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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 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.\\
+\noindent {\bf Contributions:}\\ In this work we develop a new parallel
+two-stage algorithm for large-scale clusters. Our objective is to create a mix
+between Krylov based iterative methods and the multisplitting method to improve
+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 in order
+to improve the convergence of the multisplitting algorithm.\\
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-Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting method to the linear system as follows
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where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$.
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-The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one block of processors.
+The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a block of processors.
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So, the multisplitting format of the linear system is defined as follows
where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system.
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-The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks 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 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.
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-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
\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 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.
+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 approximately 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.
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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
+architecture, a UK 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. 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 size of the 3D Poisson problem. The size is chosen in order to have
approximately 50,000 components per core. The second column represents the
-number of cores used. In brackets, one can find the decomposition used for the
-Krylov multisplitting. The third column and the sixth column respectively show
-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
+number of cores used. Between brackets, one can find the decomposition used for
+the Krylov multisplitting. The third column and the sixth column respectively
+show 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 between
brackets.
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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
+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 blocks is approximately
-equals to 115. So it is not different from GMRES.
+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 blocks is approximately equal
+to 115. So it is not different from GMRES.
\begin{figure}[htbp]
\centering
\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 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.
+\noindent {\bf Final remarks:}\\ It should be noted, on the one hand, that the
+development of a complete new code 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 quite difficult because it
+requires to establish a 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 methods. In this work, we have focused on the
+description of this method and its performances with a typical application. Many
+other investigations are required for this method as explained in the next
+section.
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methods with overlapping blocks.
\section{Acknowledgement}
-The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. This work has been partially supported by the Labex
+ACTION project (contract “ANR-11-LABX-01-01”).
+
%Other applications (=> other matrices)\\
%Larger experiments\\
