X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/4f3f422c55e11ca375bf5f17203c5d82ed9928aa..f705ba070d8b46cd38e2cdf073408f9dcfddb6ef:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
index 377a87f..dcdfcfa 100644
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+++ b/paper.tex
@@ -90,7 +90,7 @@ analysis of simulated grid-enabled numerical iterative algorithms}
 
 %% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
 
-\begin{abstract}   The behavior of multicore applications is always a challenge
+\begin{abstract}   The behavior of multi-core applications is always a challenge
 to predict, especially with a new architecture for which no experiment has been
 performed. With some applications, it is difficult, if not impossible, to build
 accurate performance models. That is why another solution is to use a simulation
@@ -101,7 +101,7 @@ applications.
 
 In this paper, we focus our attention on two parallel iterative algorithms based
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
-These algorithms  are used to  solve libear  systems. Two different  variantsof
+These algorithms  are used to  solve libear  systems. Two different  variants of
 the Multisplitting are studied: one  using synchronoous  iterations and  another
 one  with asynchronous iterations. For each algorithm we have  tested different
 parameters to see their influence.  We strongly  recommend people  interested
@@ -119,8 +119,8 @@ their  applications  using  a simulation tool before.
 
 \maketitle
 
-\section{Introduction}  The use of multi-core architectures for solving large
-scientific problems seems to  become imperative  in  a  lot  of  cases.
+\section{Introduction}  The use of multi-core architectures to solve large
+scientific problems seems to  become imperative  in  many situations.
 Whatever the scale of these architectures (distributed clusters, computational
 grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
 complex parallel applications operating on a large amount of data.
@@ -131,8 +131,7 @@ 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,
-code debugging, ability to obtain results quickly,~\ldots at the condition that
-the simulation results are in education 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
@@ -147,7 +146,7 @@ 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 times, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
+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
@@ -155,15 +154,15 @@ 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
-application on a given multi-core architecture. Finding good resource
-allocations policies under varying CPU power, network speeds and  loads is very
-challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
-problematic is even more difficult for the asynchronous scheme  where variations
-of the parameters of the execution platform can lead to very different number of
-iterations required to converge and so to very different execution times. In
-this challenging context we think that the use of a simulation tool can greatly
+Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
+consuming to find optimal configuration  and deployment requirements for a given
+application  on   a  given   multi-core  architecture.  Finding   good  resource
+allocations policies under  varying CPU power, network speeds and  loads is very
+challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
+problematic is  even more difficult  for the  asynchronous scheme where  a small
+parameter variation of the execution platform can lead to very different numbers
+of iterations to reach the converge and so to very different execution times. In
+this challenging context we think that the  use of a simulation tool can greatly
 leverage the possibility of testing various platform scenarios.
 
 The main contribution of this paper is to show that the use of a simulation tool
@@ -175,7 +174,7 @@ algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
 solver~\cite{saad86} in synchronous mode. The obtained results on different
 simulated multi-core architectures confirm the real results previously obtained
 on non simulated architectures.  We also confirm  the efficiency  of the
-asynchronous  multisplitting algorithm  comparing to the synchronous  GMRES. In
+asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. In
 this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
@@ -193,6 +192,27 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model}
 \label{sec:asynchro}
 
+Asynchronous iterative methods have been  studied for many years theoritecally 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
+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, 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
+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}
 
@@ -203,12 +223,12 @@ concluding remarks and perspectives.
 \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}
-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}
@@ -239,12 +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}
 
-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}
-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}
@@ -287,32 +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}
 
-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}
+\  \\ \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}
-	\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{}).
-	\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}
-
-
+\noindent
 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}
 
-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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
@@ -705,8 +741,7 @@ CONCLUSION
 
 \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}