Reformat source for sec.simgrid.

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 43368b77e886a80239431bbd92725e3e0708ed5a..eb8020abd087b289382b6e136d2561578dcb05e1 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,4 @@
-\documentclass[times]{cpeauth}
+ \documentclass[times]{cpeauth}

 
 \usepackage{moreverb}
 
 
 \usepackage{moreverb}
 


@@ -77,10 +77,10 @@
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{Charles Emile Ramamonjisoa\affil{1},
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{Charles Emile Ramamonjisoa\affil{1},

-    David Laiymani\affil{1},
-    Arnaud Giersch\affil{1},
-    Lilia Ziane Khodja\affil{2} and
-    Raphaël Couturier\affil{1}
+  Lilia Ziane Khodja\affil{2},
+  David Laiymani\affil{1},
+  Raphaël Couturier\affil{1} and
+  Arnaud Giersch\affil{1}

 }
 
 \address{
 }
 
 \address{


@@ -113,7 +113,7 @@
 %% execution time.
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 
 %% execution time.
 %% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
 

-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 tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
+The behavior of multi-core applications always proves quite challenging to predict, especially with a new architecture for which no experiment has yet 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 tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.

 
 In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
 
 
 In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
 


@@ -133,11 +133,11 @@ complex parallel applications operating on a large amount of data.
 Unfortunately,  users (industrials or scientists),  who need such computational
 resources, may not have an easy access to such efficient architectures. The cost
 of using the platform and/or the cost of  testing and deploying an application
 Unfortunately,  users (industrials or scientists),  who need such computational
 resources, may not have an easy access to such efficient architectures. The cost
 of using the platform and/or the cost of  testing and deploying an application

-are often very important. So, in this context it is difficult to optimize a
+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
 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\dots{} In counterpart, the simulation results need to be consistent with the real ones.
+simulation environment.  The advantages are numerous: life cycle development,
+code debugging, ability to obtain results quickly\dots{} In return, 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
 
 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


@@ -148,26 +148,26 @@ data dependencies to/from its neighbors and to iterate this process until the
 convergence of the method. Several well-known studies demonstrate the
 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies
 convergence of the method. Several well-known studies demonstrate the
 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies

-from its neighbors. We say that the iteration computation follows a
+from its neighbors. The iteration computation is said to follow a

 \textit{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 communications and computations are \textit{asynchronous}
 inducing that there is no more idle time, due to synchronizations, between two
 iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
 \textit{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 communications and computations are \textit{asynchronous}
 inducing that there is no more 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
+that we detail in Section~\ref{sec:asynchro}. Even if the number of

 iterations required to converge is generally  greater  than for the synchronous
 case, it appears that the asynchronous  iterative scheme  can significantly
 reduce  overall execution times by  suppressing idle  times due to
 synchronizations~(see~\cite{bahi07} for more details).
 
 iterations required to converge is generally  greater  than for the synchronous
 case, it appears that 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
+Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is extremely  time
+consuming to find optimal configurations  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 and of the application data can
 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 and of the application data can

-lead to very different numbers of iterations to reach the convergence and so to
+lead to very different numbers of iterations to reach the convergence and consequently 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.
 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.


@@ -180,8 +180,8 @@ validity of this approach we first compare the simulated execution of the Krylov
 multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  RESidual)
 solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
 determine  which method  to choose  for a given multi-core  architecture.
 multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  RESidual)
 solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
 determine  which method  to choose  for a given multi-core  architecture.

-Moreover the  obtained results  on different simulated  multi-core architectures
-confirm the  real results  previously obtained  on non  simulated architectures.
+Moreover, the  obtained results  on different simulated  multi-core architectures
+confirm the  real results  previously obtained  on real physical architectures.

 More precisely the simulated results are in accordance (i.e. with the same order
 of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
 the synchronous  Krylov multisplitting method  is more efficient  than GMRES  for large
 More precisely the simulated results are in accordance (i.e. with the same order
 of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
 the synchronous  Krylov multisplitting method  is more efficient  than GMRES  for large


@@ -189,8 +189,8 @@ scale  clusters.   Simulated   results  also  confirm  the   efficiency  of  the
 asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
 especially in case of geographically distant clusters.
 
 asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
 especially in case of geographically distant clusters.
 

-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
+Thus, with a simple computing architecture (a laptop) SimGrid allows us
+to run a test campaign  of  real parallel iterative  applications on

 different simulated multi-core architectures.  To our knowledge, there is no
 related work on the large-scale multi-core simulation of a real synchronous and
 asynchronous iterative application.
 different simulated multi-core architectures.  To our knowledge, there is no
 related work on the large-scale multi-core simulation of a real synchronous and
 asynchronous iterative application.


@@ -206,20 +206,20 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 
 \section{The asynchronous iteration model and the motivations of our work}
 \label{sec:asynchro}
 

-Asynchronous iterative methods have been  studied for many years theoretically and
+Asynchronous iterative methods have been  studied for many years both 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
 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,
+(i.e. problems  for which  the solution is  $x^\star =f(x^\star)$).  In practice,

 asynchronous iteration  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
+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
 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
+studied. Otherwise, there is no garantee 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
 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
+practice, only  the communications management and  the 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
 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


@@ -239,55 +239,90 @@ optimize since the financial and deployment costs on large scale multi-core
 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 produced
 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 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.
+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.

 
 \section{SimGrid}
 \label{sec:simgrid}
 
 
 \section{SimGrid}
 \label{sec:simgrid}
 

-In the scope of this paper, the Simgrid
-toolkitSimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
-an open source framework actively developped by its community, has been choosen
-to simulate the behavior of the solvers algorithms in different grid
-computational configurations. Simgrid pretends to be non-specialized in opposite
-to some other simulators which stayed to be very specific oriented-application.
-One of the well-known Simgrid advantage is its SMPI (Simulated MPI). SMPI
-purpose is to execute by simulation in a similar way as in real life, an MPI
-distributed application and to get accurate results with the detailed resources
-consumption. Several studies have demonstrated the accuracy of the simulation
-compared with execution on real physical architectures. In addition of SMPI,
-Simgrid provides other API which can be convienent for different distrbuted
-applications: computational grid applications, High Performance Computing (HPC),
-P2P but also clouds applications.In this paper we use the SMPI API. It
+In the scope of this paper, we have chosen the SimGrid
+toolkit~\cite{SimGrid,casanova+giersch+legrand+al.2014.versatile}
+to simulate the behavior of parallel iterative linear solvers on different
+computational grid configurations. In opposite to most of the simulators which
+are stayed very oriented-application, the SimGrid framework is designed to study
+the behavior of many large-scale distributed computing platforms as Grids,
+Peer-to-Peer systems, Clouds or High Performance Computation systems. It is
+still actively developed by the scientific community and distributed as an open
+source software.
+
+SimGrid provides four user interfaces which can be convenient for different
+distributed applications~\cite{casanova+legrand+quinson.2008.simgrid}. In this
+paper we are interested on the SMPI user interface (Simulator MPI) which

 implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
 modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
 implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
 modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}

