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[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index edba67f7c0fb5f612852e6fd6fe8e9c3e03d1c28..767aa599a2ec1b5f128fcf4d0795df9b731f611a 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -288,6 +288,8 @@ These two techniques can help to run simulations at a very large scale.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}

+
+\subsection{The multisplitting method}

 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  
 \begin{equation*}
 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  
 \begin{equation*}


@@ -442,6 +444,8 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re
 \end{figure}
 
 
 \end{figure}
 
 

+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+

 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


@@ -474,8 +478,8 @@ study that the results depend on the following parameters:
 \item Finally, when submitting job batches for execution, the arguments values
   passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
   algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in
 \item Finally, when submitting job batches for execution, the arguments values
   passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
   algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in

-  synchronous mode. The ratio between the execution time of synchronous
-  compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
+  synchronous mode. The ratio between the simulated execution time of synchronous GMRES algorithm
+  compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) is defined as the \emph{relative gain}. So,

   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.
 \end{itemize}
   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.
 \end{itemize}


@@ -489,14 +493,13 @@ synchronous mode allowing to get a relative gain greater than 1.  This action
 simulates the case of distant clusters linked with long distance network as in grid computing context.
 
 
 simulates the case of distant clusters linked with long distance network as in grid computing context.
 
 

-% As a first step, 
-The algorithm was run on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The algorithm convergence with a 3D
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+
+Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from

 $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
 $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =

-\text{\np{3375000}}$ entries), is obtained in asynchronous in average 2.5 times faster than in the synchronous mode.
-\AG{Expliquer comment lire les tableaux.}
-\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
+%\AG{Expliquer comment lire les tableaux.}
+%\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}

 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 \newenvironment{mytable}[1]{% #1: number of columns for data
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 \newenvironment{mytable}[1]{% #1: number of columns for data


@@ -647,7 +650,7 @@ Note that the program was run with the following parameters:
 \item Maximum number of iterations;
 \item Precisions on the residual error;
 \item Matrix size $N_x$, $N_y$ and $N_z$;
 \item Maximum number of iterations;
 \item Precisions on the residual error;
 \item Matrix size $N_x$, $N_y$ and $N_z$;

-\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix diagonal value: $6$ (See Equation~(\ref{eq:03}));

 \item Matrix off-diagonal value: $-1$;
 \item Communication mode: asynchronous.
 \end{itemize}
 \item Matrix off-diagonal value: $-1$;
 \item Communication mode: asynchronous.
 \end{itemize}