+\section{SimGrid}
+\label{sec:simgrid}
+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 ({\it Generalized Minimal RESidual})~\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}[t]
+%\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}[t]
+%\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 (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.
+
+%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}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 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 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 iterations $\MIG$ and outer iterations $\MIM$,
+ \item inner precision $\TOLG$ and outer precision $\TOLM$,
+ \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
+ \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+ \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}
+
+
+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 architecture
+has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. 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. The network has been
+designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
+latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
+(resp. inter-clusters backbone links). \\
+
+\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. \\
