+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.
+
+
+
+
+
+
+% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%
+% % SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+% % is a simulation framework to study the behavior of large-scale distributed
+% % systems. As its name suggests, it emanates from the grid computing community,
+% % but is nowadays used to study grids, clouds, HPC or peer-to-peer systems. The
+% % early versions of SimGrid date back from 1999, but it is still actively
+% % developed and distributed as an open source software. Today, it is one of the
+% % major generic tools in the field of simulation for large-scale distributed
+% % systems.
+%
+% SimGrid provides several programming interfaces: MSG to simulate Concurrent
+% Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
+% run real applications written in MPI~\cite{MPI}. Apart from the native C
+% interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
+% languages. SMPI is the interface that has been used for the work described in
+% this paper. The SMPI interface implements about \np[\%]{80} of the MPI 2.0
+% standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+% applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B).
+%
+% Within SimGrid, the execution of a distributed application is simulated by a
+% single process. The application code is really executed, but some operations,
+% like communications, are intercepted, and their running time is computed
+% according to the characteristics of the simulated execution platform. The
+% description of this target platform is given as an input for the execution, by
+% means of an XML file. It describes the properties of the platform, such as
+% the computing nodes with their computing power, the interconnection links with
+% their bandwidth and latency, and the routing strategy. The scheduling of the
+% simulated processes, as well as the simulated running time of the application
+% are computed according to these properties.
+%
+% To compute the durations of the operations in the simulated world, and to take
+% into account resource sharing (e.g. bandwidth sharing between competing
+% communications), SimGrid uses a fluid model. This allows users to run relatively fast
+% simulations, while still keeping accurate
+% results~\cite{bedaride+degomme+genaud+al.2013.toward,
+% velho+schnorr+casanova+al.2013.validity}. Moreover, depending on the
+% simulated application, SimGrid/SMPI allows to skip long lasting computations and
+% to only take their duration into account. When the real computations cannot be
+% skipped, but the results are unimportant for the simulation results, it is
+% also possible to share dynamically allocated data structures between
+% several simulated processes, and thus to reduce the whole memory consumption.
+% These two techniques can help to run simulations on a very large scale.
+%
+% The validity of simulations with SimGrid has been asserted by several studies.
+% See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+% referenced therein for the validity of the network models. Comparisons between
+% real execution of MPI applications on the one hand, and their simulation with
+% SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+% clauss+stillwell+genaud+al.2011.single,
+% bedaride+degomme+genaud+al.2013.toward}. All these works conclude that
+% SimGrid is able to simulate pretty accurately the real behavior of the
+% applications.
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage multisplitting methods}
+\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 $\mathbb{R}$:
+\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). 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}
+\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:
+\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}.
+
+\begin{figure}[htpb]
+%\begin{algorithm}[t]
+%\caption{Block Jacobi two-stage multisplitting method}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+ \State Send $x_\ell^k$ to neighboring clusters\label{send}
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+ \EndFor
+\end{algorithmic}
+\caption{Block Jacobi two-stage multisplitting method}
+\label{alg:01}
+%\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:
+\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.
+
+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}
+S=[x^1,x^2,\ldots,x^s],~s\ll n.
+\label{eq:05}
+\end{equation}
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual:
+\begin{equation}
+\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}).
+
+\begin{figure}[htbp]
+%\begin{algorithm}[t]
+%\caption{Krylov two-stage method using block Jacobi multisplitting}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
+ \State $S_{\ell,k\mod s}=x_\ell^k$
+ \If{$k\mod s = 0$}
+ \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
+ \State $\tilde{x_\ell}=S_\ell\alpha$
+ \State Send $\tilde{x_\ell}$ to neighboring clusters
+ \Else
+ \State Send $x_\ell^k$ to neighboring clusters
+ \EndIf
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
+ \EndFor
+\end{algorithmic}
+\caption{Krylov two-stage method using block Jacobi multisplitting}
+\label{alg:02}
+%\end{algorithm}
+\end{figure}
+
+\subsection{Simulation of the two-stage methods using SimGrid toolkit}
+\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 to 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 (\verb+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 and generated by
+SimGrid to simulate the grid environment.
+
+\paragraph{Parameters of the simulation in 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
+inter-cluster communications. In the following, these parameters are described:
+
+\begin{itemize}
+ \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
+nodes/processors in each cluster).
+\end{itemize}
+\noindent
+In addition, the following arguments are given to the programs at runtime:
+
+\begin{itemize}
+ \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
+ \item inner precision $\TOLG$ and outer precision $\TOLM$,
+ \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+ \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
+ \item matrix off-diagonal value is fixed to $-1.0$,
+ \item number of vectors in matrix $S$ (i.e. value of $s$),
+ \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
+ \item maximum number of iterations and precision for the classical GMRES method,
+ \item maximum number of restarts for the Arnorldi process in GMRES method,
+ \item execution mode: synchronous or asynchronous.
