X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/af016b677c3b80035457ffcfea85a9809f3ee6ce..43651bc69ac69595ccd000528fd530f485ffd7fe:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
index 59c4a67..de34cb6 100644
--- a/paper.tex
+++ b/paper.tex
@@ -581,13 +581,21 @@ convergence of the algorithm is (see the output results obtained from
 configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
 8$\times$8).
 
-The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16).
+The execution times between both algorithms is significant with different grid
+architectures. The synchronous Krylov two-stage algorithm presents better
+performances than the GMRES algorithm, even for a high number of clusters (about
+$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+observe a better sensitivity of the Krylov two-stage algorithm (compared to the
+GMRES one) when scaling up the number of the processors in the computational
+grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
+about $40\%$ better on $64$ processors (grid of 8$\times$8) than $32$ processors
+(grid of 2$\times$16).
 
 \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 matrix sizes 150$^3$ and 170$^3$}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
 \label{fig:01}
 \end{figure}
 
@@ -605,7 +613,7 @@ efficient for distributed systems with high latency networks.
 \begin{figure}[ht]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
 \LZK{CE, remplacer les ``,'' des dÃ©cimales par un ``.''}
 \RCE{ok}
 \label{fig:02}
@@ -622,7 +630,15 @@ Figure~\ref{fig:03} shows the impact of the network latency on the performances
 \end{figure}
 
 \subsubsection{Network bandwidth impacts on performances\\}
-Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
+
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
+solve a 3D Poisson problem of size $150^3$. The results of increasing the
+network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.
 
 \begin{figure}[ht]
 \centering
@@ -632,7 +648,18 @@ Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
 \end{figure}
 
 \subsubsection{Matrix size impacts on performances\\}
-In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes.  For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+
+In these experiments, the matrix size of the 3D Poisson problem is varied from
+$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
+clusters of $8$ processors each interconnected by the network $N2$ (see
+Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes.  For all problem
+sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm.
+Moreover, for this benchmark, it seems that the greater the problem size is, the
+bigger the ratio between execution times of both algorithms is. We can also
+observe that for some problem sizes, the convergence (and thus the execution
+time) of the Krylov two-stage algorithm varies quite a lot.
+%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
 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.
 
 \begin{figure}[ht]
@@ -643,7 +670,15 @@ These findings may help a lot end users to setup the best and the optimal target
 \end{figure}
 
 \subsubsection{CPU power impacts on performances\\}
-Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.
+
+Using the SimGrid simulator flexibility, we have tried to determine the impact
+of the CPU power of the processors in the different clusters on performances of
+both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The
+simulation is conducted on a grid of $2\times16$ processors interconnected by
+the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size
+$150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance
+gain, about $95\%$ for both algorithms, after improving the CPU power of
+processors.
 
 \begin{figure}[ht]
 \centering
@@ -652,11 +687,12 @@ Using the SimGrid simulator flexibility, we have tried to determine the impact o
 \label{fig:06}
 \end{figure}
 \ \\
+
 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
-with a real platform is most of  the time not possible. Moreover the behavior of
-both GMRES and  Krylov two-stage algorithms is in accordance  with larger real
-executions on large scale supercomputers~\cite{couturier15}.
+with a real platform is most of the time not possible or very costly. Moreover
+the behavior of both GMRES and  Krylov two-stage algorithms is in accordance
+with larger real executions on large scale supercomputers~\cite{couturier15}.
 
 
 \subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
@@ -669,7 +705,7 @@ classical GMRES in \textit{synchronous mode}.
 
 The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
 synchronization. With  geographically distant  clusters, this may  be essential.
-In  this case,  each  processor can  compute its  iteration  freely without  any
+In  this case,  each  processor can  compute its  iterations  freely without  any
 synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
 theoretically reduce  the overall execution  time and can improve  the algorithm
 performance.
@@ -678,8 +714,8 @@ In this section,  the SimGrid simulator is  used to compare the  behavior of the
 two-stage algorithm in  asynchronous mode  with GMRES  in synchronous  mode.  Several
 benchmarks have  been performed with  various combinations of the  grid resources
 (CPU, Network, matrix size, \ldots). The test  conditions are summarized
-in  Table~\ref{tab:02}. In  order to  compare  the execution  times, Table~\ref{tab:03}
-reports the  relative gain between both  algorithms. It is defined  by the ratio
+in  Table~\ref{tab:02}. In  order to  compare  the execution  times. Table~\ref{tab:03}
+reports the  relative gains between both  algorithms. It is defined  by the ratio
 between  the   execution  time  of   GMRES  and   the  execution  time   of  the
 multisplitting.
 \LZK{Quelle table repporte les gains relatifs?? SÃ»rement pas Table II !!}
@@ -694,7 +730,7 @@ multisplitting version is faster than GMRES.
  Grid architecture                       & 2$\times$50 totaling 100 processors\\
  Processors Power                        & 1 GFlops to 1.5 GFlops \\
  \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
-                                         & $bw$=5 Mbits, $lat=20ms$s\\
+                                         & $bw$=5 Mbits, $lat=20ms$\\
  Matrix size                             & from $62^3$ to $150^3$\\
  Residual error precision                & $10^{-5}$ to $10^{-9}$\\ \hline \\
  \end{tabular}
