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\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.\\
+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 other 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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preconditioners are not scalable when using many cores.
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-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.
+We have performed some experiments on an infiniband cluster of three Intel Xeon quad-core E5620 CPUs of 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}
+\caption{Strong scaling with 3 clusters of cores to solve a 3D Poisson problem of size $150^3$ components}
\label{fig:001}
\end{figure}
\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}
+\caption{Weak scaling with 3 clusters of cores to solve a 3D Poisson problem with approximately 280K components per core}
\label{fig:002}
\end{figure}
+The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU). Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of GMRES, classical multisplitting and Krylov multisplitting methods: strong and weak scaling respectively. We can remark from these figures that the performances of our Krylov multisplitting method are better than those of GMRES and classical multisplitting methods. In the experiments conducted in this work, our method is about twice faster than the GMRES method and about 9 times faster than the classical multisplitting method. Our multisplitting method uses a minimization step over a Krylov subspace which reduces the number of iterations and accelerates the convergence. We can also remark that the performances of the classical block Jacobi multisplitting method are the worst compared with those of two other methods. This is why in the following experiments we compare the performances of our Krylov multisplitting method with only those of the GMRES method.
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%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
-with a 3D torus.
+with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned.
Table~\ref{tab1} shows the result of the experiments. The first column shows
the size of the 3D Poisson problem. The size is chosen in order to have
the execution time for the GMRES and the Krylov multisplitting codes. The fourth
and the seventh column describe the number of iterations. For the
multisplitting code, the total number of inner iterations is represented in
-brackets. For the GMRES code (alone and in the multisplitting version) the
-restart parameter is fixed to 16. The precision of the GMRES version is fixed to
-1e-6. For the multisplitting, 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 are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
+brackets.
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\begin{table}[htbp]
\begin{center}
\end{center}
\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
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}
\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