+\end{itemize}
+
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Experimental results}
+\label{sec:expe}
+
+In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+
+\subsection{The 3D Poisson problem}
+\label{3dpoisson}
+We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
+\begin{equation}
+\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
+\label{eq:07}
+\end{equation}
+such that:
+\begin{equation*}
+\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
+\end{equation*}
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that:
+\begin{equation}
+\begin{array}{ll}
+\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
+\end{array}
+\label{eq:08}
+\end{equation}
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
+
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
+sub-problems such that $L$ is the number of clusters and $p$ is the number of
+processors in each cluster. We apply the three-dimensional partitioning instead
+of the row-by-row one in order to reduce the size of the data shared at the
+sub-problems boundaries. In this case, each processor is in charge of
+parallelepipedic block of the problem and has at most six neighbors in the same
+cluster or in distant clusters with which it shares data at boundaries.
+
+\subsection{Study setup and simulation methodology}
+
+First, to conduct our study, we propose the following methodology
+which can be reused for any grid-enabled applications.\\
+
+\textbf{Step 1}: Choose with the end users the class of algorithms or
+the application to be tested. Numerical parallel iterative algorithms
+have been chosen for the study in this paper. \\
+
+\textbf{Step 2}: Collect the software materials needed for the experimentation.
+In our case, we have two variants algorithms for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+method. In addition, the SimGrid simulator has been chosen to simulate the
+behaviors of the distributed applications. SimGrid is running in a virtual
+machine on a simple laptop. \\
+
+\textbf{Step 3}: Fix the criteria which will be used for the future
+results comparison and analysis. In the scope of this study, we retain
+on the one hand the algorithm execution mode (synchronous and asynchronous)
+and on the other hand the execution time and the number of iterations to reach the convergence. \\
+
+\textbf{Step 4}: Set up the different grid testbed environments that will be
+simulated in the simulator tool to run the program. The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
+represents the number of clusters in the grid and the second number represents
+the number of hosts (processors/cores) in each cluster. \\
+
+\textbf{Step 5}: Conduct an extensive and comprehensive testings
+within these configurations by varying the key parameters, especially
+the CPU power capacity, the network parameters and also the size of the
+input data. \\
+
+\textbf{Step 6} : Collect and analyze the output results.
+
+\subsection{Factors impacting distributed applications performance in a grid environment}
+
+When running a distributed application in a computational grid, many factors may
+have a strong impact on the performance. First of all, the architecture of the
+grid itself can obviously influence the performance results of the program. The
+performance gain might be important theoretically when the number of clusters
+and/or the number of nodes (processors/cores) in each individual cluster
+increase.
+
+Another important factor impacting the overall performance of the application
+is the network configuration. Two main network parameters can modify drastically
+the program output results:
+\begin{enumerate}
+\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
+ capacity" of the network is defined as the maximum of data that can transit
+ from one point to another in a unit of time.
+\item the network latency ($lat$ in microseconds) defined as the delay from the
+ start time to send a simple data from a source to a destination.
+\end{enumerate}
+Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
+and between distant clusters. This parameter is application dependent.
+
+ In a grid environment, it is common to distinguish, on one hand, the
+ \textit{intra-network} which refers to the links between nodes within a
+ cluster and on the other hand, the \textit{inter-network} which is the
+ backbone link between clusters. In practice, these two networks have
+ different speeds. The intra-network generally works like a high speed
+ local network with a high bandwidth and very low latency. In opposite, the
+ inter-network connects clusters sometime via heterogeneous networks components
+ through internet with a lower speed. The network between distant clusters
+ might be a bottleneck for the global performance of the application.
+
+
+\subsection{Comparison between GMRES and two-stage multisplitting algorithms in
+synchronous mode}
+In the scope of this paper, our first objective is to analyze
+when the synchronous Krylov two-stage method has better performance than the
+classical GMRES method. With a synchronous iterative method, better performance
+means a smaller number of iterations and execution time before reaching the
+convergence.
+
+Table~\ref{tab:01} summarizes the parameters used in the different simulations:
+the grid architectures (i.e. the number of clusters and the number of nodes per
+cluster), the network of inter-clusters backbone links and the matrix sizes of
+the 3D Poisson problem. However, for all simulations we fix the network
+parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
+$lat=8\mu$s. In what follows, we will present the test conditions, the output
+results and our comments.
+
+\begin{table} [ht!]
+\begin{center}
+\begin{tabular}{ll}
+\hline
+Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
+ & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
+\end{tabular}
+\caption{Parameters for the different simulations}
+\label{tab:01}
+\end{center}
+\end{table}
+
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
+
+In this section, we analyze the simulations conducted on various grid
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of the network between clusters is fixed to $N2$ (see
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
+a given matrix size of 170$^3$ elements, a non-variation in the number of
+iterations for the classical GMRES algorithm, which is not the case of the
+Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number
+of splitting (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splitting is, the slower the
+convergence of the algorithm is (see the output results obtained from
+configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
+8$\times$8).
