+
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
\algnewcommand\Output{\item[\algorithmicoutput]}
\newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
\begin{document}
-\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\title{Simulation of Asynchronous Iterative Algorithms Using SimGrid}
\author{%
\IEEEauthorblockN{%
Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
+ Lilia Ziane Khodja\IEEEauthorrefmark{2},
David Laiymani\IEEEauthorrefmark{1},
- Arnaud Giersch\IEEEauthorrefmark{1},
- Lilia Ziane Khodja\IEEEauthorrefmark{2} and
+ Arnaud Giersch\IEEEauthorrefmark{1} and
Raphaël Couturier\IEEEauthorrefmark{1}
}
\IEEEauthorblockA{\IEEEauthorrefmark{1}%
\maketitle
-\RC{Ordre des auteurs pas définitif.}
\begin{abstract}
-In recent years, the scalability of large-scale implementation in a
-distributed environment of algorithms becoming more and more complex has
-always been hampered by the limits of physical computing resources
-capacity. One solution is to run the program in a virtual environment
-simulating a real interconnected computers architecture. The results are
-convincing and useful solutions are obtained with far fewer resources
-than in a real platform. However, challenges remain for the convergence
-and efficiency of a class of algorithms that concern us here, namely
-numerical parallel iterative algorithms executed in asynchronous mode,
-especially in a large scale level. Actually, such algorithm requires a
-balance and a compromise between computation and communication time
-during the execution. Two important factors determine the success of the
-experimentation: the convergence of the iterative algorithm on a large
-scale and the execution time reduction in asynchronous mode. Once again,
-from the current work, a simulated environment like SimGrid provides
-accurate results which are difficult or even impossible to obtain in a
-physical platform by exploiting the flexibility of the simulator on the
-computing units clusters and the network structure design. Our
-experimental outputs showed a saving of up to \np[\%]{40} for the algorithm
-execution time in asynchronous mode compared to the synchronous one with
-a residual precision up to \np{E-11}. Such successful results open
-perspectives on experimentations for running the algorithm on a
-simulated large scale growing environment and with larger problem size.
+
+Synchronous iterative algorithms are often less scalable than asynchronous
+iterative ones. Performing large scale experiments with different kind of
+network parameters is not easy because with supercomputers such parameters are
+fixed. So one solution consists in using simulations first in order to analyze
+what parameters could influence or not the behaviors of an algorithm. In this
+paper, we show that it is interesting to use SimGrid to simulate the behaviors
+of asynchronous iterative algorithms. For that, we compare the behaviour of a
+synchronous GMRES algorithm with an asynchronous multisplitting one with
+simulations in which we choose some parameters. Both codes are real MPI
+codes. Simulations allow us to see when the multisplitting algorithm can be more
+efficient than the GMRES one to solve a 3D Poisson problem.
+
% no keywords for IEEE conferences
% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
increasing complexity of these requested applications combined with a continuous increase of their sizes lead to write
distributed and parallel algorithms requiring significant hardware resources (grid computing, clusters, broadband
network, etc.) but also a non-negligible CPU execution time. We consider in this paper a class of highly efficient
-parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
+parallel algorithms called \emph{iterative algorithms} executed in a distributed environment. As their name
suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
$X_{0}$ 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 numerical applications (synchronous or asynchronous) 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.
-
-To our knowledge, there is no existing work on the large-scale simulation of a
-real AIAC application. The aim of this paper is twofold. 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
-asynchronous mode algorithms by comparing their performance with the synchronous
-mode. More precisely, we had implemented a program for solving large
-non-symmetric linear system of equations by numerical method GMRES (Generalized
-Minimal Residual) []\AG[]{[]?}\LZK[]{\cite{ref1}}.\LZK{Problème traité dans le papier est symétrique ou asymétrique? (Poisson 3D symétrique?)} 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. The simulated
-results we obtained are in line with real results exposed in ??\AG[]{??}.
-SimGrid had allowed us to launch the application from a modest computing
-infrastructure by simulating different distributed architectures composed by
-clusters nodes interconnected by variable speed networks. With selected
-parameters on the network platforms (bandwidth, latency of inter cluster
-network) and on the clusters architecture (number, capacity calculation power)
-in the simulated environment, the experimental results have demonstrated not
-only the algorithm convergence within a reasonable time compared with the
-physical environment performance, but also a time saving of up to \np[\%]{40} in
-asynchronous mode.
