new

[book_gpu.git] / BookGPU / Chapters / chapter15 / ch15.tex

diff --git a/BookGPU/Chapters/chapter15/ch15.tex b/BookGPU/Chapters/chapter15/ch15.tex
index 1b3a58062544cdb21b2b1b2b73d2fd942f040eb4..7e25220e38354c0472ab23827a8bf24f8ecbb005 100644 (file)
--- a/BookGPU/Chapters/chapter15/ch15.tex
+++ b/BookGPU/Chapters/chapter15/ch15.tex
@@ -1,5 +1,5 @@
 \chapterauthor{Rachid Habel}{T\'el\'ecom SudParis, France}
 \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{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{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}


@@ -20,32 +20,32 @@ data transfers between CPU memory and GPU memory. Thus, to have good
 performance on
 GPUs, applications should be coarse-grained and have a high arithmetic
 intensity 
 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
 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), 
 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
 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
 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
 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, 
 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
 which are independent.
 Propagation equations are dominated by matrix multiplications involving sub-matrices of $\Re$.
 However, the matrix multiplications are not


@@ -54,10 +54,10 @@ columns and increases in size as the propagation proceeds \cite{VECPAR}.
 
 In a preliminary investigation PROP was selected by GENCI\footnote{GENCI: Grand Equipement National
   de Calcul Intensif, \url{www.genci.fr}} and
 
 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  
 aimed at
 deploying applications on hybrid systems based on GPUs.
 First CAPS  


@@ -66,11 +66,11 @@ For matrix products the GPU performance gain over CPU increases indeed
 with the matrix size, since the 
 CPU-GPU transfer overhead becomes less significant and since CPUs are
 still more efficient for fine computation grains. 
 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 
 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
 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


@@ -86,7 +86,7 @@ computation code of PROP on
 GPUs to avoid the overhead generated by
 data transfers 
 and we propose successive improvements
 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.
 
 
 in order to optimize the GPU code.
 
 


@@ -94,10 +94,10 @@ in order to optimize the GPU code.
 
 \section{2DRMP and the PROP program}
 \label{s:2DRMP_PROP}
 
 \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, 
 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
 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


@@ -122,22 +122,22 @@ $R$-matrix, $\Re$, from sector to sector across the internal region, as shown in
 \begin{figure}[h]
 \begin{center}
 \includegraphics*[width=0.8\linewidth]{Chapters/chapter15/figures/Domain.pdf} 
 \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
 \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
 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
 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}\\
 \begin{equation}
 \Re^{I} = \left(\begin{array}{cc}
       \Re_{II}^{I} & \Re_{IX}^{I}\\


@@ -153,7 +153,7 @@ domain $D'$, can be written as:
 
 
 
 
 
 

-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}
 associated 
 with subregion $d$, we can define
 \begin{subequations}


@@ -195,7 +195,7 @@ two linear systems are solved.
 
 \medskip
 
 
 \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
 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


@@ -204,7 +204,7 @@ strip.
 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
 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{subequations}
 \begin{eqnarray}
 \mathbf{r}_{II} = 2\mathbf{R}_{11}, \ 


@@ -237,24 +237,24 @@ diagonal.
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|}
  \hline
 \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
  \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}{0.07\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
   & & & & & \\
   \hline

-  Small & 90x90  & 4 & 10 & 360x360 & 6\\
+  Small & $90\times90$  & 4 & 10 & $360\times360$ & 6\\

  \hline
  \hline

-  Medium  & 90x90  & 4 & 10 & 360x360 & 64\\
+  Medium  & $90\times90$  & 4 & 10 & $360\times360$ & 64\\

  \hline
  \hline

-  Large  & 383x383  & 20 & 210 &  7660x7660 & 6\\
+  Large  & $383\times383$  & 20 & 210 &  $7660\times7660$ & 6\\

  \hline
  \hline

-  Huge  & 383x383  & 20 & 210 &  7660x7660 & 64\\ \hline
+  Huge  & $383\times383$  & 20 & 210 &  $7660\times7660$ & 64\\ \hline

 \end{tabular}
 \end{tabular}

-\caption{\label{data-sets}Characteristics of four data sets}
+\caption{\label{data-sets}Characteristics of four data sets.}

 \end{center}
 \end{table}
 
 \end{center}
 \end{table}
 


@@ -303,7 +303,7 @@ for the next evaluation.
   From $\Re^{I}$ and local $R$-matrices, compute $\Re^{O}$\;
  $\Re^{O}$ becomes $\Re^{I}$ for the next sector\;
  }
   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}
 }
 %\end{algorithmic}
 \end{algorithm}


@@ -318,15 +318,15 @@ $R$-matrix is computed and stored into an output file.
 In the PROP program, sectors are characterized into four types,
 depending on the computation performed: 
 \begin{itemize}
 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}
 
 
 \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 
 which are fully optimized for x86 architecture.
 This 
 program 


@@ -348,7 +348,7 @@ MPI.
 
