imperative to develop coarse-grain based algorithms to reduce the communications
in the parallel iterative solvers. Two possible solutions consists either in
using asynchronous iterative methods~\cite{ref18} or in using multisplitting
-algorithmss. In this paper, we will reconsider the use of a multisplitting
+algorithms. 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.\\
solved~\cite{o1985multi,ref18}. Furthermore, the splitting of the linear system
among several clusters of processors increases the spectral radius of the
iteration matrix, thereby slowing the convergence. In fact, the larger the
-number of splitting is, the larger the spectral radius is. In this paper, our
+number of splittings is, the larger the spectral radius is. In this paper, our
work is based on the work presented in~\cite{huang1993krylov} to increase the
convergence and improve the scalability of the multisplitting methods.
%%% MODIFIE ***********************
%%%********************************
The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+
+The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
%%%********************************
%%%********************************
-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
\begin{equation}
R\alpha=b,
\label{sec03:eq05}
%%% AJOUTE ************************
%%%********************************
-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.
+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
\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.
+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 are presented 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 the other two methods. This is why in the following experiments we compare the performances of our Krylov multisplitting method with only those of the GMRES method.
%%%********************************
%%%********************************
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,
+analyze the convergence of this method compared to other methods. 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.
%%%*******************************
We have tested our multisplitting method to solve the sparse linear system
issued from the discretization of a 3D Poisson problem. We have compared its
performances to the classical GMRES method on a supercomputer composed of 2,048
-to 8,192 cores. The experimental results showed that the multisplitting method is
+up-to 8,192 cores. The experimental results showed that the multisplitting method is
about 4 to 6 times faster than the GMRES method for different sizes of the
problem split into 2 or 4 blocks when using the multisplitting method. Indeed, the
GMRES method has difficulties to scale with many cores while the Krylov
