\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.\\
+
-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.
+%%% AJOUTE************************
+%%%*******************************
+\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.\\
+%%%*******************************
+%%%*******************************
The present paper is organized as follows. First, Section~\ref{sec:02} presents
some related works and the principle of multisplitting methods. Then, in
\section{A two-stage method with a minimization}
\label{sec:03}
-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 multisplittig method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+
+%%% MODIFIE ************************
+%%%*********************************
+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 $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+%%%*********************************
+%%%*********************************
+
+
\begin{equation}
\left\{
\begin{array}{lll}
\right.
\label{sec03:eq01}
\end{equation}
-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 $\sum_\ell n_\ell=n$. 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 cluster. So, the multisplitting format of the linear system is defined as follows
+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 $\sum_\ell n_\ell=n$.
+%%% MODIFIE ***********************
+%%%********************************
+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 cluster.
+%%%********************************
+%%%********************************
+So, the multisplitting format of the linear system is defined as follows
\begin{equation}
\forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell,
\label{sec03:eq02}
\right.
\label{sec03:eq03}
\end{equation}
-is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
+is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}).
+%%% MODIFIE ***********************
+%%%********************************
+GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace.
+%%%********************************
+%%%********************************
+%In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
It should be noted that the convergence of the inner iterative solver for the
different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the
work is based on the work presented in~\cite{huang1993krylov} to increase the
convergence and improve the scalability of the multisplitting methods.
-Krylov subspace methods are well-known for their good convergence compared to other iterative methods. In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
+%%% AJOUTE ************************
+%%%********************************
+Krylov subspace methods are well-known for their good convergence compared to other iterative methods.
+%%%********************************
+%%%********************************
+In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
\begin{equation}
S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
\label{sec03:eq04}
\end{equation}
-where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 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.
+where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system.
+%%% 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 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}
\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.
+
+%%% 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.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}
+\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{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 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.
+%%%********************************
+%%%********************************
+
+
+%%% MODIFIE ************************
+%%%*********************************
%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.
+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 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.
+%%%********************************
+%%%********************************
\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
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
