X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/7582e9f33cd2cfeaa5bc69ba88bd3039af4461b7..03fe4b6e2d3dd643cb33f0c8010415b415349fa7:/paper.tex

diff --git a/paper.tex b/paper.tex
index ca1c7e7..397decc 100644
--- a/paper.tex
+++ b/paper.tex
@@ -317,7 +317,7 @@ suppress all global variables by replacing  them with local variables or using a
 Simgrid      selector       called      "runtime       automatic      switching"
 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 effects on runtime between the threads running in the same process, generated by
-the Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?}The
+Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
 last modification on the  MPI program pointed out for some  cases, the review of
 the sequence of  the MPI\_Isend, MPI\_Irecv and  MPI\_Waitall instructions which
 might cause an infinite loop.
@@ -343,9 +343,16 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
 	\item maximum number of inner and outer iterations;
 	\item inner and outer precisions;
-	\item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
-	\item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifies, je dis ca de tête}
-	\item execution mode: synchronous or asynchronous.
+	\item maximum number of the gmres's restarts in the Arnorldi process;
+	\item maximum number of iterations qnd the tolerance threshold in classical GMRES;
+	\item tolerance threshold for outer and inner-iterations;
+	\item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
+	\item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
+	\item matrix off-diagonal value;
+	\item execution mode: synchronous or asynchronous;
+	\RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les  arguments pour CGLS ci dessous}
+	\item Size of matrix S;
+	\item Maximum number of iterations and tolerance threshold for CGLS. 
 \end{itemize}
 
 It should also be noticed that both solvers have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
@@ -356,9 +363,30 @@ It should also be noticed that both solvers have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
-In this section, experiments for both Multisplitting algorithms are reported. First the problem sued in our experiments is described.
+In this section, experiments for both Multisplitting algorithms are reported. First the problem used in our experiments is described.
 
-\RC{Lilia a toi de jouer}
+We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
+\begin{equation}
+\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
+\label{eq:07}
+\end{equation}
+such that
+\begin{equation*}
+\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
+\end{equation*}
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that      
+\begin{equation}
+\begin{array}{ll}
+\phi^\star(x,y,z)= & \frac{1}{6}(\phi(x-h,y,z)+\phi(x+h,y,z) \\
+                  & +\phi(x,y-h,z)+\phi(x,y+h,z) \\
+                  & +\phi(x,y,z-h)+\phi(x,y,z+h)\\
+                  & -h^2f(x,y,z))
+\end{array}
+\label{eq:08}
+\end{equation}
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid. 
+
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic sub-problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries. 
 
 \subsection{Study setup and Simulation Methodology}
 
@@ -372,7 +400,7 @@ have been chosen for the study in this paper. \\
 \textbf{Step 2}: Collect the software materials needed for the
 experimentation. In our case, we have two variants algorithms for the
 resolution of the 3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting method. In addition, the Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a laptop. \\
+distributed applications. Simgrid is running on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a simple laptop. \\
 
 \textbf{Step 3}: Fix the criteria which will be used for the future
 results comparison and analysis. In the scope of this study, we retain
@@ -424,7 +452,7 @@ transit between the clusters and nodes during the code execution.
  In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
  "intra-network" which refers  to the links between nodes within  a cluster and,
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
- clusters.  In   practse;  these  two   networks  have  different   speeds.  The
+ clusters.  In   practice,  these  two   networks  have  different   speeds.  The
  intra-network  generally works  like a  high speed  local network  with a  high
  bandwith and very low latency. In opposite, the inter-network connects clusters
  sometime via  heterogeneous networks components  throuth internet with  a lower
@@ -610,7 +638,7 @@ In this experimentation, the input matrix size has been set from
 N$_{x}$ = N$_{y}$ = N$_{z}$ = 40 to 200 side elements that is from 40$^{3}$ = 64.000 to
 200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 7,
 the execution time for the two algorithms convergence increases with the
-input matrix size. But the interesting results here direct on (i) the
+iinput matrix size. But the interesting results here direct on (i) the
 drastic increase (300 times) of the number of iterations needed before
 the convergence for the classical GMRES algorithm when the matrix size
 go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost
@@ -640,7 +668,7 @@ Table 6 : CPU Power impact \\
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 %\label{overflow}}
-\end{figure}
+s\end{figure}
 
 Using the Simgrid simulator flexibility, we have tried to determine the
 impact on the algorithms performance in varying the CPU power of the