\usepackage{multirow}
\usepackage{authblk}
-\algnewcommand\algorithmicinput{\textbf{Input:}}
+
+\algnewcommand\algorithmicinput{\textbf{I1nput:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
algorithmss. In this paper, we will reconsider the use of a multisplitting
method. In opposition to traditional multisplitting method that suffer from slow
convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization
-process can drastically improve the convergence.
+process can drastically improve the convergence.\\
%%% AJOUTE************************
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-In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.
+\noindent {\bf Contributions:}\\
+In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.\\
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\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 quad-core 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=0.8\textwidth]{strong_scaling_150x150x150}
+\caption{Strong scaling with 3 blocks of cores}
+\label{fig:001}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\begin{tabular}{c}
+\includegraphics[width=0.8\textwidth]{weak_scaling_280k} \\ \includegraphics[width=0.8\textwidth]{weak_scaling_280K}\\
+\end{tabular}
+\caption{Weak scaling with 3 blocks of cores}
+\label{fig:002}
+\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
\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
