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 \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
-suggests, these algorithm solves a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
+parallel algorithms called \emph{numerical 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}.
+demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
-Parallelization of such algorithms generally involved the division of the problem into several \emph{blocks} that will
+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 begin only when all nodes communications are completed,
-either \emph{asynchronous} mode where processors can continue independently without or few synchronization points. For
-instance in the \textit{Asynchronous Iterations - Asynchronous Communications (AIAC)} model \cite{bcvc06:ij}, local
+\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).
+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
+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... 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
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
+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) []. 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 ??. 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. It has been
-permitted to show With selected parameters on the network platforms (bandwidth, latency of inter cluster network) and
+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.
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
+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
+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
+\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
+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.
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
- \caption{The Asynchronous Iterations - Asynchronous Communications model }
+ \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
\label{fig:aiac}
\end{figure}
\section{SimGrid}
-SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
framework to study the behavior of large-scale distributed systems. As its name
says, it emanates from the grid computing community, but is nowadays used to
study grids, clouds, HPC or peer-to-peer systems. The early versions of SimGrid
\section{Simulation of 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
-\[
-\left(\begin{array}{ccc}
-A_{11} & \cdots & A_{1L} \\
-\vdots & \ddots & \vdots\\
-A_{L1} & \cdots & A_{LL}
-\end{array} \right)
-\times
-\left(\begin{array}{c}
-X_1 \\
-\vdots\\
-X_L
-\end{array} \right)
-=
-\left(\begin{array}{c}
-B_1 \\
-\vdots\\
-B_L
-\end{array} \right)\]
+\begin{equation*}
+ \left(\begin{array}{ccc}
+ A_{11} & \cdots & A_{1L} \\
+ \vdots & \ddots & \vdots\\
+ A_{L1} & \cdots & A_{LL}
+ \end{array} \right)
+ \times
+ \left(\begin{array}{c}
+ X_1 \\
+ \vdots\\
+ X_L
+ \end{array} \right)
+ =
+ \left(\begin{array}{c}
+ B_1 \\
+ \vdots\\
+ 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 $\sum_{l} n_l=\sum_{m} n_m=n$.
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}
-\left\{
-\begin{array}{l}
-A_{ll}X_l = Y_l \mbox{,~such that}\\
-Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
-\end{array}
-\right.
-\label{eq:4.1}
+ \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
+ \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.
\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 Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\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 $\{X_m^{k+1}\}_{m\neq l}$
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $k$}
-\State Compute local right-hand side $Y_l$: \[Y_l = B_l - \sum\nolimits^L_{\substack{m=1 \\m\neq l}}A_{lm}X_m^0\]
+\State Compute local right-hand side $Y_l$:
+ \begin{equation*}
+ Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0
+ \end{equation*}
\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
\State \Return $X_l^k$
\EndFunction
$\{X_m\}_{m\neq l}$ 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 $6$ and $7$ in Figure~\ref{algo:01}). The shared vector
-elements of the solution $x$ are exchanged by message passing using MPI
-non-blocking communication routines.
+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
\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
-\[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
+\begin{equation*}
+ (k\leq \MI) \text{ or } (\|X_l^k - X_l^{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 $X_l^{k+1}$.
\LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid}
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 units in the SimGrid architecture. Second, the alignment of certain types of variables such as "long int" had
+shared memory used by threads simulating each computing units 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
62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
\np{5211000}$ entries.
+% use the same column width for the following three tables
+\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
+\newenvironment{mytable}[1]{% #1: number of columns for data
+ \renewcommand{\arraystretch}{1.3}%
+ \begin{tabular}{|>{\bfseries}r%
+ |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
+ \end{tabular}}
+
\begin{table}[!t]
\centering
\caption{$2$ clusters, each with $50$ nodes}
\label{tab.cluster.2x50}
- \renewcommand{\arraystretch}{1.3}
- \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+ \begin{mytable}{6}
\hline
bw
& 5 & 5 & 5 & 5 & 5 & 50 \\
speedup
& 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
\hline
- \end{tabular}
+ \end{mytable}
\smallskip
- \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+ \begin{mytable}{6}
\hline
bw
& 50 & 50 & 50 & 50 & 10 & 10 \\
speedup
& 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
\hline
- \end{tabular}
+ \end{mytable}
\end{table}
Then we have changed the network configuration using three clusters containing
\centering
\caption{$3$ clusters, each with $33$ nodes}
\label{tab.cluster.3x33}
- \renewcommand{\arraystretch}{1.3}
- \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
+ \begin{mytable}{6}
\hline
bw
& 10 & 5 & 4 & 3 & 2 & 6 \\
speedup
& 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
\hline
- \end{tabular}
+ \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}.
\centering
\caption{3 clusters, each with 66 nodes}
\label{tab.cluster.3x67}
- \renewcommand{\arraystretch}{1.3}
- \begin{tabular}{|>{\bfseries}r|c|}
+ \begin{mytable}{1}
\hline
bw & 1 \\
\hline
\hline
speedup & 0.9 \\
\hline
- \end{tabular}
+ \end{mytable}
\end{table}
Note that the program was run with the following parameters:
(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 2 Mbit/s.
+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
