B_L
\end{array} \right)
\end{equation*}
-in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
+are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$ $A_{\ell
+ m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
+$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each and
+$\sum_{\ell} n_\ell=\sum_{m} n_m=n$.
The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
\begin{equation}
\label{eq:4.1}
\left\{
\begin{array}{l}
- A_{ll}X_l = Y_l \text{, such that}\\
- Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
+ A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\
+ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m
\end{array}
\right.
\end{equation}
-is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ 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 algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+is solved independently by a cluster and communications are required to update
+the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$
+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 algorithm of GMRES method~\cite{ref1} which is one of the most
+used iterative method by many researchers.
\begin{figure}[!t]
%%% IEEE instructions forbid to use an algorithm environment here, use figure
%%% instead
\begin{algorithmic}[1]
-\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
-\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\medskip
+
+\State Load $A_\ell$, $B_\ell$
\State Set the initial guess $x^0$
\For {$k=0,1,2,\ldots$ until the global convergence}
\State Restart outer iteration with $x^0=x^k$
\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
-\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $k$}
-\State Compute local right-hand side $Y_l$:
+\State Compute local right-hand side $Y_\ell$:
\begin{equation*}
- Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+ Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0
\end{equation*}
-\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
-\State \Return $X_l^k$
+\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method
+\State \Return $X_\ell^k$
\EndFunction
\end{algorithmic}
\caption{A multisplitting solver with GMRES method}
\label{algo:01}
\end{figure}
-Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines.
+Algorithm on Figure~\ref{algo:01} shows the main key points of the
+multisplitting method to solve a large sparse linear system. This algorithm is
+based on an outer-inner iteration method where the parallel synchronous GMRES
+method is used to solve the inner iteration. It is executed in parallel by each
+cluster of processors. For all $\ell,m\in\{1,\ldots,L\}$, the matrices and
+vectors with the subscript $\ell$ represent the local data for cluster $\ell$,
+while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix
+$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared
+with neighboring clusters. At every outer iteration $k$, asynchronous
+communications are performed between processors of the local cluster and those
+of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
+Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
+exchanged by message passing using MPI non-blocking communication routines.
\begin{figure}[!t]
\centering
global convergence is detected when the master of cluster 1 receives from the
master of cluster $L$ a token set to \textit{True}. In this case, the master of
cluster 1 broadcasts a stop message to masters of other clusters. In this work,
-the local convergence on each cluster $l$ is detected when the following
+the local convergence on each cluster $\ell$ is detected when the following
condition is satisfied
\begin{equation*}
- (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
+ (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon)
\end{equation*}
where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
tolerance threshold of the error computed between two successive local solution
-$X_l^k$ and $X_l^{k+1}$.
+$X_\ell^k$ and $X_\ell^{k+1}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code
Table~\ref{tab.cluster.3x67}.
\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
+\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
\LZK{Ma question est: le bw et lat sont ceux inter-clusters ou pour les deux inter et intra cluster??}
\section{Conclusion}