-
-This article is structured as follows: after this introduction, the next section will give a brief description of
-iterative asynchronous model. Then, the simulation framework SimGrid is presented with the settings to create various
-distributed architectures. The algorithm of the multisplitting method used by GMRES written with MPI primitives and
-its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results
-carried out will be presented before some concluding remarks and future works.
+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 asynchronous iterative application. {\bf The contribution of the present
+ paper can be summarized 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) method
+\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
+infrastructure by simulating different distributed architectures composed by
+clusters nodes interconnected by variable speed networks. Parameters of the
+network platforms are the bandwidth and the latency of inter cluster
+network. Parameters on the cluster's architecture are the number of machines and
+the computation power of a machine. Simulations show that the asynchronous
+multisplitting algorithm can solve the 3D Poisson problem approximately twice
+faster than GMRES with two distant clusters.
+
+
+
+This article is structured as follows: after this introduction, the next section
+will give a brief description of iterative asynchronous model. Then, the
+simulation framework SimGrid is presented with the settings to create various
+distributed architectures. Then, the multisplitting method is presented, it is
+based on GMRES to solve each block obtained of the splitting. This code is
+written with MPI primitives and its adaptation to SimGrid with SMPI (Simulated
+MPI) is detailed in the next section. At last, the simulation results carried
+out will be presented before some concluding remarks and future works.
+
\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 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, the white spaces the idle
-times and the arrows the communications. 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.
+As exposed in the introduction, parallel iterative methods are now widely used
+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 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}
-It is very challenging to develop efficient applications for large scale,
-heterogeneous and distributed platforms such as computing grids. Researchers and
-engineers have to develop techniques for maximizing application performance of
-these multi-cluster platforms, by redesigning the applications and/or by using
-novel algorithms that can account for the composite and heterogeneous nature of
-the platform. Unfortunately, the deployment of such applications on these very
-large scale systems is very costly, labor intensive and time consuming. In this
-context, it appears that the use of simulation tools to explore various platform
-scenarios at will and to run enormous numbers of experiments quickly can be very
-promising. Several works\dots{}
+%% It is very challenging to develop efficient applications for large scale,
+%% heterogeneous and distributed platforms such as computing grids. Researchers and
+%% engineers have to develop techniques for maximizing application performance of
+%% these multi-cluster platforms, by redesigning the applications and/or by using
+%% novel algorithms that can account for the composite and heterogeneous nature of
+%% the platform. Unfortunately, the deployment of such applications on these very
+%% large scale systems is very costly, labor intensive and time consuming. In this
+%% context, it appears that the use of simulation tools to explore various platform
+%% scenarios at will and to run enormous numbers of experiments quickly can be very
+%% promising. Several works\dots{}
+
+%% \AG{Several works\dots{} what?\\
+% Le paragraphe suivant se trouve déjà dans l'intro ?}
+In the context of asynchronous algorithms, the number of iterations to reach the
+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 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
+nodes, inter and intra clusrters bandwith and latency, ....) and of the
+algorithm (number of splitting with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Or course,
+the GMRES method also depends of the same parameters. As it is difficult to have
+access to many clusters, grids or supercomputers with many different network
+parameters, it is interesting to be able to simulate the behaviors of
+asynchronous iterative algoritms before being able to runs real experiments.
+
-\AG{Several works\dots{} what?\\
- Le paragraphe suivant se trouve déjà dans l'intro ?}
-In the context of AIAC algorithms, the use of simulation tools is even more
-relevant. Indeed, this class of applications is 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.
standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
Fortran, with little or no modifications.
-With SimGrid, the execution of a distributed application is simulated on a
+Within SimGrid, the execution of a distributed application is simulated on a
single machine. The application code is really executed, but some operations
-like the communications are intercepted to be simulated according to the
-characteristics of the simulated execution platform. The description of this
-target platform is given as an input for the execution, by the mean of an XML
-file. It describes the properties of the platform, such as the computing node
-with their computing power, the interconnection links with their bandwidth and
-latency, and the routing strategy. The simulated running time of the
-application is computed according to these properties.
-
-\AG{Faut-il ajouter quelque-chose ?}
-\CER{Comme tu as décrit la plateforme d'exécution, on peut ajouter éventuellement le fichier XML contenant des hosts dans les clusters formant la grille}
+like the 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
+the mean 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 simulated running
+time of the application is 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 to run relatively fast
+simulations, while still keeping accurate
+results~\cite{bedaride:hal-00919507,tomacs13}. 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 have no importance for the simulation results, there is
+also the possibility 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 at a very large scale.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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*}
B_L
\end{array} \right)
\end{equation*}
-in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
+are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$, $A_{\ell
+ m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
+$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each,
+and $\sum_{\ell} n_\ell=\sum_{m} n_m=n$.
