% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}
+\usepackage{multirow}
+
\usepackage{url}
\DeclareUrlCommand\email{\urlstyle{same}}
\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
-Asynchronous iterative methods have been studied for many years theoritecally and
+Asynchronous iterative methods have been studied for many years theoretically and
practically. Many methods have been considered and convergence results have been
proved. These methods can be used to solve, in parallel, fixed point problems
(i.e. problems for which the solution is $x^\star =f(x^\star)$. In practice,
-asynchronous iterations methods can be used to solve, for example, linear and
+asynchronous iteration methods can be used to solve, for example, linear and
non-linear systems of equations or optimization problems, interested readers are
invited to read~\cite{BT89,bahi07}.
requires very few modifications to be able to be executed in both variants. In
practice, only the communications and convergence detection are different. In
the synchronous mode, iterations are synchronized whereas in the asynchronous
-one, they are not. It should be noticed that non blocking communications can be
+one, they are not. It should be noticed that non-blocking communications can be
used in both modes. Concerning the convergence detection, synchronous variants
can use a global convergence procedure which acts as a global synchronization
point. In the asynchronous model, the convergence detection is more tricky as
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
The number of iterations required to reach the convergence is generally greater
-for the asynchronous scheme (this number depends depends on the delay of the
+for the asynchronous scheme (this number depends on the delay of the
messages). Note that, it is not the case in the synchronous mode where the
number of iterations is the same than in the sequential mode. In this way, the
set of the parameters of the platform (number of nodes, power of nodes,
-inter and intra clusters bandwidth and latency, \ldots) and of the
+inter and intra clusters bandwidth and latency,~\ldots) and of the
application can drastically change the number of iterations required to get the
convergence. It follows that asynchronous iterative algorithms are difficult to
optimize since the financial and deployment costs on large scale multi-core
-architecture are often very important. So, prior to delpoyment and tests it
+architectures are often very important. So, prior to deployment and tests it
seems very promising to be able to simulate the behavior of asynchronous
-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced
by simulation are in accordance with reality i.e. of the same order of
magnitude. To our knowledge, there is no study on this problematic.
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
\begin{figure}[t]
%\begin{algorithm}[t]
\subsection{Simulation of the two-stage methods using SimGrid toolkit}
\label{sec:04.02}
-One of our objectives when simulating the application in Simgrid is, as in real
+One of our objectives when simulating the application in SimGrid is, as in real
life, to get accurate results (solutions of the problem) but also to ensure the
test reproducibility under the same conditions. According to our experience,
-very few modifications are required to adapt a MPI program for the Simgrid
+very few modifications are required to adapt a MPI program for the SimGrid
simulator using SMPI (Simulator MPI). The first modification is to include SMPI
-libraries and related header files (smpi.h). The second modification is to
+libraries and related header files (\verb+smpi.h+). The second modification is to
suppress all global variables by replacing them with local variables or using a
-Simgrid selector called "runtime automatic switching"
+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 and generated by
-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.
+SimGrid to simulate the grid environment.
-
-\paragraph{Simgrid Simulator parameters}
-\ \\ \noindent Before running a Simgrid benchmark, many parameters for the
+\paragraph{Parameters of the simulation in SimGrid}
+\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
in which several clusters are geographically distant, so there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
\item hostfile: hosts description file.
\item platform: file describing the platform architecture: clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}).
\item archi : grid computational description (number of clusters, number of
-nodes/processors for each cluster).
+nodes/processors in each cluster).
\end{itemize}
\noindent
In addition, the following arguments are given to the programs at runtime:
\begin{itemize}
\item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
\item inner precision $\TOLG$ and outer precision $\TOLM$,
- \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
- \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+ \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+ \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
\item matrix off-diagonal value is fixed to $-1.0$,
\item number of vectors in matrix $S$ (i.e. value of $s$),
\item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
\item execution mode: synchronous or asynchronous.
\end{itemize}
-It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Experimental Results}
+\section{Experimental results}
\label{sec:expe}
-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used 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{The 3D Poisson problem}
+\label{3dpoisson}
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:
\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 in a virtual
+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 in a virtual
machine on a simple laptop. \\
\textbf{Step 3}: Fix the criteria which will be used for the future
and on the other hand the execution time and the number of iterations to reach the convergence. \\
\textbf{Step 4 }: Set up the different grid testbed environments that will be
-simulated in the simulator tool to run the program. The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+simulated in the simulator tool to run the program. The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
represents the number of clusters in the grid and the second number represents
-the number of hosts (processors/cores) in each cluster. The network has been
+the number of hosts (processors/cores) in each cluster. The network has been
designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
-(resp. inter-clusters backbone links). \\
+(resp. inter-clusters backbone links). \\
+
+\LZK{Il me semble que le bw et lat des deux réseaux varient dans les expés d'une simu à l'autre. On vire la dernière phrase?}
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
\textbf{Step 6} : Collect and analyze the output results.
