X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/eee2ce96cdbc6990b0a059a92129c0f547a3dad2..90c05cfabfecb2d354b545272590acd3051a2796:/hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 37ad4e3..da2ec91 100644
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -1,4 +1,3 @@
-
 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}
@@ -251,8 +250,8 @@ SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
 framework to study the behavior of large-scale distributed systems.  As its name
 says, it emanates from the grid computing community, but is nowadays used to
 study grids, clouds, HPC or peer-to-peer systems.  The early versions of SimGrid
-date from 1999, but it's still actively developed and distributed as an open
-source software.  Today, it's one of the major generic tools in the field of
+date from 1999, but it is still actively developed and distributed as an open
+source software.  Today, it is one of the major generic tools in the field of
 simulation for large-scale distributed systems.
 
 SimGrid provides several programming interfaces: MSG to simulate Concurrent
@@ -422,7 +421,7 @@ u =0 \text{~on~} \Gamma =\partial\Omega
 \right.
 \label{eq:02}
 \end{equation}
-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite differences scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
 \begin{equation}
 \begin{array}{l}
 u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
@@ -450,7 +449,7 @@ The parallel solving of the 3D Poisson problem with our multisplitting method re
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code 
-debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method , the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method, the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
 and with the addition of the primitive \texttt{MPI\_Test} was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
 %\CER{On voulait en fait montrer la simplicitÃ© de l'adaptation de l'algo a SimGrid. Les problÃ¨mes rencontrÃ©s dÃ©crits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais prÃ©fÃ©rÃ© avoir un peu plus de dÃ©tails sur l'adaptation de la version async} 
 %\CER{Le problÃ¨me majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a Ã©tÃ© le plantage en SMPI de Waitall aprÃ¨s un Isend et Irecv. J'avais proposÃ© un workaround en utilisant un MPI\_wait sÃ©parÃ© pour chaque Ã©change a la place d'un waitall unique pour TOUTES les Ã©changes, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais aprÃ¨s, tu as modifiÃ© le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de dÃ©tection de la convergence et du coup, l'Ã©change global avec waitall a aussi fonctionnÃ©.}
@@ -493,7 +492,7 @@ simulates the case of distant clusters linked with long distance network as in g
 
 
 Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem  ranges from $N=N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
 $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
 \text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
 %\AG{Expliquer comment lire les tableaux.}
@@ -523,7 +522,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       \\
     \hline
-    size $(n^3)$
+    size $(N)$
     & 62        & 62        & 62        & 100       & 100       \\
     \hline
     Precision
@@ -548,7 +547,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     Power (GFlops)
     & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline
-    size $(n^3)$
+    size $(N)$
     & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
     \hline
     Precision
@@ -647,11 +646,11 @@ Note that the program was run with the following parameters:
 \begin{itemize}
 \item Description of the cluster architecture matching the format <Number of
   clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
-\item Maximum number of iterations;
-\item Precisions on the residual error;
+\item Maximum numbers of outer and inner iterations;
+\item Outer and inner precisions on the residual error;
 \item Matrix size $N_x$, $N_y$ and $N_z$;
-\item Matrix diagonal value: $6$ (See Equation~(\ref{eq:03}));
-\item Matrix off-diagonal value: $-1$;
+\item Matrix diagonal value: $6$ (see Equation~(\ref{eq:03}));
+\item Matrix off-diagonal values: $-1$;
 \item Communication mode: asynchronous.
 \end{itemize}
 
@@ -664,12 +663,12 @@ asynchronous multisplitting  compared to GMRES with two distant clusters.
 With these settings, Table~\ref{tab.cluster.2x50} shows
 that after setting the bandwidth of the  inter cluster network to  \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
 of one GFlops, an efficiency of about \np[\%]{40} is
-obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+obtained in asynchronous mode for a matrix size of $62^3$ elements. It is noticed that the result remains
 stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
-increasing the matrix size up to 100 elements, it was necessary to increase the
+increasing the matrix size up to $100^3$ elements, it was necessary to increase the
 CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency.  Maintaining such processor power but increasing network throughput inter cluster up to
 \np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
-high external precision of \np{E-11} for a matrix size from 110 to 150 side
+high external precision of \np{E-11} for a matrix size from $110^3$ to $150^3$ side
 elements.
 
 %For the 3 clusters architecture including a total of 100 hosts,
@@ -704,7 +703,7 @@ reach the following two objectives:
 
 \item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
 \end{enumerate}
-Our results have shown that with two distant clusters, the asynchronous multisplitting is faster to \np[\%]{40} compared to the synchronous GMRES method
+Our results have shown that with two distant clusters, the asynchronous multisplitting method is faster to \np[\%]{40} compared to the synchronous GMRES method
 which is not negligible for solving complex practical problems with more 
 and more increasing size.
 
@@ -715,7 +714,7 @@ tool to run efficiently an iterative parallel algorithm in asynchronous
 mode in a grid architecture. 
 
 In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters. 
-We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study.
+We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study. Finally, we also plan to study other problems with the multisplitting method and other asynchronous iterative methods.
 
 \section*{Acknowledgment}
 
