_CAPS_1}-\ref{eq_CAPS_3}) as proposed by CAPS.
-Fig.~\ref{fig:speedup_1core} (respectively \ref{fig:speedup_4cores})
+Figure~\ref{fig:speedup_1core} (respectively, \ref{fig:speedup_4cores})
shows the speedup of the successive GPU versions
-over one CPU core (respectively four CPU cores).
-We use here Intel Q8200 quad-core CPUs (2.33 GHz), one C1060 GPU and
+over one CPU core (respectively, four CPU cores).
+We use here Intel Q8200 quad-core CPUs (2.33 GHz), one C1060 GPU, and
one C2050 (Fermi) GPU, located at
UPMC (Universit\'e Pierre et Marie Curie, Paris, France).
As a remark, the execution times measured on the C2050 would be the same
(e.g. the huge case) should be proportional
to those reported in Table~\ref{table:time}.
-These tests, which have been performed with CUDA 3.2, CUBLAS\index{CUBLAS} 3.2 and
+These tests, which have been performed with CUDA 3.2, CUBLAS\index{CUBLAS} 3.2, and
MAGMA 0.2,
show that the successive GPU versions of PROP offer
increasing, and at the end interesting, speedups.
does offer much more performance gain with,
for example, CUDA 3.1 and the MAGMA DGEMM\index{MAGMA functions!DGEMM}~\cite{NTD10b}: the
speedup with respect to one
-CPU core was increased from 6.3 to 8.1 on C1060, and from 9.5 to 14.3
+CPU core was increased from 6.3 to 8.1 on C1060 and from 9.5 to 14.3
on C2050.
-Indeed since CUBLAS\index{CUBLAS} 3.2 performance has been improved for non block multiple
-matrix sizes through MAGMA code~\cite{NTD10a}.
+Indeed CUBLAS\index{CUBLAS} 3.2 performance has been improved through MAGMA code %~\cite{NTD10a}.
+%for non block multiple matrix sizes through MAGMA code~\cite{NTD10a}.
+for matrix sizes which are not multiples of the inner blocking size~\cite{NTD10a}.
Although for all versions the C2050 (with its improved
double precision\index{precision!double precision} performance) offers up to almost
double speedup compared to
We detail here the execution profile on
the CPU and the GPU for the evaluation of all off-diagonal sectors
(the most representative ones) for a complete energy propagation.
- Fig.~\ref{fig:CPU-timing} and \ref{fig:profileGPU} show CPU and GPU execution times for the
+ Figures~\ref{fig:CPU-timing} and \ref{fig:profileGPU} show CPU and GPU execution times for the
171 off-diagonal sectors of the large case (see Table \ref{data-sets}).
``Copying, adding, scaling'' corresponds to the amplitude
- array construction (scaling) as well as to steps 1, 2 and 6 in
- Sect.~\ref{gpu-RO}, all implemented via CUDA kernels.
-``CPU-GPU transfers''
-aggregate transfer times for the $j$ amplitude
-arrays and the scaling factors, as well as for the correction data.
+ array construction (scaling) as well as to Steps 1, 2, and 6 in
+ Sect.~\ref{gpu-RO}, all implemented via CUDA kernels.
``Amplitude matrix product'' corresponds to the DGEMM call to
- construct the local R-matrices from the $i$ and $j$ amplitude
+ construct the local $R$-matrices from the $i$ and $j$ amplitude
arrays.
``Linear system solving'' and ``Output matrix product'' correspond
-respectively to steps 3-4 and to step 5 in Sect.~\ref{gpu-RO}.
+respectively to steps 3-4 and to step 5 in Sect.~\ref{gpu-RO}.
+``CPU-GPU transfers'' in Fig.~\ref{fig:profileGPU}
+aggregate transfer times for the $j$ amplitude
+arrays and the scaling factors, as well as for the correction data.
+
+
On one CPU core (see Fig.~\ref{fig:CPU-timing}),
matrix products for the construction of the local
matrix products for the output $R$-matrix is constant within each
strip. But as the global $R$-matrix is propagated from strip to
strip, the sizes of
-the matrices $U$ and $V$ increase, so does their multiplication time.
+the matrices $U$ and $V$ increase, and so does their multiplication time.
The time required to solve the linear system increases
slightly during the propagation.
-These three operations (``Amplitude matrix product'', ``Output matrix
-product'' and ``Linear system solving'') are clearly dominant in terms
+These three operations (``Amplitude matrix product,'' ``Output matrix
+product,'' and ``Linear system solving'') are clearly dominant in terms
of computation
time compared to the other remaining operations, which justify our
-primary focus on such three linear algebra operations.
+primary focus on these three linear algebra operations.
On the C1060 (see Fig.~\ref{GPU-timing}), we have
API} of CUBLAS\index{CUBLAS}~\cite{CUBLAS} (like MAGMA)
without thread safety problems.
