RCE : Quelques corrections

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index cea0a410d588e278172f8b1c4f6c89eda66fda5a..397decc12af88aba0e11d0f4c954cde9f731e27d 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -115,7 +115,7 @@ their  applications  using  a simulation tool before.
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 

-\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}
+\keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}

 
 \maketitle
 
 
 \maketitle
 


@@ -192,9 +192,9 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model}
 \label{sec:asynchro}
 
 \section{The asynchronous iteration model}
 \label{sec:asynchro}
 

-Asynchronous iterative methods have been  studied for many years theorecally and
+Asynchronous iterative methods have been  studied for many years theoritecally and

 practically. Many methods have been considered and convergence results have been
 practically. Many methods have been considered and convergence results have been

-proved. These  methods can  be used  to solve in  parallel fixed  point problems
+proved. These  methods can  be used  to solve, in  parallel, fixed  point problems

 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,
 asynchronous iterations  methods can be used  to solve, for example,  linear and
 non-linear systems of equations or optimization problems, interested readers are
 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,
 asynchronous iterations  methods can be used  to solve, for example,  linear and
 non-linear systems of equations or optimization problems, interested readers are


@@ -205,11 +205,13 @@ studied. Otherwise, the  application is not ensure to reach  the convergence. An
 algorithm that supports both the synchronous or the asynchronous iteration model
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In
 algorithm that supports both the synchronous or the asynchronous iteration model
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In

-the  synchronous   mode,  the  iterations   are  synchronized  whereas   in  the
-asynchronous  one,  they are  not.   It  should  be  noticed that  non  blocking
-communications can be used in  both modes. Concerning the convergence detection,
-synchronous variants  can use  a global  convergence procedure  which acts  as a
-global synchronization point. In the asynchronous model, the convergence dectection is more tricky as it must not synchronize all the processors. Interested readers can consult~\cite{myBCCV05c,bahi07,ccl09:ij}. 
+the synchronous  mode, iterations are  synchronized whereas in  the asynchronous
+one, they are not.  It should be noticed that non blocking communications can be
+used in both  modes. Concerning the convergence  detection, synchronous variants
+can use  a global convergence procedure  which acts as a  global synchronization
+point. In the  asynchronous model, the convergence detection is  more tricky as
+it   must  not   synchronize  all   the  processors.   Interested  readers   can
+consult~\cite{myBCCV05c,bahi07,ccl09:ij}.

 
 \section{SimGrid}
  \label{sec:simgrid}
 
 \section{SimGrid}
  \label{sec:simgrid}


@@ -221,12 +223,12 @@ global synchronization point. In the asynchronous model, the convergence dectect
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}

-In this paper we focus on two-stage multisplitting methods in their both versions synchronous and asynchronous~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$

 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}
 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}

-where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). The two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows

 \begin{equation}
 x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
 \label{eq:02}
 \begin{equation}
 x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
 \label{eq:02}


@@ -257,12 +259,12 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\end{algorithm}
 \end{figure}
 
 %\end{algorithm}
 \end{figure}
 

-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged

 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \end{equation}
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \end{equation}

-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 

 
 The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
 \begin{equation}
 
 The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
 \begin{equation}


@@ -305,32 +307,55 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 

-One of our objectives when simulating the application in Simgrid is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in Simgrid simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine 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.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.
+One of our objectives when simulating the  application in Simgrid is, as in real
+life, to  get accurate results  (solutions of the  problem) but also  ensure the
+test reproducibility  under the same  conditions.  According to  our experience,
+very  few modifications  are required  to adapt  a MPI  program for  the Simgrid
+simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
+libraries  and related  header files  (smpi.h).  The  second modification  is to
+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
+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}
 
 
 \paragraph{Simgrid Simulator parameters}

+\  \\ \noindent  Before running  a Simgrid  benchmark, many  parameters for  the
+computation platform must be defined. For our experiments, we consider platforms
+in which  several clusters are  geographically distant,  so there are  intra and
+inter-cluster communications. In the following, these parameters are described:

 
 \begin{itemize}
 
 \begin{itemize}

-       \item hostfile: Hosts description file.
-       \item plarform: File describing the platform architecture : clusters (CPU power,
+       \item hostfile: hosts description file.
+       \item platform: file describing the platform architecture: clusters (CPU power,

 \dots{}), intra cluster network description, inter cluster network (bandwidth bw,
 latency lat, \dots{}).
 \dots{}), intra cluster network description, inter cluster network (bandwidth bw,
 latency lat, \dots{}).

-       \item archi   : Grid computational description (Number of clusters, Number of
+       \item archi   : grid computational description (number of clusters, number of

 nodes/processors for each cluster).
 \end{itemize}
 nodes/processors for each cluster).
 \end{itemize}

-
-
+\noindent

 In addition, the following arguments are given to the programs at runtime:
 
 \begin{itemize}
 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;
-       \item Execution Mode: synchronous or asynchronous.
+       \item maximum number of inner and outer iterations;
+       \item inner and outer precisions;
+       \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}
 
 \end{itemize}
 

-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
+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.

