\usepackage{multirow}
\usepackage{authblk}
-\algnewcommand\algorithmicinput{\textbf{Input:}}
+
+\algnewcommand\algorithmicinput{\textbf{I1nput:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
\end{table}
+
+
+
From these experiments, it can be observed that the multisplitting version is
always faster than the GMRES version. The acceleration gain of the
multisplitting version ranges between 4 and 6. It can be noticed that the number of
precision with 2 clusters is slightly better but in both cases the precision is
under the specified threshold.
+
+%%% AJOUTE************************
+%%%*******************************
+In Figure~\ref{fig:01}, the number of iterations per second is reported for both
+GMRES and the multisplitting methods. It should be noted that we took only the
+inner number of iterations (i.e. the GMRES iterations) for the multisplitting
+method. Iterations of CGNR are not taken into account. From this figure, it can
+be seen that the number of iterations per second is higher with GMRES but it is
+not so different with the multisplitting method. For the case with $8,192$
+cores, the number of iterations per second with 4 clusters is approximately
+equals to 115. So it is not different from GMRES.
+
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.7\textwidth]{nb_iter_sec}
+\caption{Number of iterations per second with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
+\label{fig:01}
+\end{figure}
+
+
+\noindent {\bf Final remarks:}\\
+It should be noted, on the one hand, that the development of a complete new
+method usable with any kind of problem is a really long and fastidious task if
+one is working from scratch. On the other hand, using an existing tool for the
+inner solver is also not easy because it requires to make link between the inner
+solver and the outer one. We plan to do that later with engineers working
+specifically on that point. Moreover, we think that it is very important to
+analyze the convergence of this method compared to other method. In this work,
+we have focused on the description of this method and its performance with a
+typical application. Many other investigations are required for this method as explained in the next section.
+%%%*******************************
+%%%*******************************
+
\section{Conclusion and perspectives}
We have implemented a Krylov multisplitting method to solve sparse linear
systems on large-scale computing platforms. We have developed a synchronous
