\end{abstract}
%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
-%performance}
+%performance}
\keywords{ Performance evaluation, Simulation, SimGrid, Synchronous and asynchronous iterations, Multisplitting algorithms}
\maketitle
given application for a given architecture. In this way and in order to reduce
the access cost to these computing resources it seems very interesting to use a
simulation environment. The advantages are numerous: development life cycle,
-code debugging, ability to obtain results quickly,~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
In this paper we focus on a class of highly efficient parallel algorithms called
\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
simple. It generally involves the division of the problem into several
\emph{blocks} that will be solved in parallel on multiple processing
-units. Each processing unit has to compute an iteration, to send/receive some
+units. Each processing unit has to compute an iteration to send/receive some
data dependencies to/from its neighbors and to iterate this process until the
-convergence of the method. Several well-known methods demonstrate the
+convergence of the method. Several well-known studies demonstrate the
convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
task cannot begin a new iteration while it has not received data dependencies
-from its neighbors. We say that the iteration computation follows a synchronous
-scheme. In the asynchronous scheme a task can compute a new iteration without
-having to wait for the data dependencies coming from its neighbors. Both
-communication and computations are asynchronous inducing that there is no more
-idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
-This model presents some advantages and drawbacks that we detail in
-section~\ref{sec:asynchro} but even if the number of iterations required to
-converge is generally greater than for the synchronous case, it appears that
-the asynchronous iterative scheme can significantly reduce overall execution
-times by suppressing idle times due to synchronizations~(see~\cite{bahi07}
-for more details).
+from its neighbors. We say that the iteration computation follows a
+\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
+iteration without having to wait for the data dependencies coming from its
+neighbors. Both communication and computations are \textit{asynchronous}
+inducing that there is no more idle time, due to synchronizations, between two
+iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
+that we detail in section~\ref{sec:asynchro} but even if the number of
+iterations required to converge is generally greater than for the synchronous
+case, it appears that the asynchronous iterative scheme can significantly
+reduce overall execution times by suppressing idle times due to
+synchronizations~(see~\cite{bahi07} for more details).
Nevertheless, in both cases (synchronous or asynchronous) it is very time
consuming to find optimal configuration and deployment requirements for a given
k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
\label{eq:04}
\end{equation}
-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
\begin{equation}
Simgrid selector called "runtime automatic switching"
(smpi/privatize\_global\_variables). Indeed, global variables can generate side
effects on runtime between the threads running in the same process, generated by
-the Simgrid to simulate the grid environment. \RC{On vire cette phrase ?}The
-last modification on the MPI program pointed out for some cases, the review of
-the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which
-might cause an infinite loop.
+Simgrid to simulate the grid environment.
+
+%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
+%last modification on the MPI program pointed out for some cases, the review of
+%the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which
+%might cause an infinite loop.
\paragraph{Simgrid Simulator parameters}
\begin{itemize}
\item maximum number of inner and outer iterations;
\item inner and outer precisions;
- \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
- \item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifies, je dis ca de tête}
- \item execution mode: synchronous or asynchronous.
+ \item maximum number of the gmres's restarts in the Arnorldi process;
+ \item maximum number of iterations qnd the tolerance threshold in classical GMRES;
+ \item tolerance threshold for outer and inner-iterations;
+ \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
+ \item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
+ \item matrix off-diagonal value;
+ \item execution mode: synchronous or asynchronous;
+ \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
+ \item Size of matrix S;
+ \item Maximum number of iterations and tolerance threshold for CGLS.
\end{itemize}
It should also be noticed that both solvers have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
\section{Experimental Results}
\label{sec:expe}
-In this section, experiments for both Multisplitting algorithms are reported. First the problem sued in our experiments is described.
+In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+
+\subsection{3D Poisson}
+
-We use our two-stage algorithms to solve the well-known 3D Poisson problem $\nabla^2\phi=f$, where $\nabla^2$ is the Laplace operator. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
+We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
\begin{equation}
-\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in~}\Omega
+\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
\label{eq:07}
\end{equation}
-where the real-valued function $\phi(x,y,z)=0\mbox{~on~}\partial\Omega$ is the solution sought, $f(x,y,z)$ is a known function and the domain $\Omega=[0,1]^3$.
+such that
+\begin{equation*}
+\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
+\end{equation*}
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
+\begin{equation}
+\begin{array}{ll}
+\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
+\end{array}
+\label{eq:08}
+\end{equation}
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
+
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.
\subsection{Study setup and Simulation Methodology}
\textbf{Step 2}: Collect the software materials needed for the
experimentation. In our case, we have two variants algorithms for the
resolution of the 3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting method. In addition, the Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in the University of Franche-Comte and also in a virtual machine on a laptop. \\
+distributed applications. Simgrid is running on the Mesocentre datacenter in the University of Franche-Comte and also in a virtual machine on a simple 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 a grid environment, it is common to distinguish, on the one hand, the
"intra-network" which refers to the links between nodes within a cluster and,
on the other hand, the "inter-network" which is the backbone link between
- clusters. In practse; these two networks have different speeds. The
+ clusters. In practice, these two networks have different speeds. 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 throuth internet with a lower
speed. The network between distant clusters might be a bottleneck for the
global performance of the application.
-\subsection{Comparing GMRES and Multisplitting algorithms in
-synchronous mode}
+\subsection{Comparison of GMRES and Krylov 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
-\textit{synchronous mode}. Better algorithm performance
-should means 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.
+In the scope of this paper, our first objective is to analyze when the Krylov
+Multisplitting method has better performances than the classical GMRES
+method. With an iterative method, better performances mean a smaller number of
+iterations and 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.
The following paragraphs present the test conditions, the output results
and our comments.\\
-\textit{3.a Executing the algorithms on various computational grid
+\subsubsection{Execution of the the algorithms on various computational grid
architecture and scaling up the input matrix size}
-\\
-
+\ \\
% environment
-\begin{footnotesize}
+
+\begin{figure} [ht!]
+\begin{center}
\begin{tabular}{r c }
\hline
Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
\end{tabular}
-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}
-\end{footnotesize}
%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration 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.
+In this section, we analyze the performences of algorithms running on various
+grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
+show for all grid configuration 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.
+
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+\RC{Les légendes ne sont pas explicites...}
+
-%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
-%\label{overflow}}
+ \begin{center}
+ \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+ \end{center}
+ \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+ \label{fig:01}
\end{figure}
-%\end{wrapfigure}
+
The execution time difference between the two algorithms is important when
comparing between different grid architectures, even with the same number of
N$_{x}$ = N$_{y}$ = N$_{z}$ = 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 7,
the execution time for the two algorithms convergence increases with the
-input matrix size. But the interesting results here direct on (i) the
+iinput matrix size. But the interesting results 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 N$_{x}$=150; (ii) the classical GMRES execution time also almost
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impact on execution time}
%\label{overflow}}
-\end{figure}
+s\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.
+after adding more powerful CPU.
\subsection{Comparing GMRES in native synchronous mode and
Multisplitting algorithms in asynchronous mode}