The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
\begin{equation}
\label{eq:4.1}
\left\{
\begin{array}{l}
- A_{ll}X_l = Y_l \text{, such that}\\
- Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
+ A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\
+ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m
\end{array}
\right.
\end{equation}
-is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+is solved independently by a cluster and communications are required to update
+the right-hand side sub-vector $Y_\ell$, such that the sub-vectors $X_m$
+represent the data dependencies between the clusters. As each sub-system
+(\ref{eq:4.1}) is solved in parallel by a cluster of processors, our
+multisplitting method uses an iterative method as an inner solver which is
+easier to parallelize and more scalable than a direct method. In this work, we
+use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most
+used iterative method by many researchers.
\begin{figure}[!t]
%%% IEEE instructions forbid to use an algorithm environment here, use figure
%%% instead
\begin{algorithmic}[1]
-\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
-\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\medskip
+
+\State Load $A_\ell$, $B_\ell$
\State Set the initial guess $x^0$
\For {$k=0,1,2,\ldots$ until the global convergence}
\State Restart outer iteration with $x^0=x^k$
\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
-\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $k$}
-\State Compute local right-hand side $Y_l$:
+\State Compute local right-hand side $Y_\ell$:
\begin{equation*}
- Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+ Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0
\end{equation*}
-\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
-\State \Return $X_l^k$
+\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method
+\State \Return $X_\ell^k$
\EndFunction
\end{algorithmic}
\caption{A multisplitting solver with GMRES method}
\label{algo:01}
\end{figure}
-Algorithm on Figure~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines.
+Algorithm on Figure~\ref{algo:01} shows the main key points of the
+multisplitting method to solve a large sparse linear system. This algorithm is
+based on an outer-inner iteration method where the parallel synchronous GMRES
+method is used to solve the inner iteration. It is executed in parallel by each
+cluster of processors. For all $\ell,m\in\{1,\ldots,L\}$, the matrices and
+vectors with the subscript $\ell$ represent the local data for cluster $\ell$,
+while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix
+$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared
+with neighboring clusters. At every outer iteration $k$, asynchronous
+communications are performed between processors of the local cluster and those
+of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
+Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
+exchanged by message passing using MPI non-blocking communication routines.
\begin{figure}[!t]
\centering
\label{fig:4.1}
\end{figure}
-The global convergence of the asynchronous multisplitting solver is detected when the clusters of processors have all converged locally. We implemented the global convergence detection process as follows. On each cluster a master processor is designated (for example the processor with rank 1) and masters of all clusters are interconnected by a virtual unidirectional ring network (see Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank 1, each master processor $i$ sets the token to {\it True} if the local convergence is achieved or to {\it False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster 1 receives from the master of cluster $L$ a token set to {\it True}. In this case, the master of cluster 1 broadcasts a stop message to masters of other clusters. In this work, the local convergence on each cluster $l$ is detected when the following condition is satisfied
+The global convergence of the asynchronous multisplitting solver is detected
+when the clusters of processors have all converged locally. We implemented the
+global convergence detection process as follows. On each cluster a master
+processor is designated (for example the processor with rank 1) and masters of
+all clusters are interconnected by a virtual unidirectional ring network (see
+Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around
+the virtual ring from a master processor to another until the global convergence
+is achieved. So starting from the cluster with rank 1, each master processor $i$
+sets the token to \textit{True} if the local convergence is achieved or to
+\textit{False} otherwise, and sends it to master processor $i+1$. Finally, the
+global convergence is detected when the master of cluster 1 receives from the
+master of cluster $L$ a token set to \textit{True}. In this case, the master of
+cluster 1 broadcasts a stop message to masters of other clusters. In this work,
+the local convergence on each cluster $\ell$ is detected when the following
+condition is satisfied
\begin{equation*}
- (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)
+ (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon)
\end{equation*}
-where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$.