-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}
When running a distributed application in a computational grid, many factors may
have a strong impact on the performance. First of all, the architecture of the
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
-\item the network bandwidth (bw=bits/s) also known as "the data-carrying
+\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
capacity" of the network is defined as the maximum of data that can transit
from one point to another in a unit of time.
-\item the network latency (lat : microsecond) defined as the delay from the
+\item the network latency ($lat$ in microseconds) defined as the delay from the
start time to send a simple data from a source to a destination.
\end{enumerate}
Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
on the other hand, the "inter-network" which is the backbone link between
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
+ high bandwidth and very low latency. In opposite, the inter-network connects
+ clusters sometime via heterogeneous networks components through internet with
a lower speed. The network between distant clusters might be a bottleneck
for the global performance of the application.
-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+\subsection{Comparison of GMRES and Krylov two-stage algorithms in synchronous mode}
In the scope of this paper, our first objective is to analyze when the Krylov
-Multisplitting method has better performance than the classical GMRES
-method. With a synchronous iterative method, better performance means a
+two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means 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.
+\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}
-The following paragraphs present the test conditions, the output results
-and our comments.\\
+In what follows, we will present the test conditions, the output results and our comments.\\
-
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
+%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
\ \\
% environment
\begin{table} [ht!]
\begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll }
\hline
- Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ Network & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ \multirow{2}{*}{Matrix size} & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
+ & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
+ \end{tabular}
+\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
+\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
+ Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
- & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
\end{tabular}
-\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
+\caption{Test conditions: various grid configurations with the input matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:01}
\end{center}
\end{table}
+<<<<<<< HEAD
+In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm.
+%% First, the results in Figure~\ref{fig:01}
+%% show for all grid configurations 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...}
+=======
In this section, we analyze the performance of algorithms running on various
-grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
+grid configurations (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. First, the results in Figure~\ref{fig:01}
show for all grid configurations 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...}
-
+%\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...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\begin{figure} [ht!]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
- \caption{Various grid configurations with the input matrix size $N_{x}=150$ and $N_{x}=170$\RC{idem}
+ \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$
\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}}
+\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
\label{fig:01}
\end{figure}
-
+<<<<<<< HEAD
The execution times between the two algorithms is significant with different
grid architectures, even with the same number of processors (for example, 2x16
-and 4x8). We can observ the low sensitivity of the Krylov multisplitting method
+and 4x8). We can observe the low sensitivity of the Krylov multisplitting method
(compared with the classical GMRES) when scaling up the number of the processors
in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
-$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
+\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+=======
+
+Secondly, the execution times between the two algorithms is significant with different
+grid architectures, even with the same number of processors (for example, 2 $\times$ 16
+and 4 $\times$ 8). We can observ the sensitivity of the Krylov multisplitting method
+(compared with the classical GMRES) when scaling up the number of the processors
+in the grid: in average, the reduction of the execution time for GMRES (resp. Multisplitting) algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Simulations for two different inter-clusters network speeds \\}
\begin{table} [ht!]
\begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll}
\hline
- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ Matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
+ \end{tabular}
+\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
+ Inter Networks & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
- & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
\end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:02}
\end{center}
\end{table}
+<<<<<<< HEAD
These experiments compare the behavior of the algorithms running first on a
-speed inter-cluster network (N1) and also on a less performant network (N2). \RC{Il faut définir cela avant...}
+slow inter-cluster network (N1) and also on a more performant network (N2). \RC{Il faut définir cela avant...}
+=======
+In this section, the experiments compare the behavior of the algorithms running on a
+speeder inter-cluster network (N1) and also on a less performant network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut définir cela avant...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
Figure~\ref{fig:02} shows that end users will reduce the execution time
-for both algorithms when using a grid architecture like 4x16 or 8x8: the reduction is about $2$. The results depict also that when
+for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: the reduction is about $2$. The results depict also that when
the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%.
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2
\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
\label{fig:02}
\end{figure}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
Network & N1 : bw=1Gbs \\ %\hline
Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 4x8\\ %\hline
+ Grid Architecture & 4 $\times$ 8\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
\end{tabular}
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. It should be noticed that the same test has been done with the
-grid 2x16 leading to the same conclusion.
+grid 2 $\times$ 16 leading to the same conclusion.
\subsubsection{CPU Power impacts on performance}
\centering
\begin{tabular}{r c }
\hline
- Grid architecture & 2x16\\ %\hline
+ Grid architecture & 2 $\times$ 16\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline
Processors Power & 1 GFlops to 1.5 GFlops\\
Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
