X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/e2082ae796c90fe550e1ad8e44f43f9d5bfedf95..30a67f26a836be8200ee49398df2e69bcb9a58ac:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 17f27b3..67599b9 100644
--- a/paper.tex
+++ b/paper.tex
@@ -215,7 +215,7 @@ non-linear systems of equations or optimization problems. Interested readers are
 invited to read~\cite{BT89,bahi07}.
 
 Before  using  an  asynchronous  iterative   method,  the  convergence  must  be
-studied. Otherwise, there is no garantee that the  application will reach  the convergence. An
+studied. Otherwise, there is no guarantee that the  application will 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 management and  the convergence detection are  different. In
@@ -230,7 +230,7 @@ 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 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
+number of iterations is the same as in the sequential mode. Thus, 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
 application can drastically change the number of iterations required to get the
@@ -272,7 +272,7 @@ Section~\ref{sec:expe}). SimGrid employs a fluid model to simulate the use of
 these resources along the program execution.  This model produces accurate
 results while still running relatively
 fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
-During the simulation, the computation is really executed, but the commuications
+During the simulation, the computations are really executed, but the communications
 are intercepted and their execution time evaluated according to the parameters
 of the simulated platform. It is also possible for SimGrid/SMPI to only keep the
 duration of large computations by skipping them.  Moreover, when applicable, the
@@ -389,7 +389,7 @@ real physical architectures.
 \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 both their 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}
@@ -399,12 +399,12 @@ where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side
 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}
 \end{equation}
-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel so that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
 \begin{equation}
 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~\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 the 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. Line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using the 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}[htpb]
 %\begin{algorithm}[t]
@@ -442,7 +442,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}
 \end{equation}
-The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using the CGLS method~\cite{Hestenes52} sosuch that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 
 \begin{figure}[htbp]
 %\begin{algorithm}[t]
@@ -475,7 +475,7 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 
 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,
+test reproducibility  under similar  conditions.  According to  our experience,
 very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
 simulator using SMPI (Simulated MPI). The  first modification is to include SMPI
 libraries  and related  header files  (\verb+smpi.h+).  The  second modification  is to
@@ -485,7 +485,7 @@ SimGrid selector       called      "runtime       automatic      switching"
 effects on runtime between the threads running in the same process and generated by
 SimGrid  to simulate the  grid environment.
 
-\paragraph{Parameters of the simulation in SimGrid}
+\paragraph{Simulation parameters for 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
@@ -919,3 +919,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 %%% fill-column: 80
 %%% ispell-local-dictionary: "american"
 %%% End:
+
+%  LocalWords:  Ramamonjisoa Ziane Khodja Laiymani RaphaÃ«l Arnaud Giersch Femto
+%  LocalWords:  Franche ComtÃ© Belfort GMRES multisplitting SimGrid Krylov SMPI
+%  LocalWords:  MPI