 In order to handle larger matrices, and thus obtain better GPU  speedup, CAPS  
 recast equations (\ref{eq1}) to (\ref{eq4}) into one equation.
 
 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}
 \begin{equation}\label{eq_CAPS_1}
 \Re^{O} = \Re^{O^{\ \prime}} + U A^{-1} V, 
 \end{equation}


@@ -371,39 +371,39 @@ system $AW=V$ is solved.
 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, 
 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, 
 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 
 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 
 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  
 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. 
 
 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 
 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}. 
 
 and CPU-GPU data transfers are twice as fast as those performed in
 double precision\index{precision!double precision}. 
 


@@ -440,16 +440,16 @@ double precision\index{precision!double precision}.
 \end{comment}
 
 
 \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}
 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.
 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).
 (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,
 and the prop00X.out files. 
 To study the validity of PROP results in  single precision\index{precision!single precision},
 first,


@@ -465,7 +465,7 @@ into files in double precision\index{precision!double precision}.
 \begin{figure}[h]
 \begin{center}
 \includegraphics*[width=0.9\linewidth]{Chapters/chapter15/figures/error.pdf} 
 \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}
 
 \end{center}
 \end{figure}
 


@@ -477,23 +477,26 @@ error distribution is
 given in Fig.~\ref{fig:sp-distrib}. 
 We focus on the largest errors. 
 \begin{itemize}
 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.
 
 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
   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
   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
 \end{itemize}
 
 To study the impact of the single precision\index{precision!single precision} version of PROP on the


@@ -506,72 +509,76 @@ transitions
 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. 
 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
 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
 
 \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}
  {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, 
 \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 
 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
 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. 
  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, 
 {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
 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
 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}
  }
 \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}
 \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
 \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}
  }
 \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}
 \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
 \end{figure}
 
 As a conclusion, the medium case study shows that the execution of


@@ -579,8 +586,8 @@ PROP in single precision\index{precision!single precision} leads to a few inexac
 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. 
 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
 realistic case in Sect.~\ref{huge}. 
 \begin{comment}
 The conclusion of the medium case study is that running PROP in single


@@ -598,18 +605,18 @@ realistic case in Sect.~\ref{huge}.
 \begin{figure}[t] 
   \centering
 \includegraphics*[width=.76\linewidth]{Chapters/chapter15/figures/1s2pHT.pdf}
 \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}
 \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}).
 \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 
 corresponding to all
 atomic target states 1s \dots 7i are explored, which
 leads to 


@@ -618,10 +625,10 @@ It should be noted that in this case, over the same energy range above the ioniz
 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
 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.
 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} 
 has been carried out in double precision\index{precision!double precision}. 
 
 \section{Towards a complete deployment of PROP on GPUs} 


@@ -649,12 +656,12 @@ enable calls to CUDA kernels from PROP routines.
 \end{figure}
 
 As mentioned in Algorithm~\ref{prop-algo}, evaluating a sector
 \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},
 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
 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}
 Fig.~\ref{offdiagonal} for an off-diagonal sector.
 
 \begin{description}


@@ -662,7 +669,7 @@ Fig.~\ref{offdiagonal} for an off-diagonal sector.
   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
   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
   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


@@ -674,7 +681,7 @@ Fig.~\ref{offdiagonal} for an off-diagonal sector.
 \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 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 $U$
  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 $U$


@@ -693,7 +700,7 @@ symmetry considerations \cite{2DRMP}.
 
 \subsection{Constructing the local $R$-matrices on GPU}
 
 
 \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
  \centering
   \includegraphics[width=0.7\linewidth]{Chapters/chapter15/figures/amplitudes_nb.pdf} 
  \caption{\label{amplitudes} Constructing the local $R$-matrix R34


@@ -701,8 +708,8 @@ symmetry considerations \cite{2DRMP}.
  amplitude array associated with edge~3.}
 \end{figure}
 
  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
 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


@@ -719,8 +726,8 @@ GPU memory and the required matrix multiplications are performed on
 the GPU (via CUBLAS\index{CUBLAS} routines).
 
 
 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
 performing these matrix multiplications
 on the GPU is expected to be faster than on the CPU.
 However, this implies a greater communication volume


@@ -748,7 +755,7 @@ Therefore, we save the transfer of four $i$ amplitude arrays on
 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
 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}
 
 
 \subsection{Using double-buffering\index{double-buffering} to overlap I/O and computation}
 


@@ -774,7 +781,7 @@ intermediate.
 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 
 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 
 time.
 
 It is thus interesting to use a double-buffering\index{double-buffering} technique to overlap the 


@@ -799,9 +806,9 @@ thread mechanisms.
 
