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 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
+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
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
(SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
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
+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.
\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
\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
$\{X_m\}_{m\neq l}$ 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
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:
