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+\usepackage{algorithm}
+\usepackage{algpseudocode}
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+\algnewcommand\algorithmicinput{\textbf{Input:}}
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+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
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+
\begin{document}
%
Décrire SimGrid (Arnaud)
-\section{Simulation of the multi-splitting method}
-Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Simulation of 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 $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping
+\[
+\left(\begin{array}{ccc}
+A_{11} & \cdots & A_{1L} \\
+\vdots & \ddots & \vdots\\
+A_{L1} & \cdots & A_{LL}
+\end{array} \right)
+\times
+\left(\begin{array}{c}
+X_1 \\
+\vdots\\
+X_L
+\end{array} \right)
+=
+\left(\begin{array}{c}
+Y_1 \\
+\vdots\\
+Y_L
+\end{array} \right)\]
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.
+
+The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ clusters of processors, in such a way each sub-system
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{eq:4.1}
+\end{equation}
+is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+
+\begin{algorithm}
+\caption{A multisplitting solver with inner iteration GMRES method}
+\begin{algorithmic}[1]
+\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$
+\State Initialize the shared vector $\hat{x}=x^0$
+\For {$k=1,2,3,\ldots$ until the global convergence}
+\State $x^0=\hat{x}$
+\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
+\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
+\EndFor
+
+\Statex
+
+\Function {InnerSolver}{$x^0$, $k$}
+\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
+\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
+\State \Return $X_l^k$
+\EndFunction
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+
+
\section{Experimental results}
-{\raggedright
+
When the ``real'' application runs in the simulation environment and produces
the expected results, varying the input parameters and the program arguments
allows us to compare outputs from the code execution. We have noticed from this
simulation in having an execution time in asynchronous less than in synchronous
mode, in others words, in having a ``speedup'' less than 1 (Speedup = Execution
time in synchronous mode / Execution time in asynchronous mode).
-}
-{\raggedright
A priori, obtaining a speedup less than 1 would be difficult in a local area
network configuration where the synchronous mode will take advantage on the rapid
exchange of information on such high-speed links. Thus, the methodology adopted
degrading the inter-cluster network performance will "penalize" the synchronous
mode allowing to get a speedup lower than 1. This action simulates the case of
clusters linked with long distance network like Internet.
-}
-{\raggedright
As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
-the above factors have providing the results shown in Table 1 with a matrix size
+the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
ranging from Nx = Ny = Nz = 62 to 171 elements or from 62$^{3}$ = 238328 to
171$^{3}$ = 5,211,000 entries.
-}
-{\raggedright
Then we have changed the network configuration using three clusters containing
respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table 2 which shows the speedups less than 1 with
+permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the speedups less than 1 with
a matrix size from 62 to 100 elements.
-}
-{\raggedright
In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in Table 3.
-}
+configuration but increasing by two hundreds hosts has been recorded in Table~\ref{tab.cluster.3x67}.
-{\raggedright
Note that the program was run with the following parameters:
-}
-%{\raggedright
-\textbullet{} \textbf {SMPI parameters:}
-%}
+\paragraph*{SMPI parameters}
\begin{itemize}
\item HOSTFILE : Hosts file description.
\end{itemize}
-%{\raggedright
-\textbullet{} \textbf {Arguments of the program:}
-%}
+\paragraph*{Arguments of the program}
\begin{itemize}
\item Description of the cluster architecture;
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
-\textbf{Table 1}
-
-\textit{{\scriptsize 2 clusters X 50 nodes}}
-\includegraphics[width=209pt]{img-1.eps}
+\begin{table}
+ \centering
+ \caption{2 clusters X 50 nodes}
+ \label{tab.cluster.2x50}
+ \includegraphics[width=209pt]{img-1.eps}
+\end{table}
+
+\begin{table}
+ \centering
+ \caption{3 clusters X 33 n\oe{}uds}
+ \label{tab.cluster.3x33}
+ \includegraphics[width=209pt]{img-1.eps}
+\end{table}
+
+\begin{table}
+ \centering
+ \caption{3 clusters X 67 noeuds}
+ \label{tab.cluster.3x67}
+ \includegraphics[width=128pt]{img-2.eps}
+\end{table}
+
+\paragraph*{Interpretations and comments}
-\textbf{Table 2}
-
-\textit{{\scriptsize 3 clusters X 33 n\oe{}uds}}
-\includegraphics[width=209pt]{img-1.eps}
-\textbf{Table 3}
-
-\textit{{\scriptsize 3 clusters X 67 noeuds}}
-\includegraphics[width=128pt]{img-2.eps}
-
-{\raggedright
-\textbf{Interpretations and comments}
-}
-
-{\raggedright
After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables 1 and 2), some combinations of the
+clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50} and~\ref{tab.cluster.3x33}), some combinations of the
used parameters affecting the results have given a speedup less than 1, showing
the effectiveness of the asynchronous performance compared to the synchronous
mode.
-}
-{\raggedright
-In the case of a two clusters configuration, Table 1 shows that with a
+In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
deterioration of inter cluster network set with 5 Mbits/s of bandwidth, a latency
in order of a hundredth of a millisecond and a system power of one GFlops, an
efficiency of about 40\% in asynchronous mode is obtained for a matrix size of 62
inter cluster up to 50 Mbits /s, the result of efficiency of about 40\% is
obtained with high external precision of E-11 for a matrix size from 110 to 150
side elements .
-}
-{\raggedright
-For the 3 clusters architecture including a total of 100 hosts, Table 2 shows
+For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows
that it was difficult to have a combination which gives an efficiency of
asynchronous below 80 \%. Indeed, for a matrix size of 62 elements, equality
between the performance of the two modes (synchronous and asynchronous) is
achieved with an inter cluster of 10 Mbits/s and a latency of E- 01 ms. To
challenge an efficiency by 78\% with a matrix size of 100 points, it was
necessary to degrade the inter cluster network bandwidth from 5 to 2 Mbit/s.
-}
-{\raggedright
A last attempt was made for a configuration of three clusters but more power
with 200 nodes in total. The convergence with a speedup of 90 \% was obtained
-with a bandwidth of 1 Mbits/s as shown in Table 3.
-}
+with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.
\section{Conclusion}
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