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\begin{document}

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

\author{%
  \IEEEauthorblockN{%
    Raphaël Couturier,
    Arnaud Giersch,
    David Laiymani and
    Charles Emile Ramamonjisoa
  }
  \IEEEauthorblockA{%
    Femto-ST Institute - DISC Department\\
    Université de Franche-Comté\\
    Belfort\\
    Email: \email{raphael.couturier@univ-fcomte.fr}
  }
}

\maketitle

\begin{abstract}
The abstract goes here.
\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\dots{}) 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}

Décrire le modèle asynchrone. Je m'en charge (DL)

\section{SimGrid}

Décrire 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 $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning 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  
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\vdots & \ddots & \vdots\\
A_{L1} & \cdots & A_{LL}
\end{array} \right)
\times 
\left(\begin{array}{c}
X_1 \\
\vdots\\
X_L
\end{array} \right)
=
\left(\begin{array}{c}
Y_1 \\
\vdots\\
Y_L
\end{array} \right)\] 
in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.

The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ 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_{i=1,i\neq l}^{L}A_{li}X_i,
\end{array}
\right.
\label{eq:4.1}
\end{equation}
is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ 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 GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. 

\begin{algorithm}
\caption{A multisplitting solver with inner iteration GMRES method}
\begin{algorithmic}[1]
\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
\Output $X_l$ (local solution vector)\vspace{0.2cm}
\State Load $A_l$, $B_l$, $x^0$
\State Initialize the shared vector $\hat{x}=x^0$
\For {$k=1,2,3,\ldots$ until the global convergence}
\State $x^0=\hat{x}$
\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
\EndFor

\Statex

\Function {InnerSolver}{$x^0$, $k$}
\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
\State \Return $X_l^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








\section{Experimental results}

When the ``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 ``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).

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 Nx = Ny = Nz = 62 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 NX, NY and NZ;
        \item Matrix diagonal value = 6.0;
        \item Execution Mode: synchronous or asynchronous.
\end{itemize}

\begin{table}
  \centering
  \caption{2 clusters X 50 nodes}
  \label{tab.cluster.2x50}
  \includegraphics[width=209pt]{img1.jpg}
\end{table}

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

\begin{table}
  \centering
  \caption{3 clusters X 67 nodes}
  \label{tab.cluster.3x67}
%  \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{E-1} ms. 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 power
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}

\section*{Acknowledgment}


The authors would like to thank\dots{}


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