X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/5319c6448989a67e062c412732f84404b3195ae0..154021e5b0685f238d989478d56a2a6c2d55dc84:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 094f1aa..efbda8a 100644
--- a/paper.tex
+++ b/paper.tex
@@ -204,11 +204,11 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 
-Asynchronous iterative methods have been  studied for many years theoritecally and
+Asynchronous iterative methods have been  studied for many years theoretically and
 practically. Many methods have been considered and convergence results have been
 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
+asynchronous iteration  methods can be used  to solve, for example,  linear and
 non-linear systems of equations or optimization problems, interested readers are
 invited to read~\cite{BT89,bahi07}.
 
@@ -218,7 +218,7 @@ 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, iterations are  synchronized whereas in  the asynchronous
-one, they are not.  It should be noticed that non blocking communications can be
+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
@@ -226,17 +226,17 @@ it   must  not   synchronize  all   the  processors.   Interested  readers   can
 consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 
 The number of iterations required to reach the convergence is generally greater
-for the asynchronous scheme (this number depends depends on  the delay of the
+for the asynchronous scheme (this number depends on  the delay of the
 messages). Note that, it is not the case in the synchronous mode where the
 number of iterations is the same than in the sequential mode. In this way, the
 set of the parameters  of the  platform (number  of nodes,  power of nodes,
-inter and  intra clusters  bandwidth  and  latency, \ldots) and  of  the
+inter and  intra clusters  bandwidth  and  latency,~\ldots) and  of  the
 application can drastically change the number of iterations required to get the
 convergence. It follows that asynchronous iterative algorithms are difficult to
 optimize since the financial and deployment costs on large scale multi-core
-architecture are often very important. So, prior to delpoyment and tests it
+architectures are often very important. So, prior to deployment and tests it
 seems very promising to be able to simulate the behavior of asynchronous
-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced
 by simulation are in accordance with reality i.e. of the same order of
 magnitude. To our knowledge, there is no study on this problematic.
 
@@ -317,7 +317,7 @@ where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
 
 \begin{figure}[t]
 %\begin{algorithm}[t]
@@ -386,26 +386,20 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of the two-stage methods using SimGrid toolkit}
 \label{sec:04.02}
 
-One of our objectives when simulating the  application in Simgrid is, as in real
+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 to 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
+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
+libraries  and related  header files  (\verb+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"
+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 and generated by
-Simgrid  to simulate the  grid environment.
+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}
-\  \\ \noindent  Before running  a Simgrid  benchmark, many  parameters for  the
+\paragraph{Parameters of the simulation in SimGrid}
+\  \\ \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:
@@ -413,10 +407,10 @@ inter-cluster communications. In the following, these parameters are described:
 \begin{itemize}
 	\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
-nodes/processors for each cluster).
+nodes/processors in each cluster).
 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:
@@ -424,8 +418,8 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
 	\item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
 	\item inner precision $\TOLG$ and outer precision $\TOLM$,
-	\item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
-	\item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+	\item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+	\item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
 	\item matrix off-diagonal value is fixed to $-1.0$,
 	\item number of vectors in matrix $S$ (i.e. value of $s$),
 	\item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
@@ -434,7 +428,7 @@ In addition, the following arguments are given to the programs at runtime:
       	\item execution mode: synchronous or asynchronous.
 \end{itemize}
 
-It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-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 \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
@@ -445,6 +439,7 @@ 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.
 
 \subsection{The 3D Poisson problem}
+\label{3dpoisson}
 
 
 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:
