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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 \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
$X_{0}$ 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{}. 

Parallelization of such algorithms generally involved the division of the problem 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 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. 

% DL : reprendre correction ici 
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{Motivations and scientific context}

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{bcvc02:ip}). 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
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 grey 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
algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
in a grid computing context.

\begin{figure}[htbp]
  \centering
    \includegraphics[width=8cm]{AIAC.pdf}
  \caption{The Asynchronous Iterations - Asynchronous Communications model } 
  \label{fig:aiac}
\end{figure}


It is very challenging to develop efficient applications for large scale, heterogeneous and distributed platforms such
as computing grids. Researchers and engineers have to develop techniques for maximizing application performance of these
multi-cluster platforms, by redesigning the applications and/or by using novel algorithms that can account for the
composite and heterogeneous nature of the platform. Unfortunately, the deployment of such applications on these very
large scale systems is very costly, labor intensive and time consuming. In this context, it appears that the use of
simulation tools to explore various platform scenarios at will and to run enormous numbers of experiments quickly can be
very promising. Several works...

In the context of AIAC algorithms, the use of simulation tools is even more relevant. Indeed, this class of applications
is very sensible to the execution environment context. For instance, variations in the network bandwith (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.




\section{SimGrid}

SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
framework to sudy 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.
%- open source, developped since 1999, one of the major solution in the field
%
SimGrid provides several programming interfaces: MSG to simulate Concurrent
Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
run real applications written in MPI~\cite{MPI}.  Apart from the native C
interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
languages.  The SMPI interface supports applications written in C or Fortran,
with little or no modifications.
%- implements most of MPI-2 \cite{ref} standard [CHECK]

%%% explain simulation
%- simulated processes folded in one real process
%- simulates interactions on the network, fluid model
%- able to skip long-lasting computations
%- traces + visu?

%%% platforms
%- describe resources and their interconnection, with their properties
%- XML files

%%% validation + refs

\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,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{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 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}
\caption{A multisplitting solver with GMRES method}
\label{algo:01}
\end{figure}

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
  \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 I 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.

\begin{table}[h!]
 \centering

 \tiny
 
\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|} 
 \hline 
 \bf bw & 5 &5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 & 10 & 10\\ 
 \hline
 \bf lat & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01\\ 
 \hline 
 \bf power & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5\\ \hline    \bf size & 62 & 62 & 62 & 100 & 100 & 110 & 120& 130 & 140 & 150 & 171 & 171\\ \hline
 \bf Prec/Eprec & 10$^{-5}$ & 10$^{-8}$ & 10$^{-9}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
 \bf speedup & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778\\ \hline 
 \end{tabular}
 \smallskip 
 \caption{2 Clusters x 50 nodes each} \label{tab1} 
\end{table} 
  
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 II which shows the speedups less than 1 with
a matrix size from 62 to 100 elements.

\begin{table}[h!]
 \centering

 \tiny
 
\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|} 
 \hline 
 \bf bw & 10 &5 & 4 & 3 & 2 & 6\\ \hline
 \bf lat & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\\ 
 \hline 
 \bf power & 1 & 1 & 1 & 1 & 1 & 1\\ \hline    
 \bf size & 62 & 100 & 100 & 100 & 100 & 171\\ \hline
 \bf Prec/Eprec & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
 \bf speedup & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99\\ 
 \hline 
 \end{tabular}
 \smallskip 
 \caption{3 Clusters x 33 nodes each} \label{tab2} 
\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 III.

\begin{table}[h!]
 \centering
 \tiny
\begin{tabular}{|M{0.55cm}|M{0.25cm}|} 
 \hline 
 \bf bw & 1\\ \hline
 \bf lat & 0.02\\ 
 \hline 
 \bf power & 1\\ 
 \hline    
 \bf size & 62\\ 
 \hline
 \bf Prec/Eprec & 10$^{-5}$\\ 
 \hline 
 \bf speedup & 0.9\\ 
 \hline 
 \end{tabular}
 \smallskip 
 \caption{3 Clusters x 66 nodes each} \label{tab3} 
\end{table} 

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}


\paragraph*{Interpretations and comments}

After analyzing the outputs, generally, for the configuration with two or three
clusters including one hundred hosts (Table I and II), 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 I 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 II 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 III.

\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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