+
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
$X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
-Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will
-be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
-iteration starts and until the approximate solution is reached. These parallel computations can be performed either in
-\emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
-or in \emph{asynchronous} mode where processors can continue independently with few or no synchronization points. For
-instance in the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model~\cite{bcvc06:ij}, local
-computations do not need to wait for required data. Processors can then perform their iterations with the data present
-at that time. Even if the number of iterations required before the convergence is generally greater than for the
-synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
-synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details).
-
-Parallel (synchronous or asynchronous) applications may have different
-configuration and deployment requirements. Quantifying their resource
-allocation policies and application scheduling algorithms in grid computing
-environments under varying load, CPU power and network speeds is very costly,
-very labor intensive and very time
-consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC
-algorithms is even more problematic since they are very sensible to the
-execution environment context. For instance, variations in the network bandwidth
-(intra and inter-clusters), in the number and the power of nodes, in the number
-of clusters\dots{} can lead to very different number of iterations and so to
-very different execution times. Then, it appears that the use of simulation
-tools to explore various platform scenarios and to run large numbers of
-experiments quickly can be very promising. In this way, the use of a simulation
-environment to execute parallel iterative algorithms found some interests in
-reducing the highly cost of access to computing resources: (1) for the
-applications development life cycle and in code debugging (2) and in production
-to get results in a reasonable execution time with a simulated infrastructure
-not accessible with physical resources. Indeed, the launch of distributed
-iterative asynchronous algorithms to solve a given problem on a large-scale
-simulated environment challenges to find optimal configurations giving the best
-results with a lowest residual error and in the best of execution time.
+Parallelization of such algorithms generally involve the division of the problem
+into several \emph{blocks} that will be solved in parallel on multiple
+processing units. The latter will communicate each intermediate results before a
+new iteration starts and until the approximate solution is reached. These
+parallel computations can be performed either in \emph{synchronous} mode where a
+new iteration begins only when all nodes communications are completed, or in
+\emph{asynchronous} mode where processors can continue independently with no
+synchronization points~\cite{bcvc06:ij}. In this case, local computations do not
+need to wait for required data. Processors can then perform their iterations
+with the data present at that time. Even if the number of iterations required
+before the convergence is generally greater than for the synchronous case,
+asynchronous iterative algorithms can significantly reduce overall execution
+times by suppressing idle times due to synchronizations especially in a grid
+computing context (see~\cite{Bahi07} for more details).
+
+Parallel applications based on a (synchronous or asynchronous) iteration model
+may have different configuration and deployment requirements. Quantifying their
+resource allocation policies and application scheduling algorithms in grid
+computing environments under varying load, CPU power and network speeds is very
+costly, very labor intensive and very time
+consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of asynchronous
+iterative algorithms is even more problematic since they are very sensible to
+the execution environment context. For instance, variations in the network
+bandwidth (intra and inter-clusters), in the number and the power of nodes, in
+the number of clusters\dots{} can lead to very different number of iterations
+and so to very different execution times. Then, it appears that the use of
+simulation tools to explore various platform scenarios and to run large numbers
+of experiments quickly can be very promising. In this way, the use of a
+simulation environment to execute parallel iterative algorithms found some
+interests in reducing the highly cost of access to computing resources: (1) for
+the applications development life cycle and in code debugging (2) and in
+production to get results in a reasonable execution time with a simulated
+infrastructure not accessible with physical resources. Indeed, the launch of
+distributed iterative asynchronous algorithms to solve a given problem on a
+large-scale simulated environment challenges to find optimal configurations
+giving the best results with a lowest residual error and in the best of
+execution time.
To our knowledge, there is no existing work on the large-scale simulation of a
-real AIAC application. {\bf The contribution of the present paper can be
- summarised in two main points}. First we give a first approach of the
-simulation of AIAC algorithms using a simulation tool (i.e. the SimGrid
-toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of the
-asynchronous multisplitting algorithm by comparing its performance with the
-synchronous GMRES (Generalized Minimal Residual) \cite{ref1}. Both these codes
-can be used to solve large linear systems. In this paper, we focus on a 3D
-Poisson problem. We show, that with minor modifications of the initial MPI
-code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC
-application on different computing architectures.
+real asynchronous iterative application. {\bf The contribution of the present
+ paper can be summarised in two main points}. First we give a first approach
+of the simulation of asynchronous iterative algorithms using a simulation tool
+(i.e. the SimGrid toolkit~\cite{SimGrid}). Second, we confirm the
+effectiveness of the asynchronous multisplitting algorithm by comparing its
+performance with the synchronous GMRES (Generalized Minimal Residual)
+\cite{ref1}. Both these codes can be used to solve large linear systems. In
+this paper, we focus on a 3D Poisson problem. We show, that with minor
+modifications of the initial MPI code, the SimGrid toolkit allows us to perform
+a test campaign of a real asynchronous iterative application on different
+computing architectures.
% The simulated results we
%obtained are in line with real results exposed in ??\AG[]{ref?}.