 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%


@@ -338,78 +363,101 @@ At last, note that the two solver algorithms have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
 \section{Experimental Results}
 \label{sec:expe}
 

+In this section, experiments for both Multisplitting algorithms are reported. First the problem used in our experiments is described.
+
+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}
 
 
 \subsection{Study setup and Simulation Methodology}
 

-To conduct our study, we have put in place 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.\\

 
 

-\textbf{Step 1} : Choose with the end users the class of algorithms or
+\textbf{Step 1}: Choose with the end users the class of algorithms or

 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
+\textbf{Step 2}: Collect the software materials needed for the

 experimentation. In our case, we have two variants algorithms for the
 experimentation. In our case, we have two variants algorithms for the

-resolution of the 3D-Poisson problem: (1) using the classical GMRES (Algo-1); (2) and the multisplitting method (Algo-2). In addition, Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
+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 3} : Fix the criteria which will be used for the future
+\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
 results comparison and analysis. In the scope of this study, we retain

-in one hand the algorithm execution mode (synchronous and asynchronous)
-and in the other hand the execution time and the number of iterations of
-the application before obtaining the convergence. \\
-
-\textbf{Step 4 }: Set up the different grid testbed environments
-which will be simulated in the simulator tool to run the program. The
-following architecture has been configured in Simgrid : 2x16 - that is a
-grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8,
-4x16, 8x8 and 2x50. The network has been designed to operate with a
-bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8.10$^{-6}$
-microseconds (resp. 5.10$^{-5}$) for the intra-clusters links (resp.
-inter-clusters backbone links). \\
+on the  one hand the algorithm execution mode (synchronous and asynchronous)
+and on the other hand the execution time and the number of iterations to reach the convergence. \\
+
+\textbf{Step 4  }: Set up the  different grid testbed environments  that will be
+simulated in the  simulator tool to run the program.  The following architecture
+has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+represents the number  of clusters in the grid and  the second number represents
+the number  of hosts (processors/cores)  in each  cluster. The network  has been
+designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
+latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
+(resp.  inter-clusters backbone links). \\

 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings

-within these configurations in varying the key parameters, especially
+within these configurations by varying the key parameters, especially

 the CPU power capacity, the network parameters and also the size of the
 the CPU power capacity, the network parameters and also the size of the

-input matrix. Note that some parameters like some program input arguments should be fixed to be invariant to allow the comparison. \\
+input data.  \\

 
 \textbf{Step 6} : Collect and analyze the output results.
 
 \subsection{Factors impacting distributed applications performance in
 a grid environment}
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 
 \subsection{Factors impacting distributed applications performance in
 a grid environment}
 

-From our previous experience on running distributed application in a
-computational grid, many factors are identified to have an impact on the
-program behavior and performance on this specific environment. Mainly,
-first of all, the architecture of the grid itself can obviously
-influence the performance results of the program. The performance gain
-might be important theoretically when the number of clusters and/or the
-number of nodes (processors/cores) in each individual cluster increase.
-
-Another important factor impacting the overall performance of the
-application is the network configuration. Two main network parameters
-can modify drastically the program output results : (i) the network
-bandwidth (bw=bits/s) also known as "the data-carrying capacity"
-of the network is defined as the maximum of data that can pass
-from one point to another in a unit of time. (ii) 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. 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.
-
- In a grid environment, it is common to distinguish in one hand, the
-"\,intra-network" which refers to the links between nodes within a
-cluster and in the other hand, the "\,inter-network" which is the
-backbone link between clusters. By design, these two networks perform
-with different speed. 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 thru internet with a lower speed. The network
-between distant clusters might be a bottleneck for the global
-performance of the application.
+When running a distributed application in a computational grid, many factors may
+have a strong impact on the performances.  First of all, the architecture of the
+grid itself can obviously influence the  performance results of the program. The
+performance gain  might be important  theoretically when the number  of clusters
+and/or  the  number  of  nodes (processors/cores)  in  each  individual  cluster
+increase.
+
+Another important factor  impacting the overall performances  of the application
+is the network configuration. Two main network parameters can modify drastically
+the program output results:
+\begin{enumerate}
+\item  the network  bandwidth  (bw=bits/s) also  known  as "the  data-carrying
+    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.
+\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.
+
+ 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   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{Comparing GMRES and Multisplitting algorithms in
 synchronous mode}


@@ -590,7 +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
 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
 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


@@ -620,7 +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}}
 \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
 
 Using the Simgrid simulator flexibility, we have tried to determine the
 impact on the algorithms performance in varying the CPU power of the


@@ -723,8 +771,7 @@ CONCLUSION
 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}
 

-
-The authors would like to thank\dots{}
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

 
 
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