\section{Experiments}
\label{sec:04}
-In order to illustrate the interest of our algorithm, we have compared our
-algorithm with the GMRES method which is a commonly used method in many
-situations. We have chosen to focus on only one problem which is very simple to
-implement: a 3 dimension Poisson problem.
+%%% MODIFIE ***********************
+%%%********************************
+In order to illustrate the interest of our Krylov multisplitting algorithm, we have compared its performances with those of a classical block Jacobi multisplitting method and those of the GMRES method which is a commonly used method in many situations.
+%%%********************************
+%%%********************************
+ We have chosen to focus on only one problem which is very simple to implement: a 3 dimension Poisson problem.
\begin{equation}
\left\{
preconditioner it is possible to reduce the number of iterations but
preconditioners are not scalable when using many cores.
+
+%%% MODIFIE ***********************
+%%%********************************
+We have performed some experiments on an infiniband cluster of 3 nodes of Intel Xeon CPU E5620 2.40 GHz and 12 GB of memory. For the GMRES code (alone and in both multisplitting versions) the restart parameter is fixed to 16. The precision of the GMRES version is fixed to 1e-6. For the multisplitting versions, there are two precisions, one for the external solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It should be noted that a high precision is used but we also fixed a maximum number of iterations for each internal step. In practice, we limit the number of iterations in the internal step to 10. So an internal iteration is finished when the precision is reached or when the maximum internal number of iterations is reached. The precision and the maximum number of iterations of CGNR method used by our Krylov multisplitting algorithm are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis S is fixed to 10 vectors.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.7\textwidth]{strong_scaling_150x150x150}
+\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}
+
+%%%********************************
+%%%********************************
+
+
%Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.
In the following we present some experiments we could achieve out on the Hector
architecture, a UK's high-end computing resource, funded by the UK Research
Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two
-16-core AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected
+16-core AMD Opteron 2.3 GHz and 32 GB of memory. Machines are interconnected
with a 3D torus.
Table~\ref{tab1} shows the result of the experiments. The first column shows
Reviewer #5: In this paper, the authors have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. The technical approach and analysis of this paper is reasonable and the paper is clear, logical, and understandable.
,----
However, the paper does not take into considerate account relevant current and past research on the topic.
-`----
+`---
+%Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.-
****************************************************
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