X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/dad13e36a03a1d7b58cfff7721b3b81d7537968b..feda1be54f4c3fca6e201e6791bd285548372ede:/paper.tex

diff --git a/paper.tex b/paper.tex
index 827e7e6..f1f47bf 100644
--- a/paper.tex
+++ b/paper.tex
@@ -317,10 +317,12 @@ 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
-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.
+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.
 
 
 \paragraph{Simgrid Simulator parameters}
@@ -343,9 +345,16 @@ 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} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
+	\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.
@@ -356,14 +365,30 @@ 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 3D Poisson problem used in our experiments is described.
+
+\subsection{3D Poisson}
+
 
-We use our two-stage algorithms to solve the well-known 3D Poisson problem $\nabla^2\phi=f$, where $\nabla^2$ is the Laplace operator. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
+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~}\Omega
+\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}
-where the real-valued function $\phi(x,y,z)=0\mbox{~on~}\partial\Omega$ is the solution sought, $f(x,y,z)$ is a known function and the domain $\Omega=[0,1]^3$. 
+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,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\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 block of the 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}
 
@@ -377,7 +402,7 @@ 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
@@ -429,37 +454,35 @@ 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
  speed.  The network  between distant  clusters might  be a  bottleneck for  the
  global performance of the application.
 
-\subsection{Comparing GMRES and Multisplitting algorithms in
-synchronous mode}
+\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
 
-In the scope of this paper, our first objective is to demonstrate the
-Algo-2 (Multisplitting method) shows a better performance in grid
-architecture compared with Algo-1 (Classical GMRES) both running in
-\textit{synchronous mode}. Better algorithm performance
-should means a less number of iterations output and a less execution time
-before reaching the convergence. For a systematic study, the experiments
-should figure out that, for various grid parameters values, the
-simulator will confirm the targeted outcomes, particularly for poor and
-slow networks, focusing on the impact on the communication performance
-on the chosen class of algorithm.
+In the scope  of this paper, our  first objective is to analyze  when the Krylov
+Multisplitting  method   has  better  performances  than   the  classical  GMRES
+method. With an  iterative method, better performances mean a  smaller number of
+iterations and execution time before reaching the convergence.  For a systematic
+study,  the experiments  should figure  out  that, for  various grid  parameters
+values, the simulator will confirm  the targeted outcomes, particularly for poor
+and slow  networks, focusing on the  impact on the communication  performance on
+the chosen class of algorithm.
 
 The following paragraphs present the test conditions, the output results
 and our comments.\\
 
 
-\textit{3.a Executing the algorithms on various computational grid
+\subsubsection{Execution of the the algorithms on various computational grid
 architecture and scaling up the input matrix size}
-\\
-
+\ \\
 % environment
-\begin{footnotesize}
+
+\begin{figure} [ht!]
+\begin{center}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
@@ -467,26 +490,34 @@ architecture and scaling up the input matrix size}
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
  \end{tabular}
-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}
 
-\end{footnotesize}
 
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 
-In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration the non-variation of the number of iterations of classical GMRES for a given input matrix size; it is not
-the case for the multisplitting method.
+In this  section, we analyze the  performences of algorithms running  on various
+grid configuration  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+show for all grid configuration the non-variation of the number of iterations of
+classical  GMRES for  a given  input matrix  size; it  is not  the case  for the
+multisplitting method.
+
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+\RC{Les légendes ne sont pas explicites...}
+
 
-%\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
-%\label{overflow}}
+  \begin{center}
+    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+  \end{center}
+  \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+  \label{fig:01}
 \end{figure}
-%\end{wrapfigure}
+
 
 The execution time difference between the two algorithms is important when
 comparing between different grid architectures, even with the same number of
@@ -615,7 +646,7 @@ 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
@@ -645,7 +676,7 @@ 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