-
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
of asynchronous iterative algorithms. For that, we compare the behaviour of a
synchronous GMRES algorithm with an asynchronous multisplitting one with
simulations which let us easily choose some parameters. Both codes are real MPI
-codes ans simulations allow us to see when the asynchronous multisplitting algorithm can be more
+codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more
efficient than the GMRES one to solve a 3D Poisson problem.
$X_{0}$ 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{BT89,Bahi07}.
-Parallelization of such algorithms generally involve the division of the problem
+Parallelization of such algorithms generally involves the division of the problem
into several \emph{blocks} that will be solved in parallel on multiple
processing units. The latter will communicate each intermediate results before a
new iteration starts and until the approximate solution is reached. These
convergence depends on the delay of messages. With synchronous iterations, the
number of iterations is exactly the same than in the sequential mode (if the
parallelization process does not change the algorithm). So the difficulty with
-asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
+asynchronous iterative algorithms comes from the fact it is necessary to run the algorithm
with real data. In fact, from an execution to another the order of messages will
change and the number of iterations to reach the convergence will also change.
According to all the parameters of the platform (number of nodes, power of
-nodes, inter and intra clusrters bandwith and latency, ....) and of the
-algorithm (number of splitting with the multisplitting algorithm), the
-multisplitting code will obtain the solution more or less quickly. Or course,
+nodes, inter and intra clusrters bandwith and latency, etc.) and of the
+algorithm (number of splittings with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Of course,
the GMRES method also depends of the same parameters. As it is difficult to have
access to many clusters, grids or supercomputers with many different network
parameters, it is interesting to be able to simulate the behaviors of
framework to study 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. The early versions of SimGrid
-date from 1999, but it's still actively developed and distributed as an open
-source software. Today, it's one of the major generic tools in the field of
+date from 1999, but it is still actively developed and distributed as an open
+source software. Today, it is one of the major generic tools in the field of
simulation for large-scale distributed systems.
SimGrid provides several programming interfaces: MSG to simulate Concurrent
\begin{figure}[!t]
\centering
- \includegraphics[width=60mm,keepaspectratio]{clustering2}
-\caption{Example of two distant clusters of processors.}
+ \includegraphics[width=60mm,keepaspectratio]{clustering}
+\caption{Example of three distant clusters of processors.}
\label{fig:4.1}
\end{figure}
\right.
\label{eq:02}
\end{equation}
-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite differences scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
\begin{equation}
\begin{array}{l}
u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SimGrid unless some code
-debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method , the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
+debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between processors within a cluster or between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. For the synchronous GMRES method, the execution of the program raised no particular issue but in the asynchronous multisplitting method, the review of the sequence of \texttt{MPI\_Isend, MPI\_Irecv} and \texttt{MPI\_Waitall} instructions
and with the addition of the primitive \texttt{MPI\_Test} was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm.
%\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async}
%\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N=N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one.
%\AG{Expliquer comment lire les tableaux.}
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 \\
\hline
- size $(n^3)$
+ size $(N)$
& 62 & 62 & 62 & 100 & 100 \\
\hline
Precision
Power (GFlops)
& 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
\hline
- size $(n^3)$
+ size $(N)$
& 110 & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
\hline
Precision
\begin{itemize}
\item Description of the cluster architecture matching the format <Number of
clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
-\item Maximum number of iterations;
-\item Precisions on the residual error;
+\item Maximum numbers of outer and inner iterations;
+\item Outer and inner precisions on the residual error;
\item Matrix size $N_x$, $N_y$ and $N_z$;
-\item Matrix diagonal value: $6$ (See Equation~(\ref{eq:03}));
-\item Matrix off-diagonal value: $-1$;
+\item Matrix diagonal value: $6$ (see Equation~(\ref{eq:03}));
+\item Matrix off-diagonal values: $-1$;
\item Communication mode: asynchronous.
\end{itemize}
With these settings, Table~\ref{tab.cluster.2x50} shows
that after setting the bandwidth of the inter cluster network to \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
of one GFlops, an efficiency of about \np[\%]{40} is
-obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+obtained in asynchronous mode for a matrix size of $62^3$ elements. It is noticed that the result remains
stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
-increasing the matrix size up to 100 elements, it was necessary to increase the
+increasing the matrix size up to $100^3$ elements, it was necessary to increase the
CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency. Maintaining such processor power but increasing network throughput inter cluster up to
\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
-high external precision of \np{E-11} for a matrix size from 110 to 150 side
+high external precision of \np{E-11} for a matrix size from $110^3$ to $150^3$ side
elements.
%For the 3 clusters architecture including a total of 100 hosts,
\item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
\end{enumerate}
-Our results have shown that with two distant clusters, the asynchronous multisplitting is faster to \np[\%]{40} compared to the synchronous GMRES method
+Our results have shown that with two distant clusters, the asynchronous multisplitting method is faster to \np[\%]{40} compared to the synchronous GMRES method
which is not negligible for solving complex practical problems with more
and more increasing size.
mode in a grid architecture.
In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters.
-We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study.
+We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study. Finally, we also plan to study other problems with the multisplitting method and other asynchronous iterative methods.
\section*{Acknowledgment}