+
+The execution times between both algorithms is significant with different grid
+architectures. The synchronous Krylov two-stage algorithm presents better
+performances than the GMRES algorithm, even for a high number of clusters (about
+$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+observe a better sensitivity of the Krylov two-stage algorithm (compared to the
+GMRES one) when scaling up the number of the processors in the computational
+grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
+about $40\%$ better on $64$ processors (grid of 8$\times$8) than $32$ processors
+(grid of 2$\times$16).
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\end{center}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
+\label{fig:01}
+\end{figure}
+
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In Figure~\ref{fig:02} we present the execution times of both algorithms to
+solve a 3D Poisson problem of size $150^3$ on two different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure that the Krylov two-stage algorithm is sensitive to the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an interesting behavior of the Krylov two-stage algorithm. It is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the GMRES algorithms. This means that the multisplitting methods are more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]
+\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}
+\label{fig:02}
+\end{figure}
+
+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impacts on performances}
+\label{fig:03}
+\end{figure}
+
+\subsubsection{Network bandwidth impacts on performances\\}
+
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
+solve a 3D Poisson problem of size $150^3$. The results of increasing the
+network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impacts on performances}
+\label{fig:04}
+\end{figure}
+
+\subsubsection{Matrix size impacts on performances\\}
+
+In these experiments, the matrix size of the 3D Poisson problem is varied from
+$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
+clusters of $8$ processors each interconnected by the network $N2$ (see
+Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes. For all problem
+sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm.
+Moreover, for this benchmark, it seems that the greater the problem size is, the
+bigger the ratio between execution times of both algorithms is. We can also
+observe that for some problem sizes, the convergence (and thus the execution
+time) of the Krylov two-stage algorithm varies quite a lot.
+%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Problem size impacts on performances}
+\label{fig:05}
+\end{figure}
+
+\subsubsection{CPU power impacts on performances\\}
+
+Using the SimGrid simulator flexibility, we have tried to determine the impact
+of the CPU power of the processors in the different clusters on performances of
+both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The
+simulation is conducted on a grid of $2\times16$ processors interconnected by
+the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size
+$150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance
+gain, about $95\%$ for both algorithms, after improving the CPU power of
+processors.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
+\caption{CPU Power impacts on performances}
+\label{fig:06}
+\end{figure}
+\ \\
+
+To conclude these series of experiments, with SimGrid we have been able to make
+many simulations with many parameters variations. Doing all these experiments
+with a real platform is most of the time not possible or very costly. Moreover
+the behavior of both GMRES and Krylov two-stage algorithms is in accordance
+with larger real executions on large scale supercomputers~\cite{couturier15}.
+
+
+\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
+
+The previous paragraphs put in evidence the interests to simulate the behavior
+of the application before any deployment in a real environment. In this
+section, following the same previous methodology, our goal is to compare the
+efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the
+classical GMRES in \textit{synchronous mode}.
+
+The interest of using an asynchronous algorithm is that there is no more
+synchronization. With geographically distant clusters, this may be essential.
+In this case, each processor can compute its iterations freely without any
+synchronization with the other processors. Thus, the asynchronous may
+theoretically reduce the overall execution time and can improve the algorithm
+performance.
+
+In this section, the SimGrid simulator is used to compare the behavior of the
+two-stage algorithm in asynchronous mode with GMRES in synchronous mode.
+Several benchmarks have been performed with various combinations of the grid
+resources (CPU, Network, matrix size, \ldots). The test conditions are
+summarized in Table~\ref{tab:02}.
+
+
+
+%\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+%\RCE{Table III avec la nouvelle numerotation}
+
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{ll}
+ \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, $lat=20ms$\\
+ Matrix size & from $62^3$ to $150^3$\\
+ Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\
+ \end{tabular}
+\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\label{tab:02}
+\end{table}
+
+
+% use the same column width for the following three tables
+\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
+\newenvironment{mytable}[1]{% #1: number of columns for data
+ \renewcommand{\arraystretch}{1.3}%
+ \begin{tabular}{|>{\bfseries}r%
+ |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
+ \end{tabular}}
+
+
+\begin{table}[!t]
+\centering
+%\begin{table}
+% \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+% \label{"Table 7"}
+ \begin{mytable}{11}
+ \hline
+ bandwidth (Mbit/s)
+ & 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
+ \hline
+ latency (ms)
+ & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 \\
+ \hline
+ power (GFlops)
+ & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
+ \hline
+ size ($N^3$)
+ & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
+ \hline
+ Precision
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
+ \hline
+ Relative gain
+ & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\
+ \hline
+ \end{mytable}
+%\end{table}
+ \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \label{tab:03}
+\end{table}
+
+
+Table~\ref{tab:03} reports the relative gains between both algorithms. It is
+defined by the ratio between the execution time of GMRES and the execution time
+of the multisplitting. The ratio is greater than one because the asynchronous
+multisplitting version is faster than GMRES. In average, the two-stage
+multisplitting algorithm to be more than $2.5$ times faster than the classical
+GMRES. These experiments also show the relative tolerance of the multisplitting
+algorithm when using a low speed network as usually observed with geographically
+distant clusters through the internet.
