petites modifs intro

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 6affca8845a1f083b1bb76c328a147986eda8e22..25bf4d15fef4fe511ebb2e022e2a6d5083f80554 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -114,7 +114,7 @@ their  applications  using  a simulation tool before.
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;

-%performance} 
+%performance}

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


@@ -131,28 +131,28 @@ are often very important. So, in this context it is difficult to optimize a
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,

-code debugging, ability to obtain results quickly,~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.

 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite
 simple. It generally involves the division of the problem into  several
 \emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing
 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite
 simple. It generally involves the division of the problem into  several
 \emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing

-units.  Each processing unit has to compute an iteration, to send/receive some
+units.  Each processing unit has to compute an iteration to send/receive some

 data dependencies to/from its neighbors and to iterate this process until the
 data dependencies to/from its neighbors and to iterate this process until the

-convergence of the method. Several well-known methods demonstrate the
+convergence of the method. Several well-known studies demonstrate the

 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies
 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies

-from its neighbors. We say that the iteration computation follows a synchronous
-scheme. In the asynchronous scheme a task can compute a new iteration without
-having to wait for the data dependencies coming from its neighbors. Both
-communication and computations are asynchronous inducing that there is no more
-idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
-This model presents some advantages and drawbacks that we detail in
-section~\ref{sec:asynchro} but even if the number of iterations required to
-converge is generally  greater  than for the synchronous  case, it appears that
-the asynchronous  iterative scheme  can significantly  reduce  overall execution
-times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
-for more details).
+from its neighbors. We say that the iteration computation follows a
+\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
+iteration without having to wait for the data dependencies coming from its
+neighbors. Both communication and computations are \textit{asynchronous}
+inducing that there is no more idle time, due to synchronizations, between two
+iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
+that we detail in section~\ref{sec:asynchro} but even if the number of
+iterations required to converge is generally  greater  than for the synchronous
+case, it appears that the asynchronous  iterative scheme  can significantly
+reduce  overall execution times by  suppressing idle  times due to
+synchronizations~(see~\cite{bahi07} for more details).

 
 Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
 consuming to find optimal configuration  and deployment requirements for a given
 
 Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
 consuming to find optimal configuration  and deployment requirements for a given


@@ -264,7 +264,7 @@ In this paper, we propose two algorithms of two-stage multisplitting methods. Th
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \end{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}


@@ -354,7 +354,7 @@ In addition, the following arguments are given to the programs at runtime:
        \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 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. 
+       \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.
 \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.


@@ -379,16 +379,16 @@ 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))
 \end{array}
 \label{eq:08}
 \end{equation}
 \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. 
+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. 
+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}
 


@@ -461,8 +461,7 @@ transit between the clusters and nodes during the code execution.
  speed.  The network  between distant  clusters might  be a  bottleneck for  the
  global performance of the application.
 
  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}

 
 In the scope  of this paper, our  first objective is to analyze  when the Krylov
 Multisplitting  method   has  better  performances  than   the  classical  GMRES
 
 In the scope  of this paper, our  first objective is to analyze  when the Krylov
 Multisplitting  method   has  better  performances  than   the  classical  GMRES


@@ -481,7 +480,9 @@ and our comments.\\
 architecture and scaling up the input matrix size}
 \ \\
 % environment
 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
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline


@@ -489,26 +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}
  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}
 
 
 
 
 
 %\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!]
 \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{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
 
 The execution time difference between the two algorithms is important when
 comparing between different grid architectures, even with the same number of


@@ -673,7 +682,7 @@ 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,
 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. 
+after adding more powerful CPU.

 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}
 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}