+where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
+tolerance threshold of the error computed between two successive local solution
+$X_\ell^k$ and $X_\ell^{k+1}$.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code
-debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between the six neighbors of each point (left,right,front,behind,top,down) in a cubic partitionned submatrix within a cluster or between clusters, \CER{J'ai rajouté quelques précisions mais serait-il nécessaire de décrire a ce niveau la discrétisation 3D ?}
-\LZK{Non ce n'est pas nécessaire. A ce niveau, on décrit l'algorithme général de multisplitting. Donc, je pense qu'il est préférable de ne pas préciser le schéma de communication qui peut changer selon le type de problème. \\ {\bf Par exemple: Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters}}
-the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous
-mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
-and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
-\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}
-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, all declared
-global variables have been moved to local variables for each subroutine. In fact, 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, the alignment of certain types of variables such as ``long int'' had
-also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
-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 tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating
-Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}.
-
-
-
-\section{Experimental results}
-
-When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input
-parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this
-study that the results depend on the following parameters:
-\begin{itemize}
-\item At the network level, we found that the most critical values are the
- bandwidth (bw) and the network latency (lat).
-\item Hosts power (GFlops) can also influence on the results.
-\item Finally, when submitting job batches for execution, the arguments values
- passed to the program like the maximum number of iterations or the
- \textit{external} precision are critical. They allow to ensure not only the
- convergence of the algorithm but also to get the main objective of the
- experimentation of the simulation in having an execution time in asynchronous
- less than in synchronous mode (i.e. speed-up less than 1).
-\end{itemize}
-\LZK{Propositions pour remplacer le terme ``speedup'': acceleration ratio ou relative gain}
-A priori, obtaining a speedup less than 1 would be difficult in a local area
-network configuration where the synchronous mode will take advantage on the
-rapid exchange of information on such high-speed links. Thus, the methodology
-adopted was to launch the application on clustered network. In this last
-configuration, degrading the inter-cluster network performance will
-\textit{penalize} the synchronous mode allowing to get a speedup lower than 1.
-This action simulates the case of clusters linked with long distance network
-like Internet.
In this paper, we solve the 3D Poisson problem whose the mathematical model is
\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 choose 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 of in distant clusters with which it shares data at sub-domain boundaries.
+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.
\begin{figure}[!t]
\centering
\end{figure}
-As a first step, the algorithm was run on a network consisting of two clusters
-containing 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a
-matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
-$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
-\text{\np{5211000}}$ entries.
+\subsection{Simulation of the multisplitting method using SimGrid and SMPI}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code
+debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method , the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+and with the addition of the primitive \texttt{MPI\_Test} was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
+%\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. 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?}
+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 for the synchronous GMRES algorithm compared with our asynchronous multisplitting algorithm after few modifications.
+
+
+
+\section{Simulation results}
+
+When the \textit{real} application runs in the simulation environment and produces the expected results, varying the input
+parameters and the program arguments allows us to compare outputs from the code execution. We have noticed from this
+study that the results depend on the following parameters:
+\begin{itemize}
+\item At the network level, we found that the most critical values are the
+ bandwidth and the network latency.
+\item Hosts processors power (GFlops) can also influence on the results.
+\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 with the asynchronous multisplitting less than with synchronous GMRES.
+ \end{itemize}
+
+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.
+A priori, obtaining a relative gain greater than 1 would be difficult in a local
+area network configuration where the synchronous GMRES method will take advantage on the
+rapid exchange of information on such high-speed links. Thus, the methodology
+adopted was to launch the application on a clustered network. In this
+configuration, degrading the inter-cluster network performance will penalize the
+synchronous mode allowing to get a relative gain greater than 1. This action
+simulates the case of distant clusters linked with long distance network as in grid computing context.
+
+
+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). 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
\begin{table}[!t]
\centering
- \caption{2 clusters, each with 50 nodes}
+ \caption{Relative gain of the multisplitting algorithm compared to GMRES for
+ different configurations with 2 clusters, each one composed of 50 nodes.}
\label{tab.cluster.2x50}
- \begin{mytable}{6}
+ \begin{mytable}{5}
\hline
- bw
- & 5 & 5 & 5 & 5 & 5 & 50 \\
+ bandwidth (Mbit/s)
+ & 5 & 5 & 5 & 5 & 5 \\
\hline
- lat
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ latency (ms)
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
\hline
- power
- & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\
+ power (GFlops)
+ & 1 & 1 & 1 & 1.5 & 1.5 \\
\hline
- size
- & 62 & 62 & 62 & 100 & 100 & 110 \\
+ size $(n^3)$
+ & 62 & 62 & 62 & 100 & 100 \\
\hline
- Prec/Eprec
- & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ Precision
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} \\
\hline
- speedup
- & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
\hline
- \end{mytable}
-
- \smallskip
-
- \begin{mytable}{6}
- \hline
- bw
- & 50 & 50 & 50 & 50 & 10 & 10 \\
- \hline
- lat
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\
- \hline
- power
- & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 \\
- \hline
- size
- & 120 & 130 & 140 & 150 & 171 & 171 \\
- \hline
- Prec/Eprec
- & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\
- \hline
- speedup
- & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
+ Relative gain
+ & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 \\
\hline
\end{mytable}
-\end{table}
-
-Then we have changed the network configuration using three clusters containing
-respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
-clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
-speedups less than 1 with a matrix size from 62 to 100 elements.
