\algnewcommand\Output{\item[\algorithmicoutput]}
\newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
\begin{document}
\RC{Ordre des auteurs pas définitif.}
\begin{abstract}
+\AG{L'abstract est AMHA incompréhensible et ne donne pas envie de lire la suite.}
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
perspectives on experimentations for running the algorithm on a
simulated large scale growing environment and with larger problem size.
+\LZK{Long\ldots}
+
% no keywords for IEEE conferences
% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
\end{abstract}
Minimal Residual) \cite{ref1}. 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[]{??}.
+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. With selected
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.
+\AG{Il faudrait revoir la phrase précédente (couper en deux?). Là, on peut
+ avoir l'impression que le gain de \np[\%]{40} est entre une exécution réelle
+ et une exécution simulée!}
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
+distributed architectures. The algorithm of the multisplitting method used by GMRES \LZK{??? GMRES n'utilise pas la méthode de multisplitting! Sinon ne doit on pas expliquer le choix d'une méthode de multisplitting?} 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.
\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
+times and the arrows the communications.
+\AG{There are no ``white spaces'' on the figure.}
+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.
+in a grid computing context.\LZK{Répétition par rapport à l'intro}
\begin{figure}[!t]
\centering
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}
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
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.
+also to be reviewed.
+\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
+ 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}.
+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.
\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).
+ less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater 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
+A priori, obtaining a relative gain greater 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
+\textit{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
like Internet.
In this paper, we solve the 3D Poisson problem whose the mathematical model is
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.
+\text{\np{5000211}}$ entries.
+\AG{Expliquer comment lire les tableaux.}
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
Prec/Eprec
& \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
\hline
- speedup
- & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
+ Relative gain
+ & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 \\
\hline
\end{mytable}
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
+ & 2.51 & 2.58 & 2.55 & 2.54 & 1.59 & 1.29 \\
\hline
\end{mytable}
\end{table}
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.
+relative gains greater than 1 with a matrix size from 62 to 100 elements.
\begin{table}[!t]
\centering
Prec/Eprec
& \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
\hline
- speedup
- & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
+ Relative gain
+ & 1.003 & 1.01 & 1.08 & 1.19 & 1.28 & 1.01 \\
\hline
\end{mytable}
\end{table}
\hline
Prec/Eprec & \np{E-5} \\
\hline
- speedup & 0.9 \\
+ Relative gain & 1.11 \\
\hline
\end{mytable}
\end{table}
\paragraph*{SMPI parameters}
+~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}
\begin{itemize}
\item HOSTFILE: Hosts file description.
\item PLATFORM: file description of the platform architecture : clusters (CPU power,
\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 Matrix off-diagonal value: \np{-1.0};
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
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
+the results have given a relative gain more than 2.5, 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
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
+increasing the matrix 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
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5 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
+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 by
-\np[\%]{78} with a matrix size of 100 points, it was necessary to degrade the
+\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}.
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
+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}.
-\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}
+\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 bw et lat sont ceux inter-clusters ou pour les deux inter et intra cluster??}
\section{Conclusion}
The experimental results on executing a parallel iterative algorithm in
