suite expe

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 93f215d5bab1aad50151e30a5ad25dab9d926025..82777e31dc256726f9b307977e12fda64c3610c0 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -367,19 +367,19 @@ It should also be noticed that both solvers have been executed with the Simgrid
 
 In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used 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}
+\subsection{The 3D Poisson problem}

 
 
 
 

-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
+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}
 \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
+such that:

 \begin{equation*}
 \phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
 \end{equation*}
 \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
+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))
 \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))


@@ -390,7 +390,7 @@ until convergence where $h$ is the grid spacing between two adjacent elements in
 
 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.
 
 
 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}
+\subsection{Study setup and simulation methodology}

 
 First, to conduct our study, we propose the following methodology
 which can be reused for any grid-enabled applications.\\
 
 First, to conduct our study, we propose the following methodology
 which can be reused for any grid-enabled applications.\\


@@ -399,10 +399,12 @@ which can be reused for any grid-enabled applications.\\
 the application to be tested. Numerical parallel iterative algorithms
 have been chosen for the study in this paper. \\
 
 the application to be tested. Numerical parallel iterative algorithms
 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 simple laptop. \\
+\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 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
 
 \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


@@ -443,23 +445,20 @@ the program output results:
     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
 \item the  network latency  (lat :  microsecond) defined as  the delay  from the
     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
 \item the  network latency  (lat :  microsecond) defined as  the delay  from the

-  start time to send  the data from a source and the  final time the destination
-  have finished to receive it.
+  start time to send  a simple data from a source to a destination.

 \end{enumerate}
 \end{enumerate}

-Upon  the   network  characteristics,  another  impacting   factor  is  the
-application dependent volume of data exchanged  between the nodes in the cluster
-and  between distant  clusters.  Large volume  of data  can  be transferred  and
-transit between the clusters and nodes during the code execution.
+Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
+and  between distant  clusters.  This parameter is application dependent.

 
  In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
 
  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,
+ "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
  on  the other  hand, the  "inter-network" which  is the  backbone link  between

- 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.
+ 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{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
 
 
 \subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
 


@@ -519,26 +518,28 @@ multisplitting method.
 \end{figure}
 
 
 \end{figure}
 
 

-The execution time difference between the two algorithms is important when
-comparing between different grid architectures, even with the same number of
-processors (like 2x16 and 4x8 = 32 processors for example). The
-experiment concludes the low sensitivity of the multisplitting method
-(compared with the classical GMRES) when scaling up the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs 40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.
+The execution  times between  the two algorithms  is significant  with different
+grid architectures, even  with the same number of processors  (for example, 2x16
+and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+(compared with the classical GMRES) when scaling up the number of the processors
+in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs
+40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.

 
 

-\textit{\\3.b Running on two different speed cluster inter-networks\\}
+\subsubsection{Running on two different speed cluster inter-networks}
+\ \\

 
 

-% environment
-\begin{footnotesize}
+\begin{figure} [ht!]
+\begin{center}

 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\

- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline 

  \end{tabular}
  \end{tabular}

-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
+\caption{Clusters x Nodes - Networks N1 x N2}
+\end{center}
+\end{figure}

 
 
 
 
 
 


@@ -547,31 +548,30 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Cluster x Nodes N1 x N2}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Cluster x Nodes N1 x N2}

-%\label{overflow}}
+\label{fig:02}

 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and also on a less performant network (N2).
-Figure 4 shows that end users will gain to reduce the execution time
-for both algorithms in using a grid architecture like 4x16 or 8x8: the
-performance was increased in a factor of 2. The results depict also that
-when the network speed drops down (12.5\%), the difference between the execution
-times can reach more than 25\%.
-
-\textit{\\3.c Network latency impacts on performance\\}
+These experiments  compare the  behavior of  the algorithms  running first  on a
+speed inter-cluster  network (N1) and  also on  a less performant  network (N2).
+Figure~\ref{fig:02} shows that end users will  gain to reduce the execution time
+for  both  algorithms  in using  a  grid  architecture  like  4x16 or  8x8:  the
+performance was increased  in a factor of  2. The results depict  also that when
+the  network speed  drops down  (12.5\%), the  difference between  the execution
+times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?}

