X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/755621fb88b1cb7c7236ed679094c722a6104f71..7509f7bc4a76338951fc145f7c10a932fe8144cc:/hpcc.tex?ds=inline

diff --git a/hpcc.tex b/hpcc.tex
index 47480f8..98a55ff 100644
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -1,3 +1,4 @@
+
 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}
@@ -287,6 +288,8 @@ These two techniques can help to run simulations at a very large scale.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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*}
@@ -419,17 +422,17 @@ u =0 \text{~on~} \Gamma =\partial\Omega
 \right.
 \label{eq:02}
 \end{equation}
-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as 
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
 \begin{equation}
-\begin{array}{ll}
-u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
-               & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
-               & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
-               & u^k(x,y,z-1) + u^k(x,y,z+1)),
+\begin{array}{l}
+u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ %         & u(x,y-1,z) + u(x,y+1,z) + \\
+  %        & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
 \end{array}
 \label{eq:03}
 \end{equation} 
-where the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
+where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 
 
 The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 
 
@@ -441,6 +444,8 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re
 \end{figure}
 
 
+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -473,8 +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
-  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}
@@ -488,14 +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.
 
 
-% 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 $\text{150}^\text{3} =
-\text{\np{3375000}}$ 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
@@ -646,8 +650,8 @@ 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 Matrix diagonal value: \np{1.0} (See~(\ref{eq:03}));
-\item Matrix off-diagonal value: \np{-1}/\np{6} (See~(\ref{eq:03}));
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
 \item Communication mode: asynchronous.
 \end{itemize}