-\begin{table}[!t]
- \centering
- \caption{3 clusters, each with 33 nodes}
- \label{tab.cluster.3x33}
+ \bigskip
- \begin{mytable}{6}
- \hline
- bw
- & 10 & 5 & 4 & 3 & 2 & 6 \\
+ \begin{mytable}{5}
\hline
- lat
- & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ bandwidth (Mbit/s)
+ & 50 & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
\hline
- power
- & 1 & 1 & 1 & 1 & 1 & 1 \\
+ latency (ms)
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ % & 0.03 & 0.01 \\
\hline
- size
- & 62 & 100 & 100 & 100 & 100 & 171 \\
+ Power (GFlops)
+ & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
\hline
- Prec/Eprec
- & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
+ size $(n^3)$
+ & 110 & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
\hline
- speedup
- & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
+ Precision
+ & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5} & \np{E-5} \\
\hline
- \end{mytable}
-\end{table}
-
-In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in
-Table~\ref{tab.cluster.3x67}.
-
-\begin{table}[!t]
- \centering
- \caption{3 clusters, each with 66 nodes}
- \label{tab.cluster.3x67}
-
- \begin{mytable}{1}
\hline
- bw & 1 \\
- \hline
- lat & 0.02 \\
- \hline
- power & 1 \\
- \hline
- size & 62 \\
- \hline
- Prec/Eprec & \np{E-5} \\
- \hline
- speedup & 0.9 \\
+ Relative gain
+ & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\ % & 1.59 & 1.29 \\
\hline
\end{mytable}
\end{table}
+
+%Then we have changed the network configuration using three clusters containing
+%respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
+%clusters. In the same way as above, a judicious choice of key parameters has
+%permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
+%relative gains greater than 1 with a matrix size from 62 to 100 elements.
+
+\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
+%\begin{table}[!t]
+% \centering
+% \caption{3 clusters, each with 33 nodes}
+% \label{tab.cluster.3x33}
+%
+% \begin{mytable}{6}
+% \hline
+% bandwidth
+% & 10 & 5 & 4 & 3 & 2 & 6 \\
+% \hline
+% latency
+% & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+% \hline
+% power
+% & 1 & 1 & 1 & 1 & 1 & 1 \\
+% \hline
+% size
+% & 62 & 100 & 100 & 100 & 100 & 171 \\
+% \hline
+% Prec/Eprec
+% & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
+% \hline
+% \hline
+% Relative gain
+% & 1.003 & 1.01 & 1.08 & 1.19 & 1.28 & 1.01 \\
+% \hline
+% \end{mytable}
+%\end{table}
+
+%In a final step, results of an execution attempt to scale up the three clustered
+%configuration but increasing by two hundreds hosts has been recorded in
+%Table~\ref{tab.cluster.3x67}.
+
+%\begin{table}[!t]
+% \centering
+% \caption{3 clusters, each with 66 nodes}
+% \label{tab.cluster.3x67}
+%
+% \begin{mytable}{1}
+% \hline
+% bandwidth & 1 \\
+% \hline
+% latency & 0.02 \\
+% \hline
+% power & 1 \\
+% \hline
+% size & 62 \\
+% \hline
+% Prec/Eprec & \np{E-5} \\
+% \hline
+% \hline
+% Relative gain & 1.11 \\
+% \hline
+% \end{mytable}
+%\end{table}
Note that the program was run with the following parameters:
\paragraph*{SMPI parameters}
\begin{itemize}
- \item HOSTFILE: Hosts file description.
- \item PLATFORM: file description of the platform architecture : clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-lat latency, \dots{}).