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline

- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline

  \end{tabular}
  \end{tabular}

-Table 3 : Network latency impact \\
-
-\end{footnotesize}
+\caption{Network latency impact}
+\end{figure}

 
 
 
 
 
 


@@ -579,142 +579,141 @@ Table 3 : Network latency impact \\
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impact on execution time}
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impact on execution time}

-%\label{overflow}}
+\label{fig:03}

 \end{figure}
 
 
 \end{figure}
 
 

-According the results in figure 5, degradation of the network
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time
-increase more than 75\% (resp. 82\%) of the execution for the classical
-GMRES (resp. multisplitting) algorithm. In addition, it appears that the
-multisplitting method tolerates more the network latency variation with
-a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-5
-}$), the execution time for GMRES is almost the double of the time for
-the multisplitting, even though, the performance was on the same order
-of magnitude with a latency of 8.10$^{-6}$.
-
-\textit{\\3.d Network bandwidth impacts on performance\\}
+According  the results  in  Figure~\ref{fig:03}, a  degradation  of the  network
+latency from 8.10$^{-6}$  to 6.10$^{-5}$ implies an absolute  time increase more
+than 75\%  (resp. 82\%) of the  execution for the classical  GMRES (resp. Krylov
+multisplitting)   algorithm.   In   addition,   it  appears   that  the   Krylov
+multisplitting method tolerates  more the network latency variation  with a less
+rate  increase  of  the  execution   time.   Consequently,  in  the  worst  case
+(lat=6.10$^{-5 }$), the  execution time for GMRES is almost  the double than the
+time of the Krylov multisplitting, even  though, the performance was on the same
+order of magnitude with a latency of 8.10$^{-6}$.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
  \end{tabular}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
  \end{tabular}

-Table 4 : Network bandwidth impact \\
-
-\end{footnotesize}
+\caption{Network bandwidth impact}
+\end{figure}

 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impact on execution time}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impact on execution time}

-%\label{overflow}
+\label{fig:04}

 \end{figure}
 
 
 
 \end{figure}
 
 
 

-The results of increasing the network bandwidth show the improvement
-of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.
-
-\textit{\\3.e Input matrix size impacts on performance\\}
+The results  of increasing  the network  bandwidth show  the improvement  of the
+performance  for   both  algorithms   by  reducing   the  execution   time  (see
+Figure~\ref{fig:04}). However,  in this  case, the Krylov  multisplitting method
+presents a better  performance in the considered bandwidth interval  with a gain
+of 40\% which is only around 24\% for classical GMRES.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 4x8\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 4x8\\ %\hline

- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ 
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline

  \end{tabular}
  \end{tabular}

-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Input matrix size impact}
+\end{figure}

 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}

-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impact on execution time}
+\label{fig:05}

 \end{figure}
 
 \end{figure}
 

-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
-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
-the double from N$_{x}$=140 compared with the convergence time of the
-multisplitting method. These findings may help a lot end users to setup
-the best and the optimal targeted environment for the application
-deployment when focusing on the problem size scale up. Note that the
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{\\3.f CPU Power impact on performance\\}
+In these experiments, 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 Figure~\ref{fig:05},  the execution
+time for  both algorithms increases when  the input matrix size  also increases.
+But the interesting results are:
+\begin{enumerate}
+  \item the drastic increase (300 times) \RC{Je ne vois pas cela sur la figure}
+of the  number of  iterations needed  to reach the  convergence for  the classical
+GMRES algorithm when  the matrix size go beyond N$_{x}$=150;
+\item the  classical GMRES execution time  is almost the double  for N$_{x}$=140
+  compared with the Krylov multisplitting method.
+\end{enumerate}

 
 

-% environment
-\begin{footnotesize}
+These  findings may  help a  lot end  users to  setup the  best and  the optimal
+targeted environment for the application deployment when focusing on the problem
+size scale up.  It  should be noticed that the same test has  been done with the
+grid 2x16 leading to the same conclusion.
+
+\subsubsection{CPU Power impact on performance}
+
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}

-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
-
+\caption{CPU Power impact}
+\end{figure}

