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\begin{document}
\RCE{Titre a confirmer.}
\title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
%\itshape{\journalnamelc}\footnotemark[2]}

\author{    Charles Emile Ramamonjisoa and
    David Laiymani and
    Arnaud Giersch and
    Lilia Ziane Khodja and
    Raphaël Couturier
}

\address{
        \centering    
    Femto-ST Institute - DISC Department\\
    Université de Franche-Comté\\
    Belfort\\
    Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
}

%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be

\begin{abstract}
ABSTRACT
\end{abstract}

\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}

\maketitle

\section{Introduction}

\section{The asynchronous iteration model}

\section{SimGrid}

%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two-stage splitting methods}
\label{sec:04}
\subsection{Multisplitting methods for sparse linear systems}
\label{sec:04.01}
Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
\begin{equation}
Ax=b,
\label{eq:01}
\end{equation}
where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
\begin{equation}
x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
\label{eq:02}
\end{equation}
where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$~\cite{O'leary85,White86}, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system   
\begin{equation}
M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b,
\label{eq:03}
\end{equation}
then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
\begin{equation}
x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell.
\label{eq:04}
\end{equation}
The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition  
%\begin{equation}
%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
%\label{eq:05}
%\end{equation}
%where $\rho$ is the spectral radius of the square matrix. 
The multisplitting methods are convergent: 
\begin{itemize}
\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or
\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous.
\end{itemize}
The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method).

In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form 
\begin{equation}
A_{\ell\ell} x_\ell^{k+1} = b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m},\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
\label{eq:05}
\end{equation}
where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting
\begin{equation}
A_{\ell\ell} x_\ell = c_\ell,
\label{eq:06}
\end{equation}
is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\ref{alg:01} shows the main key points of the block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of iterations and the tolerance threshold respectively.  

\begin{algorithm}[t]
\caption{Block Jacobi two-stage method}
\begin{algorithmic}[1]
  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence}
    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
    \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$ \label{solve}
    \State Send $x_\ell^k$ to neighboring clusters
    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
  \EndFor
\end{algorithmic}
\label{alg:01}
\end{algorithm}

\subsection{Simulation of two-stage methods using SimGrid framework}

%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Experimental, Results and Comments}


\textbf{V.1. Setup study and Methodology}

To conduct our study, we have put in place the following methodology 
which can be reused with any grid-enabled applications.

\textbf{Step 1} : Choose with the end users the class of algorithms or 
the application to be tested. Numerical parallel iterative algorithms 
have been chosen for the study in the paper. 

\textbf{Step 2} : Collect the software materials needed for the 
experimentation. In our case, we have three variants algorithms for the 
resolution of three 3D-Poisson problem: (1) using the classical GMRES alias Algo-1 in this 
paper, (2) using the multisplitting method alias Algo-2 and (3) an 
enhanced version of the multisplitting method as Algo-3. In addition, 
SIMGRID simulator has been chosen to simulate the behaviors of the 
distributed applications. SIMGRID is running on the Mesocentre 
datacenter in Franche-Comte University $[$10$]$ but also in a virtual 
machine on a laptop.

\textbf{Step 3} : Fix the criteria which will be used for the future 
results comparison and analysis. In the scope of this study, we retain 
in one hand the algorithm execution mode (synchronous and asynchronous) 
and in the other hand the execution time and the number of iterations of 
the application before obtaining the convergence.

\textbf{Step 4 }: Setup up the different grid testbeds environment 
which will be simulated in the simulator tool to run the program. The 
following architecture has been configured in Simgrid : 2x16 - that is a 
grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8, 
4x16, 8x8 and 2x50. The network has been designed to operate with a 
bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8E-6 
microseconds (resp. 5E-5) for the intra-clusters links (resp. 
inter-clusters backbone links).

\textbf{Step 5}: Process an extensive and comprehensive testings 
within these configurations in varying the key parameters, especially 
the CPU power capacity, the network parameters and also the size of the 
input matrix. Note that some parameters should be invariant to allow the 
comparison like some program input arguments.

\textbf{Step 6} : Collect and analyze the output results.

\textbf{ V.2. Factors impacting distributed applications performance in 
a grid environment}

From our previous experience on running distributed application in a 
computational grid, many factors are identified to have an impact on the 
program behavior and performance on this specific environment. Mainly, 
first of all, the architecture of the grid itself can obviously 
influence the performance results of the program. The performance gain 
might be important theoretically when the number of clusters and/or the 
number of nodes (processors/cores) in each individual cluster increase. 