A {\em breadth first} issue order is used for kernel
-launchs \cite{CUDA_stream}: for a given GPU kernel, all kernel launchs
+launches \cite{CUDA_stream}: for a given GPU kernel, all kernel launchs
are indeed issued together in the host thread, using one stream for each
concurrent energy, in order to maximize concurrent kernel
execution\index{concurrent kernel execution}.
in order to enable concurrent executions among the required kernels.
-\begin{table}[t]
+\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c||c|c|c|c|c|}
\hline
- \multirow{4}{0.18\linewidth}{Medium case} & Number of &
+ \multirow{4}{0.18\linewidth}{Medium Case} & Number of &
\multirow{2}{0.07\linewidth}{\centering 1} &
\multirow{2}{0.07\linewidth}{\centering 2} &
\multirow{2}{0.07\linewidth}{\centering 4} &
\multirow{2}{0.07\linewidth}{\centering 8} &
\multirow{2}{0.07\linewidth}{\centering 16} \\
- & energies & & & & & \\
+ & Energies & & & & & \\
\cline{2-7}
& Time (s) & 11.18 & 6.87 & 5.32 & 4.96 & 4.76 \\
\cline{2-7}
& Speedup & - & 1.63 & 2.10 & 2.26 & 2.35 \\
\hline
\hline
- \multirow{4}{0.18\linewidth}{Large case} & Number of &
+ \multirow{4}{0.18\linewidth}{Large Case} & Number of &
\multirow{2}{0.07\linewidth}{\centering 1} &
\multicolumn{2}{c|}{\multirow{2}{0.07\linewidth}{\centering 2}} &
\multicolumn{2}{c|}{\multirow{2}{0.07\linewidth}{\centering 3}} \\
- & energies & & \multicolumn{2}{c|}{~} & \multicolumn{2}{c|}{~} \\
+ & Energies & & \multicolumn{2}{c|}{~} & \multicolumn{2}{c|}{~} \\
\cline{2-7}
& Time (s) & 509.51 & \multicolumn{2}{c|}{451.49} & \multicolumn{2}{c|}{436.72} \\
\cline{2-7}
kernels\index{concurrent kernel execution} is also limited by other operations on the
GPU \cite{CUDA_ProgGuide,CUDA_stream} and by the current MAGMA code which
prevents concurrent MAGMA calls in different streams.
-Better speedups can be here expected on the latest Kepler GPUs which
+Better speedups can be expected on the latest Kepler GPUs which
offer additional compute power, and whose {\em Hyper-Q} feature may help
improve further the GPU utilization with concurrent energies.
-On the contrary, the same code on the C1060 shows no speedup
+To the contrary, the same code on the C1060 shows no speedup
since the concurrent kernel launches are serialized on this previous GPU architecture.
therefore deployed the PROP code in double precision\index{precision!double precision} on
a GPU, with successive improvements. The different GPU versions
each offer increasing speedups over the CPU version.
-Compared to the single (respectively four) core(s) CPU version, the
+Compared to the single (respectively, four) core(s) CPU version, the
optimal GPU implementation
-gives a speedup of 8.2 (resp. 4.6) on one C1060 GPU,
-and a speedup of 15.9 (resp. 9.0) on one
-C2050 GPU with improved double precision\index{precision!double precision} performance.
+gives a speedup of 8.2 (resp., 4.6) on one C1060 GPU,
+and a speedup of 15.9 (resp., 9.0) on one
+C2050 GPU with improved double-precision\index{precision!double precision} performance.
An additional gain of around 15\%
can also be obtained on one Fermi GPU
with large memory (C2070) thanks to concurrent kernel execution.
different and the CUBLAS\index{CUBLAS} versions are more recent.
However, the programming effort required
progressively to deploy PROP on GPUs clearly offers improved and interesting speedups for this
-real-life application in double precision\index{precision!double precision} with varying-size matrices.
+real-life application in double precision\index{precision!double precision} with varying-sized matrices.
We are currently working on a hybrid CPU-GPU version that spreads the
computations of the independent energies on both the CPU
and the GPU. This will enable
multiple energy execution on the CPU, with
-one or several core(s) dedicated to each energy (via multi-threaded
+one or several core(s) dedicated to each energy (via multithreaded
BLAS\index{BLAS} libraries). Multiple
concurrent energies may also be propagated on each Fermi GPU.
By merging this work with the current MPI PROP program, we will
obtain a scalable hybrid CPU-GPU version.
-This final version will offer an additional level of parallelism
+This final version will offer an additional level of parallelism,
thanks to the MPI
-standard in order to exploit multiple
+standard, in order to exploit multiple
nodes with multiple CPU cores and possibly multiple GPU cards.
\putbib[Chapters/chapter15/biblio]