SimGrid had allowed us to launch the application from a modest computing
\section{Motivations and scientific context}
As exposed in the introduction, parallel iterative methods are now widely used
-in many scientific domains. They can be classified in three main classes
+in many scientific domains. They can be classified in three main classes
depending on how iterations and communications are managed (for more details
-readers can refer to~\cite{bcvc06:ij}). In the \textit{Synchronous Iterations~--
- Synchronous Communications (SISC)} model data are exchanged at the end of each
-iteration. All the processors must begin the same iteration at the same time and
-important idle times on processors are generated. The \textit{Synchronous
- Iterations~-- Asynchronous Communications (SIAC)} model can be compared to the
-previous one except that data required on another processor are sent
-asynchronously i.e. without stopping current computations. This technique
-allows to partially overlap communications by computations but unfortunately,
-the overlapping is only partial and important idle times remain. It is clear
-that, in a grid computing context, where the number of computational nodes is
-large, heterogeneous and widely distributed, the idle times generated by
-synchronizations are very penalizing. One way to overcome this problem is to use
-the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)}
-model. Here, local computations do not need to wait for required
-data. Processors can then perform their iterations with the data present at that
-time. Figure~\ref{fig:aiac} illustrates this model where the gray blocks
-represent the computation phases. With this algorithmic model, the number of
-iterations required before the convergence is generally greater than for the two
-former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC algorithms can
-significantly reduce overall execution times by suppressing idle times due to
-synchronizations especially in a grid computing context.
-%\LZK{Répétition par rapport à l'intro}
+readers can refer to~\cite{bcvc06:ij}). In the synchronous iterations model,
+data are exchanged at the end of each iteration. All the processors must begin
+the same iteration at the same time and important idle times on processors are
+generated. It is possible to use asynchronous communications, in this case, the
+model can be compared to the previous one except that data required on another
+processor are sent asynchronously i.e. without stopping current computations.
+This technique allows to partially overlap communications by computations but
+unfortunately, the overlapping is only partial and important idle times remain.
+It is clear that, in a grid computing context, where the number of computational
+nodes is large, heterogeneous and widely distributed, the idle times generated
+by synchronizations are very penalizing. One way to overcome this problem is to
+use the asynchronous iterations model. Here, local computations do not need to
+wait for required data. Processors can then perform their iterations with the
+data present at that time. Figure~\ref{fig:aiac} illustrates this model where
+the gray blocks represent the computation phases. With this algorithmic model,
+the number of iterations required before the convergence is generally greater
+than for the two former classes. But, and as detailed in~\cite{bcvc06:ij},
+asynchronous iterative algorithms can significantly reduce overall execution
+times by suppressing idle times due to synchronizations especially in a grid
+computing context.
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
- \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
+ \caption{The asynchronous iterations model}
\label{fig:aiac}
\end{figure}
-\RC{Je serais partant de virer AIAC et laisser asynchronous algorithms... à voir}
%% It is very challenging to develop efficient applications for large scale,
%% heterogeneous and distributed platforms such as computing grids. Researchers and
convergence depends on the delay of messages. With synchronous iterations, the
number of iterations is exactly the same than in the sequential mode (if the
parallelization process does not change the algorithm). So the difficulty with
-asynchronous algorithms comes from the fact it is necessary to run the algorithm
+asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
with real data. In fact, from an execution to another the order of messages will
change and the number of iterations to reach the convergence will also change.
According to all the parameters of the platform (number of nodes, power of
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
+
+\subsection{The multisplitting method}
%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping
\begin{equation*}
\right.
\label{eq:02}
\end{equation}
-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
\begin{equation}
-\begin{array}{ll}
-u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
- & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
- & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
- & u^k(x,y,z-1) + u^k(x,y,z+1)),
+\begin{array}{l}
+u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ % & u(x,y-1,z) + u(x,y+1,z) + \\
+ % & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
\end{array}
\label{eq:03}
\end{equation}
-where the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite.
+where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite.
The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries.
\end{figure}
+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async}
%\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
Note here that the use of SMPI functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.
-As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, the scope of all declared
+As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. The scope of all declared
global variables have been moved to local to subroutine. Indeed, global variables generate side effects arising from the concurrent access of
shared memory used by threads simulating each computing unit in the SimGrid architecture.
-Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
-\AG{compilation or run-time error?}
+%Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
+%\AG{compilation or run-time error?}
In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real
environment. We have successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications.
\item Finally, when submitting job batches for execution, the arguments values
passed to the program like the maximum number of iterations or the precision are critical. They allow us to ensure not only the convergence of the
algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in
- synchronous mode. The ratio between the execution time of synchronous
- compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
+ synchronous mode. The ratio between the simulated execution time of synchronous GMRES algorithm
+ compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) is defined as the \emph{relative gain}. So,
our objective running the algorithm in SimGrid is to obtain a relative gain
greater than 1.
\end{itemize}
simulates the case of distant clusters linked with long distance network as in grid computing context.
-% As a first step,
-The algorithm was run on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The algorithm convergence with a 3D
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+
+Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
-\text{\np{3375000}}$ entries), is obtained in asynchronous in average 2.5 times faster than in the synchronous mode.
-\AG{Expliquer comment lire les tableaux.}
-\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one.
+%\AG{Expliquer comment lire les tableaux.}
+%\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
% 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
\item Maximum number of iterations;
\item Precisions on the residual error;
\item Matrix size $N_x$, $N_y$ and $N_z$;
-\item Matrix diagonal value: \np{1.0} (See~(\ref{eq:03}));
-\item Matrix off-diagonal value: \np{-1}/\np{6} (See~(\ref{eq:03}));
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
\item Communication mode: asynchronous.
\end{itemize}