Another important factor impacting the overall performance of the 
application is the network configuration. Two main network parameters 
can modify drastically the program output results : (i) the network 
bandwidth (bw=bits/s) also known as "the data-carrying capacity" 
$[$13$]$ of the network is defined as the maximum of data that can pass 
from one point to another in a unit of time. (ii) the network latency 
(lat : microsecond) defined as the delay from the start time to send the 
data from a source and the final time the destination have finished to 
receive it. Upon the network characteristics, another impacting factor 
is the application dependent volume of data exchanged between the nodes 
in the cluster and between distant clusters. Large volume of data can be 
transferred in transit between the clusters and nodes during the code 
execution. 

 In a grid environment, it is common to distinguish in one hand, the 
"\,intra-network" which refers to the links between nodes within a 
cluster and in the other hand, the "\,inter-network" which is the 
backbone link between clusters. By design, these two networks perform 
with different speed. The intra-network generally works like a high 
speed local network with a high bandwith and very low latency. In 
opposite, the inter-network connects clusters sometime via heterogeneous 
networks components thru internet with a lower speed. The network 
between distant clusters might be a bottleneck for the global 
performance of the application. 

\textbf{V.3 Comparing GMRES and Multisplitting algorithms in 
synchronous mode}

In the scope of this paper, our first objective is to demonstrate the 
Algo-2 (Multisplitting method) shows a better performance in grid 
architecture compared with Algo-1 (Classical GMRES) both running in 
\textbf{\textit{synchronous mode}}. Better algorithm performance 
should mean a less number of iterations output and a less execution time 
before reaching the convergence. For a systematic study, the experiments 
should figure out that, for various grid parameters values, the 
simulator will confirm the targeted outcomes, particularly for poor and 
slow networks, focusing on the impact on the communication performance 
on the chosen class of algorithm $[$12$]$.

The following paragraphs present the test conditions, the output results 
and our comments.


\textit{3.a Executing the algorithms on various computational grid 
architecture scaling up the input matrix size}
\\

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
 Network & N2 : bw=1Gbs-lat=5E-05 \\ %\hline
 Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
 - & N$_{x}$ =170 x 170 x 170    \\ \hline
 \end{tabular}
\end{footnotesize}


 Table 1 : Clusters x Nodes with NX=150 or NX=170

\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}


The results in figure 1 show the non-variation of the number of 
iterations of classical GMRES for a given input matrix size; it is not 
the case for the multisplitting method. 

%\begin{wrapfigure}{l}{60mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\caption{Cluster x Nodes NX=150 and NX=170} 
%\label{overflow}}
\end{figure}
%\end{wrapfigure}

Unless the 8x8 cluster, the time 
execution difference between the two algorithms is important when 
comparing between different grid architectures, even with the same number of 
processors (like 2x16 and 4x8 = 32 processors for example). The 
experiment concludes the low sensitivity of the multisplitting method 
(compared with the classical GMRES) when scaling up to higher input 
matrix size. 

\textit{3.b Running on various computational grid architecture}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x16, 4x8\\ %\hline
 Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
 - & N2 : bw=1Gbs-lat=5E-05 \\
 Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
 \end{tabular}
\end{footnotesize}

%Table 2 : Clusters x Nodes - Networks N1 x N2
%\RCE{idem pour tous les tableaux de donnees}


%\begin{wrapfigure}{l}{60mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Cluster x Nodes N1 x N2}
%\label{overflow}}
\end{figure}
%\end{wrapfigure}

The experiments compare the behavior of the algorithms running first on 
speed inter- cluster network (N1) and a less performant network (N2). 
The figure 2 shows that end users will gain to reduce the execution time 
for both algorithms in using a grid architecture like 4x16 or 8x8: the 
performance was increased in a factor of 2. The results depict also that 
when the network speed drops down, the difference between the execution 
times can reach more than 25\%. 

\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x16\\ %\hline
 Network & N1 : bw=1Gbs \\ %\hline
 Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
 \end{tabular}
\end{footnotesize}

Table 3 : Network latency impact


\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
\caption{Network latency impact on execution time}
%\label{overflow}}
\end{figure}


According the results in table and figure 3, degradation of the network 
latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time 
increase more than 75\% (resp. 82\%) of the execution for the classical 
GMRES (resp. multisplitting) algorithm. In addition, it appears that the 
multisplitting method tolerates more the network latency variation with 
a less rate increase. Consequently, in the worst case (lat=6.10$^{-5
}$), the execution time for GMRES is almost the double of the time for 
the multisplitting, even though, the performance was on the same order 
of magnitude with a latency of 8.10$^{-6}$. 

\textit{3.d Network bandwidth impacts on performance}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x16\\ %\hline
 Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
 Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
 \end{tabular}
\end{footnotesize}

Table 4 : Network bandwidth impact

\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impact on execution time}
%\label{overflow}
\end{figure}



The results of increasing the network bandwidth depict the improvement 
of the performance by reducing the execution time for both of the two 
algorithms. However, and again in this case, the multisplitting method 
presents a better performance in the considered bandwidth interval with 
a gain of 40\% which is only around 24\% for classical GMRES.

