11-04-2014 bis

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index e7edb13e6ae5791438131f8b2a57e90600e4c5ca..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}
+\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}


@@ -138,7 +141,7 @@ Décrire le modèle asynchrone. Je m'en charge (DL)
 
 \section{SimGrid}
 
 
 \section{SimGrid}
 

-Décrire SimGrid (Arnaud)
+Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid} (Arnaud)

 
 
 
 
 
 


@@ -149,7 +152,7 @@ Décrire SimGrid (Arnaud)
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting 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 $b$ 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  
+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. In this case, we apply a row-by-row splitting without overlapping  

 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\
 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\


@@ -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}{$x^0$, $k+1$}


@@ -207,7 +210,7 @@ is solved independently by a cluster and communication are required to update th
 \label{algo:01}
 \end{algorithm}
 
 \label{algo:01}
 \end{algorithm}
 

-Algorithm~\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 such that the parallel 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 $6$ and $7$ in Algorithm~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. 
+Algorithm~\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 $6$ and $7$ in Algorithm~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. 

 
 \begin{figure}
 \centering
 
 \begin{figure}
 \centering


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

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