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\usepackage{amsmath}
-\usepackage{algorithm}
+%\usepackage{algorithm}
\usepackage{algpseudocode}
%\usepackage{amsthm}
\usepackage{graphicx}
\RC{Ordre des autheurs pas définitif.}
\begin{abstract}
-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
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
+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
simulated large scale growing environment and with larger problem size.
% no keywords for IEEE conferences
-% Keywords: Algorithm distributed iterative asynchronous simulation simgrid
+% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
\end{abstract}
\section{Introduction}
the field. Indeed, the 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\dots{}) but
+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 iterative executed
in a distributed environment. As their name suggests, these algorithm
-solves a given problem that might be NP- complete complex by successive
+solves a given problem that might be NP-complete complex 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
section will give a brief description of iterative asynchronous model.
Then, the simulation framework SimGrid will be presented with the
settings to create various distributed architectures. The algorithm of
-the multi -splitting method used by GMRES written with MPI primitives
+the multi-splitting method used by GMRES written with MPI primitives
and its adaptation to SimGrid with SMPI (Simulated MPI) will be in the
next section. At last, the experiments results carried out will be
presented before the conclusion which we will announce the opening of
\section{SimGrid}
-\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid} (Arnaud)}
+\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} (Arnaud)}
+%%% brief history?
+%%% programming interfaces: MSG, SimDAG, SMPI
+%%% platforms
+%%% validation?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
\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.
-\begin{algorithm}
-\caption{A multisplitting solver with GMRES method}
+\begin{figure}
+ %%% 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 \Return $X_l^k$
\EndFunction
\end{algorithmic}
+\caption{A multisplitting solver with GMRES method}
\label{algo:01}
-\end{algorithm}
+\end{figure}
-Algorithm~\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 $6$ and $7$ in Algorithm~\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 $l,m\in\{1,\ldots,L\}$, 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 $6$ 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.
\begin{figure}
\centering
\section{Experimental results}
-When the ``real'' application runs in the simulation environment and produces
+When the \emph{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: (1) at the network
network latency (lat). (2) Hosts power (GFlops) can also influence on the
results. And finally, (3) when submitting job batches for execution, the
arguments values passed to the program like the maximum number of iterations or
-the ``external'' precision are critical to ensure not only the convergence of the
+the \emph{external} precision are critical 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, in others words, in having a ``speedup'' less than 1 (Speedup = Execution
-time in synchronous mode / Execution time in asynchronous mode).
+mode, in others words, in having a \emph{speedup} less than 1
+({speedup}${}={}${execution time in synchronous mode}${}/{}${execution time in
+asynchronous mode}).
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
As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
-ranging from Nx = Ny = Nz = 62 to 171 elements or from $62^{3} = \np{238328}$ to
+ranging from $N_x = N_y = N_z = 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to
$171^{3} = \np{5211000}$ entries.
Then we have changed the network configuration using three clusters containing
\item Description of the cluster architecture;
\item Maximum number of internal and external iterations;
\item Internal and external precisions;
- \item Matrix size NX, NY and NZ;
- \item Matrix diagonal value = 6.0;
+ \item Matrix size $N_x$, $N_y$ and $N_z$;
+ \item Matrix diagonal value: \np{6.0};
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
with a bandwidth of \np[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.
\section{Conclusion}
-CONCLUSION
-
The experimental results on executing a parallel iterative algorithm in
asynchronous mode on an environment simulating a large scale of virtual
computers organized with interconnected clusters have been presented.
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% the document is modified later
\bibliographystyle{IEEEtran}
-\bibliography{hpccBib}
+\bibliography{IEEEabrv,hpccBib}
\end{document}