-(see section 4.2).
-
-
- Provided as an input to the simulator, at least $3$ XML files describe the
- computational grid resources: number of clusters in the grid, number of
- processors/cores in each cluster, detailed description of the intra and inter
- networks and the list of the hosts in each cluster (see the details in
- paragraph ?). Simgrid uses a fluid model to simulate the program execution.
- This gives several simulation modes which produce accurate
- results~\cite{bedaride+degomme+genaud+al.2013.toward,
- velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode
- really executes the computation but "intercepts" the communications (running
- time is then evaluated according to the parameters of the simulated platform).
- It is also possible for SimGrid/SMPI to only keep duration of large
- computations by skipping them. Moreover the application can be run "in vitro"
- by sharing some in-memory structures between the simulated processes and
- thus allowing the use of very large data scale.
-
-
-The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized
-by the results obtained by several studies to validate, in real environments,
-the behavior of different network models simulated in
-Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
-the comparison between real MPI executions  and SimGrid/SMPI
-ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single,
-bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of
-SimGrid simulations.
+(see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the
+execution, as in the real life, of an unmodified MPI distributed application,
+and gets accurate results with the detailed resources consumption.
+
+SimGrid simulator uses at least three XML input files describing the
+computational grid resources: the number of clusters in the grid, the number of
+processors/cores in each cluster, the detailed description of the intra and
+inter networks and the list of the hosts in each cluster (see the details in
+Section~\ref{sec:expe}). SimGrid uses a fluid model to simulate the program
+execution. It allows several simulation modes which produce accurate
+results~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}. For
+instance, the "in vivo" mode really executes the computation but "intercepts"
+the communications (the execution time is then 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 the
+application can be run "in vitro" mode by sharing some in-memory structures
+between the simulated processes and thus allowing the use of very large-scale
+data.
+
+The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized
+by the results obtained by several studies to validate, in the real
+environments, the behavior of different network models simulated in
+SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
+the comparison between the real MPI application executions and the SimGrid/SMPI
+ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These
+works show the accuracy of SimGrid simulations compared to the executions on
+real physical architectures.
+
+%% In the scope of this paper, the SimGrid toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
+%% an open source framework actively developed by its scientific community, has been chosen to simulate the behavior of iterative linear solvers in different computational grid configurations. SimGrid pretends to be non-specialized in opposite to some other simulators which stayed to be very specific oriented-application. One of the well-known SimGrid advantage is its SMPI (Simulated MPI) user interface. SMPI purpose is to execute by simulation in a similar way as in real life, an MPI distributed application and to get accurate results with the detailed resources
+%% consumption.Several studies have demonstrated the accuracy of the simulation
+%% compared with execution on real physical architectures. In addition of SMPI,
+%% Simgrid provides other API which can be convienent for different distrbuted
+%% applications: computational grid applications, High Performance Computing (HPC),
+%% P2P but also clouds applications. In this paper we use the SMPI API. It
+%% implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
+%% modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
+%% (see Section~\ref{sec:04.02}).
+
+
+%%  Provided as an input to the simulator, at least $3$ XML files describe the
+%%  computational grid resources: number of clusters in the grid, number of
+%%  processors/cores in each cluster, detailed description of the intra and inter
+%%  networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). Simgrid uses a fluid model to simulate the program execution.
+%%  This gives several simulation modes which produce accurate
+%%  results~\cite{bedaride+degomme+genaud+al.2013.toward,
+%%  velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode
+%%  really executes the computation but "intercepts" the communications (running
+%%  time is then evaluated according to the parameters of the simulated platform).
+%%  It is also possible for SimGrid/SMPI to only keep duration of large
+%%  computations by skipping them. Moreover the application can be run "in vitro"
+%%  by sharing some in-memory structures between the simulated processes and
+%%  thus allowing the use of very large data scale.
+
+
+%% The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized
+%% by the results obtained by several studies to validate, in real environments,
+%% the behavior of different network models simulated in
+%% Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
+%% the comparison between real MPI executions  and SimGrid/SMPI
+%% ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single,
+%% bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of
+%% SimGrid simulations.

 
 
 
 
 
 


@@ -390,7 +425,7 @@ 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 the asynchronous model which allows 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 scheme which allows 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}
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}


@@ -664,8 +699,8 @@ efficient for distributed systems with high latency networks.
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}

-\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
-\RCE{ok}
+%\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+%\RCE{ok}

 \label{fig:02}
 \end{figure}
 
 \label{fig:02}
 \end{figure}
 


@@ -778,12 +813,12 @@ summarized in Table~\ref{tab:02}.
  \hline
  Grid architecture                       & 2$\times$50 totaling 100 processors\\
  Processors Power                        & 1 GFlops to 1.5 GFlops \\
  \hline
  Grid architecture                       & 2$\times$50 totaling 100 processors\\
  Processors Power                        & 1 GFlops to 1.5 GFlops \\

- \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
-                                         & $bw$=5 Mbits, $lat=20ms$\\
+ \multirow{2}{*}{Network inter-clusters} & $bw$: 5 Mbits to 50 Mbits\\
+                                         & $lat$: 20 ms\\

  Matrix size                             & from $62^3$ to $150^3$\\
  Matrix size                             & from $62^3$ to $150^3$\\

- Residual error precision                & $10^{-5}$ to $10^{-9}$\\ \hline \\
+ Residual error precision                & $10^{-5}$ to $10^{-11}$\\ \hline \\

  \end{tabular}
  \end{tabular}

-\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}

 \label{tab:02}
 \end{table}
 
 \label{tab:02}
 \end{table}
 


@@ -824,7 +859,7 @@ summarized in Table~\ref{tab:02}.
     \hline
   \end{mytable}
 %\end{table}
     \hline
   \end{mytable}
 %\end{table}

- \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \caption{Relative gains of the asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}

  \label{tab:03}
 \end{table}
 
  \label{tab:03}
 \end{table}