From 2367662d760a73d98ca68d37a3912c211972afe3 Mon Sep 17 00:00:00 2001
From: lilia <lilia@mondomaine.fr>
Date: Fri, 11 Apr 2014 00:06:45 +0200
Subject: [PATCH] 11-04-2014

---
 hpcc.tex    | 54 ++++++++++++++++++++++++++++++++++++++++++++++++++---
 hpccBib.bib | 10 ++++++++++
 2 files changed, 61 insertions(+), 3 deletions(-)
 create mode 100644 hpccBib.bib

diff --git a/hpcc.tex b/hpcc.tex
index b4d0be3..968b235 100644
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -320,6 +320,8 @@
 
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
+\usepackage{amsfonts,amssymb}
+\usepackage{amsmath}
 %\usepackage{amsmath}
 %\usepackage{amsthm}
 %\usepackage{amsfonts}
@@ -409,9 +411,55 @@ Décrire le modèle asynchrone. Je m'en charge (DL)
 
 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. 
+%%%%%
+
+
+
+
+
+
+
 
 \section{Experimental results}
 
@@ -650,7 +698,7 @@ The authors would like to thank...
 % http://www.michaelshell.org/tex/ieeetran/bibtex/
 \bibliographystyle{IEEEtran}
 % argument is your BibTeX string definitions and bibliography database(s)
-\bibliography{bib/hpccBib}
+\bibliography{hpccBib}
 %
 % <OR> manually copy in the resultant .bbl file
 % set second argument of \begin to the number of references
diff --git a/hpccBib.bib b/hpccBib.bib
new file mode 100644
index 0000000..310aef0
--- /dev/null
+++ b/hpccBib.bib
@@ -0,0 +1,10 @@
+@article{ref1,
+title = {{GMRES}: {A} {G}eneralized {M}inimal {R}esidual {A}lgorithm for {S}olving {N}onsymmetric {L}inear {S}ystems},
+author = {Saad, Y. and Schultz, M. H.},
+journal = {SIAM Journal on Scientific and Statistical Computing},
+volume = {7},
+number = {3},
+pages = {856--869},
+year = {1986}
+}
+
-- 
2.39.5