 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}

-%\label{overflow}}
-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
-clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6
-confirm the performance gain, around 95\% for both of the two methods,
-after adding more powerful CPU.
-
-\subsection{Comparing GMRES in native synchronous mode and
-Multisplitting algorithms in asynchronous mode}
-
-The previous paragraphs put in evidence the interests to simulate the
-behavior of the application before any deployment in a real environment.
-We have focused the study on analyzing the performance in varying the
-key factors impacting the results. The study compares
-the performance of the two proposed algorithms both in \textit{synchronous mode
-}. In this section, following the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textit{
-asynchronous mode} compared with the classical GMRES staying in
-\textit{synchronous mode}.
-
-Note that the interest of using the asynchronous mode for data exchange
-is mainly, in opposite of the synchronous mode, the non-wait aspects of
-the current computation after a communication operation like sending
-some data between nodes. Each processor can continue their local
-calculation without waiting for the end of the communication. Thus, the
-asynchronous may theoretically reduce the overall execution time and can
-improve the algorithm performance.
-
-As stated supra, Simgrid simulator tool has been used to prove the
-efficiency of the multisplitting in asynchronous mode and to find the
-best combination of the grid resources (CPU, Network, input matrix size,
-\ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+\label{fig:06}
+\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  clusters nodes
+from 1  to 19 GFlops.  The outputs  depicted in Figure~\ref{fig:06}  confirm the
+performance gain,  around 95\% for  both of the  two methods, after  adding more
+powerful CPU.
+
+\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+
+The previous paragraphs  put in evidence the interests to  simulate the behavior
+of  the application  before  any  deployment in  a  real  environment.  In  this
+section, following  the same previous  methodology, our  goal is to  compare the
+efficiency of the multisplitting method  in \textit{ asynchronous mode} with the
+classical GMRES in \textit{synchronous mode}.
+
+The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
+synchronization. With  geographically distant  clusters, this may  be essential.
+In  this case,  each  processor can  compute its  iteration  freely without  any
+synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
+theoretically reduce  the overall execution  time and can improve  the algorithm
+performance.
+
+\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
+As stated before, the Simgrid simulator tool has been successfully used to show
+the efficiency of  the multisplitting in asynchronous mode and  to find the best
+combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to
+get    the   highest    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
+exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.

 
 
 The test conditions are summarized in the table below : \\
 
 
 
 The test conditions are summarized in the table below : \\
 

-% environment
-\begin{footnotesize}
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline


@@ -724,15 +723,17 @@ The test conditions are summarized in the table below : \\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}

-\end{footnotesize}
+\end{figure}

 
 

-Again, comprehensive and extensive tests have been conducted varying the
-CPU power and the network parameters (bandwidth and latency) in the
-simulator tool with different problem size. The relative gains greater
-than 1 between the two algorithms have been captured after each step of
-the test. Table 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
+Again,  comprehensive and  extensive tests  have been  conducted with  different
+parametes as  the CPU power, the  network parameters (bandwidth and  latency) in
+the simulator tool  and with different problem size. The  relative gains greater
+than 1  between the  two algorithms have  been captured after  each step  of the
+test.   In  Figure~\ref{table:01}  are  reported the  best  grid  configurations
+allowing the  multisplitting method to  be more than  2.5 times faster  than the
+classical  GMRES.  These  experiments also  show the  relative tolerance  of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters throuth the internet.

 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}


@@ -743,14 +744,12 @@ the classical GMRES execution and convergence time. The experimentation has demo
     \end{tabular}}
 
 
     \end{tabular}}
 
 

-\begin{table}[!t]
-  \centering
+\begin{figure}[!t]
+\centering
+%\begin{table}

 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 %  \label{"Table 7"}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 %  \label{"Table 7"}

-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
-  \begin{mytable}{11}
+ \begin{mytable}{11}

     \hline
     bandwidth (Mbit/s)
     & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\
     \hline
     bandwidth (Mbit/s)
     & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\


@@ -771,7 +770,11 @@ the classical GMRES \\
     & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
     \hline
   \end{mytable}
     & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
     \hline
   \end{mytable}

-\end{table}
+%\end{table}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{table:01}
+\end{figure}
+

 
 \section{Conclusion}
 CONCLUSION
 
 \section{Conclusion}
 CONCLUSION