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Femto-ST Institute - DISC Department\\
Université de Franche-Comté\\
Belfort\\
- Email: raphael.couturier@univ-fcomte.fr
+ Email: \email{raphael.couturier@univ-fcomte.fr}
}
}
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
+resources (grid computing, clusters, broadband network, etc\dots{}) but
+also a non-negligible CPU execution time. We consider in this paper a
class of highly efficient parallel algorithms called iterative executed
in a distributed environment. As their name suggests, these algorithm
solves a given problem that might be NP- complete complex 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
+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. Generally, to reduce the complexity and the
execution time, the problem is divided into several "pieces" that will
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
+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
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
+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 40 \% in asynchronous mode.
+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
+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 (Simulation MPI ) will be in the
-next section . At last, the experiments results carried out will be
+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.
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~\ref{tab.cluster.2x50} with a matrix size
-ranging from Nx = Ny = Nz = 62 to 171 elements or from 62$^{3}$ = 238328 to
-171$^{3}$ = 5,211,000 entries.
+ranging from Nx = Ny = Nz = 62 to 171 elements or from $62^{3} = \np{238328}$ to
+$171^{3} = \np{5211000}$ entries.
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
\paragraph*{SMPI parameters}
\begin{itemize}
- \item HOSTFILE : Hosts file description.
+ \item HOSTFILE: Hosts file description.
\item PLATFORM: file description of the platform architecture : clusters (CPU power,
-... ) , intra cluster network description, inter cluster network (bandwidth bw ,
-lat latency , ... ).
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+lat latency, \dots{}).
\end{itemize}
\item Description of the cluster architecture;
\item Maximum number of internal and external iterations;
\item Internal and external precisions;
- \item Matrix size NX , NY and NZ;
+ \item Matrix size NX, NY and NZ;
\item Matrix diagonal value = 6.0;
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
mode.
In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
-deterioration of inter cluster network set with 5 Mbits/s of bandwidth, a latency
+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 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 E -05 to E-09. By increasing the problem size up to 100
-elements, it was necessary to increase the CPU power of 50 \% to 1.5 GFlops for a
+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 50 Mbits /s, the result of efficiency of about 40\% is
-obtained with high external precision of E-11 for a matrix size from 110 to 150
-side elements .
+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~\ref{tab.cluster.3x33} shows
that it was difficult to have a combination which gives an efficiency of
-asynchronous below 80 \%. Indeed, for a matrix size of 62 elements, equality
+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 10 Mbits/s and a latency of E- 01 ms. To
-challenge an efficiency by 78\% with a matrix size of 100 points, it was
+achieved with an inter cluster of \np[Mbits/s]{10} and a latency of \np{E-1} ms. 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 power
-with 200 nodes in total. The convergence with a speedup of 90 \% was obtained
-with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.
+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~\ref{tab.cluster.3x67}.
\section{Conclusion}
\section*{Acknowledgment}
-The authors would like to thank...
+The authors would like to thank\dots{}
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