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\begin{document}

\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}

\author{%
  \IEEEauthorblockN{%
    Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
    David Laiymani\IEEEauthorrefmark{1},
    Arnaud Giersch\IEEEauthorrefmark{1},
    Lilia Ziane Khodja\IEEEauthorrefmark{2} and
    Raphaël Couturier\IEEEauthorrefmark{1}
  }
  \IEEEauthorblockA{\IEEEauthorrefmark{1}%
    Femto-ST Institute -- DISC Department\\
    Université de Franche-Comté,
    IUT de Belfort-Montbéliard\\
    19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
    Email: \email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}
  }
  \IEEEauthorblockA{\IEEEauthorrefmark{2}%
    Inria Bordeaux Sud-Ouest\\
    200 avenue de la Vieille Tour, 33405 Talence cedex, France \\
    Email: \email{lilia.ziane@inria.fr}
  }
}

\maketitle

\RC{Ordre des autheurs 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. 

% no keywords for IEEE conferences
% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid
\end{abstract}

\section{Introduction}

Parallel computing and high performance computing (HPC) are becoming 
more and more imperative for solving various problems raised by 
researchers on various scientific disciplines but also by industrial in 
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.) 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
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. Generally, to reduce the complexity and the 
execution time, the problem is divided into several \emph{pieces} that will
be solved in parallel on multiple processing units. The latter will 
communicate each intermediate results before a new iteration starts 
until the approximate solution is reached. These distributed parallel 
computations can be performed either in \emph{synchronous} communication 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. Despite the 
effectiveness of iterative approach, a major drawback of the method is 
the requirement of huge resources in terms of computing capacity, 
storage and high speed communication network. Indeed, limited physical 
resources are blocking factors for large-scale deployment of parallel 
algorithms. 

In recent years, 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. According our knowledge, no testing of large-scale 
simulation of the class of algorithm solving to achieve real results has 
been undertaken to date. We had in the scope of this work implemented a 
program for solving large non-symmetric linear system of equations by 
numerical method GMRES (Generalized Minimal Residual) in the simulation
environment SimGrid. The simulated platform 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. In addition, it has been 
permitted to show the effectiveness of asynchronous mode algorithm by 
comparing its performance with the synchronous mode time. 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 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
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 
our future work after the results.
 
\section{The asynchronous iteration model}

\DL{Décrire le modèle asynchrone. Je m'en charge}

\section{SimGrid}

\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid} (Arnaud)}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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. In this case, we apply a row-by-row splitting without overlapping  
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\vdots & \ddots & \vdots\\
A_{L1} & \cdots & A_{LL}
\end{array} \right)
\times 
\left(\begin{array}{c}
X_1 \\
\vdots\\
X_L
\end{array} \right)
=
\left(\begin{array}{c}
B_1 \\
\vdots\\
B_L
\end{array} \right)\] 
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}
\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{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 $A_l$, $B_l$
\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 Send shared elements of $X_l^{k+1}$ to neighboring clusters
\State 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 Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
\State \Return $X_l^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}

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. 

\begin{figure}
\centering
  \includegraphics[width=60mm,keepaspectratio]{clustering}
\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
\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
\[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








\section{Experimental results}

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
level, we found that the most critical values are the bandwidth (bw) and the
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 \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 \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
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 \emph{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.

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 $N_x = N_y = N_z = 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to
$171^{3} = \np{5211000}$ entries.

Then we have changed the network configuration using three clusters containing
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.

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}.

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{}).
\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 $N_z$;
        \item Matrix diagonal value: \np{6.0};
        \item Execution Mode: synchronous or asynchronous.
\end{itemize}

\begin{table}
  \centering
  \caption{2 clusters X 50 nodes}
  \label{tab.cluster.2x50}
  \AG{Ces tableaux (\ref{tab.cluster.2x50}, \ref{tab.cluster.3x33} et
    \ref{tab.cluster.3x67}) sont affreux. Utiliser un format vectoriel (eps ou
    pdf) ou, mieux, les réécrire en \LaTeX{}. Réécrire les légendes proprement
    également (\texttt{\textbackslash{}times} au lieu de \texttt{X} par ex.)}
  \includegraphics[width=209pt]{img1.jpg}
\end{table}

\begin{table}
  \centering
  \caption{3 clusters X 33 nodes}
  \label{tab.cluster.3x33}
  \AG{Refaire le tableau.}
  \includegraphics[width=209pt]{img2.jpg}
\end{table}

\begin{table}
  \centering
  \caption{3 clusters X 67 nodes}
  \label{tab.cluster.3x67}
  \AG{Refaire le tableau.}
%  \includegraphics[width=160pt]{img3.jpg}
  \includegraphics[scale=0.5]{img3.jpg}
\end{table}

\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[Mbits/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[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} 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[Mbits/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.

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[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.

\section{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. 
Our work has demonstrated that using such a simulation tool allow us to 
reach the following three objectives: 

\newcounter{numberedCntD}
\begin{enumerate}
\item To have a flexible configurable execution platform resolving the 
hard exercise to access to very limited but so solicited physical 
resources;
\item to ensure the algorithm convergence with a raisonnable time and 
iteration number ;
\item and finally and more importantly, to find the correct combination 
of the cluster and network specifications permitting to save time in 
executing the algorithm in asynchronous mode.
\setcounter{numberedCntD}{\theenumi}
\end{enumerate}
Our results have shown that in certain conditions, asynchronous mode is 
speeder up to \np[\%]{40} than executing the algorithm in synchronous mode
which is not negligible for solving complex practical problems with more 
and more increasing size.

 Several studies have already addressed the performance execution time of 
this class of algorithm. The work presented in this paper has 
demonstrated an original solution to optimize the use of a simulation 
tool to run efficiently an iterative parallel algorithm in asynchronous 
mode in a grid architecture. 

\section*{Acknowledgment}


The authors would like to thank\dots{}


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