11-04-2014 bis

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index e1720c5ec864fb962a3bb0392092ab67edff27d4..47d810260f2a6cb3e4d6ce6eca5110ecc62fdb5b 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -27,6 +27,8 @@
   \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
   \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}

+\newcommand{\LZK}[2][inline]{%
+  \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -59,7 +61,8 @@
 
 \maketitle
 
 
 \maketitle
 

-\RC{Ordre des autheurs pas définitif.\\ Adresse de Lilia: Inria Bordeaux Sud-Ouest, 200 Avenue de la Vieille Tour, 33405 Talence Cedex, France \\ Email: lilia.ziane@inria.fr}
+\RC{Ordre des autheurs pas définitif.}
+\LZK{Adresse de Lilia: Inria Bordeaux Sud-Ouest, 200 Avenue de la Vieille Tour, 33405 Talence Cedex, France \\ Email: lilia.ziane@inria.fr}

 \begin{abstract}
 The abstract goes here.
 \end{abstract}
 \begin{abstract}
 The abstract goes here.
 \end{abstract}


@@ -180,7 +183,7 @@ Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
 \right.
 \label{eq:4.1}
 \end{equation}
 \right.
 \label{eq:4.1}
 \end{equation}

-is solved independently by a cluster and communication 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 $Y_l$, 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. 

 
 \begin{algorithm}
 \caption{A multisplitting solver with GMRES method}
 
 \begin{algorithm}
 \caption{A multisplitting solver with GMRES method}


@@ -188,7 +191,7 @@ is solved independently by a cluster and communication are required to update th
 \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 $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
 \Output $X_l$ (solution sub-vector)\vspace{0.2cm}
 \State Load $A_l$, $B_l$

-\State Initialize the solution vector $x^0$
+\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}{$x^0$, $k+1$}
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}


@@ -216,9 +219,11 @@ Algorithm~\ref{algo:01} shows the main key points of the multisplitting method t
 \label{fig:4.1}
 \end{figure}
 
 \label{fig:4.1}
 \end{figure}
 

-The global convergence of the asynchronous multisplitting solver is detected when the clusters of processors have all converged locally. We implemented the global convergence detection process as follows. On each cluster a master processor is designated (for example the processor with rank $1$) and masters of all clusters are interconnected by a virtual unidirectional ring network (see Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank $1$, each master processor $i$ sets the token to {\it True} if the local convergence is achieved or to {\it False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster $1$ receive from the master of cluster $L$ a token set to {\it True}. In this case, the master of cluster $1$ sends a stop message to masters of other clusters. In this work, the local convergence on each cluster $l$ is detected when the following condition is satisfied
+The global convergence of the asynchronous multisplitting solver is detected when the clusters of processors have all converged locally. We implemented the global convergence detection process as follows. On each cluster a master processor is designated (for example the processor with rank $1$) and masters of all clusters are interconnected by a virtual unidirectional ring network (see Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank $1$, each master processor $i$ sets the token to {\it True} if the local convergence is achieved or to {\it False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster $1$ receives from the master of cluster $L$ a token set to {\it 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 condition is satisfied

 \[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
 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 $X_l^{k+1}$. 
 \[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
 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 $X_l^{k+1}$. 

+
+\LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid}

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