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
\section{SimGrid}
\label{sec:simgrid}
-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.
+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. Contrary to most simulators which remain very application-oriented, 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} (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 provides four user interfaces which can be convenient for different distributed applications~\cite{casanova+legrand+quinson.2008.simgrid}. In this paper we are interested in 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} (see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the execution, as in 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 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 an "in vitro" mode by sharing some 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.
+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 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.
\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}
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]
\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]
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 (Simulator MPI). The first modification is to include SMPI
libraries and related header files (\verb+smpi.h+). The second modification is to
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
