\end{table*}
-In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
-
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie
+architecture are reported. For this application, we fixed $\alpha=0.6$. As it
+can be seen in that Table, the size of the problem has a strong influence on the
+number of iterations to reach the convergence. That is why we have preferred to
+change the threshold. If we set it to $1e-3$ as with the previous application,
+only one iteration is necessray to reach the convergence. So Table~\ref{tab:04}
+shows the results of differents executions with differents number of cores and
+differents thresholds. As with the previous example, we can observe that TSIRM
+is faster than FGMRES. The ratio greatly depends on the number of iterations for
+FMGRES to reach the threshold. The greater the number of iterations to reach the
+convergence is, the better the ratio between our algorithm and FMGRES is. This
+experiment is also a weak scaling with approximately $25,000$ components per
+core. It can also be observed that the difference between CGLS and LSQR is not
+significant. Both can be good but it seems not possible to know in advance which
+one will be the best.
+
+Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
+Curie architecture. So in this case, the number of unknownws is fixed to
+$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
+of two. The threshold is fixed to $5e-5$ and only the $mg$ preconditioner has
+been tested. Here again we can see that TSIRM is faster that FGMRES. Efficiecy
+of each algorithms is reported. It can be noticed that FGMRES is more efficient
+than TSIRM except with $8,192$ cores and that its efficiency is greater that one
+whereas the efficiency of TSIRM is lower than one. Nevertheless, the ratio of
+TSIRM with any version of the least-squares method is always faster. With
+$8,192$ cores when the number of iterations is far more important for FGMRES, we
+can see that it is only slightly more important for TSIRM.
+
+In Figure~\ref{fig:02} we report the number of iterations per second for
+experiments reported in Table~\ref{tab:05}. This Figure highlights that the
+number of iterations per seconds is more of less the same for FGMRES and TSIRM
+with a little advantage for FGMRES. It can be explained by the fact that, as we
+have previously explained, that the iterations of the least-sqaure steps are not
+taken into account with TSIRM.
\begin{table*}[htbp]
\begin{center}
\label{fig:02}
\end{figure}
+
+Concerning the experiments some other remarks are interesting. We can tested
+other examples of PETSc (ex29, ex45, ex49). For all these examples, we also
+obtained similar gain between GMRES and TSIRM but those examples are not
+scalable with many cores. In general, we had some problems with more than
+$4,096$ cores.
%%%*********************************************************
%%%*********************************************************
5 or 7 times faster than GMRES.
-For future work, the authors' intention is to investigate
-other kinds of matrices, problems, and inner solvers. The
-influence of all parameters must be tested too, while
-other methods to minimize the residuals must be regarded.
-The number of outer iterations to minimize should become
-adaptative to improve the overall performances of the proposal.
-Finally, this solver will be implemented inside PETSc.
+For future work, the authors' intention is to investigate other kinds of
+matrices, problems, and inner solvers. The influence of all parameters must be
+tested too, while other methods to minimize the residuals must be regarded. The
+number of outer iterations to minimize should become adaptative to improve the
+overall performances of the proposal. Finally, this solver will be implemented
+inside PETSc. This would be very interesting because it would allow us to test
+all the non-linear examples and compare our algorithm with the other algorithm
+implemented in PETSc.
% conference papers do not normally have an appendix
