relecture jusqu'à fin de section 4

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index cea0a410d588e278172f8b1c4f6c89eda66fda5a..dcdfcfadc2b21bf2c56f4f7767c0dea08aafcca5 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -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 +307,48 @@ 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
+the Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?}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 matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
+       \item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifie, je dis ca de tête}
+       \item execution mode: synchronous or asynchronous.

 \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.

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


@@ -723,8 +741,7 @@ 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).

 
 
 \bibliographystyle{wileyj}
 
 
 \bibliographystyle{wileyj}