+\item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;
+\item PLATFORM: XML file description of the platform architecture whith the following characteristics: %two clusters (cluster1 and cluster2) with the following characteristics :
+ \begin{itemize}
+ \item 2 clusters of 50 hosts each;
+ \item Processor unit power: \np[GFlops]{1} or \np[GFlops]{1.5};
+ \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{0.05};
+ \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[$\mu$s]{20};
+ \end{itemize}
\end{itemize}
\paragraph*{Arguments of the program}
\begin{itemize}
- \item Description of the cluster architecture;
- \item Maximum number of internal and external iterations;
- \item Internal and external precisions;
- \item Matrix size $N_x$, $N_y$ and $N_z$;
-%<<<<<<< HEAD
- \item Matrix diagonal value: \np{6.0};
- \item Matrix Off-diagonal value: \np{-1.0};
-%=======
-%>>>>>>> 5fb6769d88c1720b6480a28521119ef010462fa6
- \item Execution Mode: synchronous or asynchronous.
+\item Description of the cluster architecture matching the format <Number of
+ clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
+\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: $6$ (See Equation~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
+\item Communication mode: asynchronous.
\end{itemize}
\paragraph*{Interpretations and comments}
-After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50}
-and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting
-the results have given a speedup less than 1, showing the effectiveness of the
-asynchronous performance compared to the synchronous mode.
-
-In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows
-that with a deterioration of inter cluster network set with \np[Mbit/s]{5} of
-bandwidth, a latency in order of a hundredth of a millisecond and a system power
-of one GFlops, an efficiency of about \np[\%]{40} in asynchronous mode is
-obtained for a matrix size of 62 elements. It is noticed that the result remains
-stable even if we vary the external precision from \np{E-5} to \np{E-9}. By
-increasing the problem size up to 100 elements, it was necessary to increase the
-CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a convergence of the algorithm
-with the same order of asynchronous mode efficiency. Maintaining such a system
-power but this time, increasing network throughput inter cluster up to
-\np[Mbit/s]{50}, the result of efficiency of about \np[\%]{40} is obtained with
+After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting
+the results have given a relative gain more than 2.5, showing the effectiveness of the
+asynchronous multiplsitting compared to GMRES with two distant clusters.
+
+With these settings, Table~\ref{tab.cluster.2x50} shows
+that after setting the bandwidth of the inter cluster network to \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
+of one GFlops, an efficiency of about \np[\%]{40} is
+obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
+increasing the matrix size up to 100 elements, it was necessary to increase the
+CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency. Maintaining such processor power but increasing network throughput inter cluster up to
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
high external precision of \np{E-11} for a matrix size from 110 to 150 side
elements.
-For the 3 clusters architecture including a total of 100 hosts,
-Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
-which gives an efficiency of asynchronous below \np[\%]{80}. Indeed, for a
-matrix size of 62 elements, equality between the performance of the two modes
-(synchronous and asynchronous) is achieved with an inter cluster of
-\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency by
-\np[\%]{78} with a matrix size of 100 points, it was necessary to degrade the
-inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
-
-A last attempt was made for a configuration of three clusters but more powerful
-with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was
-obtained with a bandwidth of \np[Mbit/s]{1} as shown in
-Table~\ref{tab.cluster.3x67}.
-
-\LZK{Dans le papier, on compare les deux versions synchrone et asycnhrone du multisplitting. Y a t il des résultats pour comparer gmres parallèle classique avec multisplitting asynchrone? Ca permettra de montrer l'intérêt du multisplitting asynchrone sur des clusters distants}
-
+%For the 3 clusters architecture including a total of 100 hosts,
+%Table~\ref{tab.cluster.3x33} shows that it was difficult to have a combination
+%which gives a relative gain of asynchronous mode more than 1.2. Indeed, for a
+%matrix size of 62 elements, equality between the performance of the two modes
+%(synchronous and asynchronous) is achieved with an inter cluster of
+%\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
+%inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
+\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
+ Quelle est la perte de perfs en faisant ça ?}
+
+%A last attempt was made for a configuration of three clusters but more powerful
+%with 200 nodes in total. The convergence with a relative gain around 1.1 was
+%obtained with a bandwidth of \np[Mbit/s]{1} as shown in
+%Table~\ref{tab.cluster.3x67}.
+
+%\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
+%\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
+%\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
+%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
\section{Conclusion}
The experimental results on executing a parallel iterative algorithm in
asynchronous mode on an environment simulating a large scale of virtual
% LocalWords: Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
% LocalWords: Fortran GFlops priori Mbit de du fcomte multisplitting scalable
% LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib
+% LocalWords: intra durations nonsingular Waitall discretization discretized
+% LocalWords: InnerSolver Isend Irecv