 
 \subsection{Matrix padding\index{padding}}
 
 
 \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  
 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  


@@ -811,18 +818,16 @@ the matrices are therefore padded with $0.0$ (and $1.0$ on the diagonal for the
 so that their sizes are 
 multiples of 64.
 This corresponds indeed to the optimal size for the matrix product on the
 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
 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). 
 
  GPU memory). 
 

-
-

 \section{Performance results}
 \subsection{PROP deployment on GPU}
 
 \section{Performance results}
 \subsection{PROP deployment on GPU}
 


@@ -830,14 +835,14 @@ as the matrices increase in size during the propagation
 \begin{center}
 \begin{tabular}{|c||c|c||}
  \hline
 \begin{center}
 \begin{tabular}{|c||c|c||}
  \hline

-  PROP version & \multicolumn{2}{c|}{Execution time} \\
+  PROP Version & \multicolumn{2}{c|}{Execution Time} \\

   \hline
   \hline
   \hline
   \hline

-  CPU version: 1  core & \multicolumn{2}{c|}{201m32s} \\
+  CPU Version: 1 Core & \multicolumn{2}{c|}{201m32s} \\

   \hline
   \hline

-  CPU version: 4  cores &  \multicolumn{2}{c|}{113m28s} \\
+  CPU Version: 4 Cores &  \multicolumn{2}{c|}{113m28s} \\

   \hline \hline
   \hline \hline

-GPU version  & C1060 & C2050 \\
+GPU Version  & C1060 & C2050 \\

   \hline\hline
   GPU V1 (\S~\ref{gpuv1}) & 79m25s & 66m22s  \\
   \hline
   \hline\hline
   GPU V1 (\S~\ref{gpuv1}) & 79m25s & 66m22s  \\
   \hline


@@ -851,7 +856,7 @@ GPU version  & C1060 & C2050 \\
   \hline
 \end{tabular}
 \end{center}
   \hline
 \end{tabular}
 \end{center}

-\caption{Execution time of PROP on CPU and GPU}
+\caption{Execution time of PROP on CPU and GPU.}

 \label{table:time} 
 \end{table}
 
 \label{table:time} 
 \end{table}
 


@@ -904,10 +909,10 @@ the execution times
 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. 
 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
 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 
 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 


@@ -921,7 +926,7 @@ Since energies are independent, execution times for more energies
 (e.g. the huge case) should be proportional
 to those reported in Table~\ref{table:time}.  
 
 (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.
 MAGMA 0.2, 
 show that the successive GPU versions of PROP offer 
 increasing, and at the end interesting, speedups.


@@ -943,10 +948,11 @@ is noticed that the padding\index{padding}
 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 
 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. 
 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 
 Although for all versions the C2050 (with its improved
 double precision\index{precision!double precision} performance) offers up to almost 
 double speedup compared to 


@@ -980,19 +986,21 @@ the GPU for such an application.
 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. 
 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
 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
  ``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
  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
 
 On one CPU core (see  Fig.~\ref{fig:CPU-timing}), 
  matrix products for the construction of the local


@@ -1001,14 +1009,14 @@ computation time during the whole propagation. Likewise the CPU time required by
  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
  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.
 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
 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
 
 
 On the C1060 (see Fig.~\ref{GPU-timing}), we have


@@ -1046,7 +1054,7 @@ launched by the same host thread. We therefore rely here on the {\em legacy
 API} of CUBLAS\index{CUBLAS}~\cite{CUBLAS} (like MAGMA)
 without thread safety problems. 
 A {\em breadth first} issue order is used for kernel
 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}.  
 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}.  


@@ -1057,28 +1065,28 @@ propagation of each single energy
 in order to enable concurrent executions among the required kernels.
 
 
 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
 \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} \\  
   \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
   \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}} \\
   \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}
   \cline{2-7}
   & Time (s) & 509.51 & \multicolumn{2}{c|}{451.49} & \multicolumn{2}{c|}{436.72}  \\  
   \cline{2-7}


@@ -1110,10 +1118,10 @@ with only one energy on the GPU. The concurrent execution of multiple
 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. 
 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. 
 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. 
 
 
  since the concurrent kernel launches are serialized on this previous GPU architecture. 
 
 


@@ -1144,11 +1152,11 @@ We have
 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.
 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 
 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. 
 An additional gain of around 15\% 
 can also be obtained on one Fermi GPU
 with large memory (C2070) thanks to concurrent kernel execution. 


@@ -1159,21 +1167,21 @@ obtained with the HMPP\index{HMPP} version, since in our tests the CPUs are
 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 
 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 
 
 
 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.
 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
 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]
 nodes with multiple CPU cores and possibly multiple GPU cards. 
 
 \putbib[Chapters/chapter15/biblio]