\textit{3.e Input matrix size impacts on performance}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 4x8\\ %\hline
 Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
 Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
 \end{tabular}
\end{footnotesize}

Table 5 : Input matrix size impact

\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf}
\caption{Pb size impact on execution time}
%\label{overflow}}
\end{figure}

In this experimentation, the input matrix size has been set from 
Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 5, 
the execution time for the algorithms convergence increases with the 
input matrix size. But the interesting result here direct on (i) the 
drastic increase (300 times) of the number of iterations needed before 
the convergence for the classical GMRES algorithm when the matrix size 
go beyond Nx=150; (ii) the classical GMRES execution time also almost 
the double from Nx=140 compared with the convergence time of the 
multisplitting method. These findings may help a lot end users to setup 
the best and the optimal targeted environment for the application 
deployment when focusing on the problem size scale up. Note that the 
same test has been done with the grid 2x16 getting the same conclusion.

\textit{3.f CPU Power impact on performance}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x16\\ %\hline
 Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
 Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
 \end{tabular}
\end{footnotesize}

Table 6 : CPU Power impact

\begin{figure} [ht!]
\centering
\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impact on execution time}
%\label{overflow}}
\end{figure}

Using the SIMGRID simulator flexibility, we have tried to determine the 
impact on the algorithms performance in varying the CPU power of the 
clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6 
confirm the performance gain, around 95\% for both of the two methods, 
after adding more powerful CPU. Note that the execution time axis in the 
figure is in logarithmic scale.

 \textbf{V.4 Comparing GMRES in native synchronous mode and 
Multisplitting algorithms in asynchronous mode}

The previous paragraphs put in evidence the interests to simulate the 
behavior of the application before any deployment in a real environment. 
We have focused the study on analyzing the performance in varying the 
key factors impacting the results. In the same line, the study compares 
the performance of the two proposed methods in \textbf{synchronous mode
}. In this section, with the same previous methodology, the goal is to 
demonstrate the efficiency of the multisplitting method in \textbf{
asynchronous mode} compare with the classical GMRES staying in the 
synchronous mode.

Note that the interest of using the asynchronous mode for data exchange 
is mainly, in opposite of the synchronous mode, the non-wait aspects of 
the current computation after a communication operation like sending 
some data between nodes. Each processor can continue their local 
calculation without waiting for the end of the communication. Thus, the 
asynchronous may theoretically reduce the overall execution time and can 
improve the algorithm performance.

As stated supra, SIMGRID simulator tool has been used to prove the 
efficiency of the multisplitting in asynchronous mode and to find the 
best combination of the grid resources (CPU, Network, input matrix size, 
\ldots ) to get the highest "\,relative gain" in comparison with the 
classical GMRES time. 


The test conditions are summarized in the table below : 

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline  
 Grid & 2x50 totaling 100 processors\\ %\hline
 Processors & 1 GFlops to 1.5 GFlops\\
   Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
   Inter-Network & bw=5 Mbits - lat=2E-02\\
 Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
 Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline
 \end{tabular}
\end{footnotesize}

Again, comprehensive and extensive tests have been conducted varying the 
CPU power and the network parameters (bandwidth and latency) in the 
simulator tool with different problem size. The relative gains greater 
than 1 between the two algorithms have been captured after each step of 
the test. Table I below has recorded the best grid configurations 
allowing a multiplitting method time more than 2.5 times lower than 
classical GMRES execution and convergence time. The finding thru this 
experimentation is the tolerance of the multisplitting method under a 
low speed network that we encounter usually with distant clusters thru the 
internet.

% 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{Relative gain of the multisplitting algorithm compared with 
the classical GMRES}
  \label{tab.cluster.2x50}

  \begin{mytable}{6}
    \hline
    bw
    & 5         & 5         & 5         & 5         & 5         & 50 \\
    \hline
    lat
    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
    \hline
    power
    & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
    \hline
    size
    & 62        & 62        & 62        & 100       & 100       & 110 \\
    \hline
    Prec/Eprec
    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
    \hline
    speedup
    & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
    \hline
  \end{mytable}

  \smallskip

  \begin{mytable}{6}
    \hline
    bw
    & 50        & 50        & 50        & 50        & 10        & 10 \\
    \hline
    lat
    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
    \hline
    power
    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
    \hline
    size
    & 120       & 130       & 140       & 150       & 171       & 171 \\
    \hline
    Prec/Eprec
    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
    \hline
    speedup
    & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
    \hline
  \end{mytable}
\end{table}

\section{Conclusion}
CONCLUSION


\section*{Acknowledgment}


The authors would like to thank\dots{}


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