11-04-2014

diff --git a/hpcc.tex b/hpcc.tex
index 968b235a7d890f16722e62668fd7ac55f17eef7c..fc793914b518df5239ca4c9b56621e385e645a33 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -322,9 +322,9 @@
 \usepackage[utf8]{inputenc}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
 \usepackage[utf8]{inputenc}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}

-%\usepackage{amsmath}
+\usepackage{algorithm}
+\usepackage{algpseudocode}

 %\usepackage{amsthm}
 %\usepackage{amsthm}

-%\usepackage{amsfonts}

 %\usepackage{graphicx}
 %\usepackage{xspace}
 \usepackage[american]{babel}
 %\usepackage{graphicx}
 %\usepackage{xspace}
 \usepackage[american]{babel}


@@ -335,6 +335,14 @@
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
 
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
 

+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+
+

 
 \begin{document}
 %
 
 \begin{document}
 %


@@ -417,7 +425,7 @@ Décrire SimGrid (Arnaud)
 
 
 
 
 
 

-%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \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  
 \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  


@@ -452,7 +460,31 @@ Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
 \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. 
 \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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%