\begin{abstract}
In this paper we revisit the Krylov multisplitting algorithm presented in
-\cite{huang1993krylov} which uses a scalar method to minimize the Krylov
+\cite{huang1993krylov} which uses a sequential method to minimize the Krylov
iterations computed by a multisplitting algorithm. Our new algorithm is based on
a parallel multisplitting algorithm with few blocks of large size using a
parallel GMRES method inside each block and on a parallel Krylov minimization in
proposed in~\cite{huang1993krylov}, the use of a minimization process can
drastically improve the convergence.
+The paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. The, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem.
+
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Related works and presention of the multisplitting method}
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}
A general framework for studying parallel multisplitting has been presented in~\cite{o1985multi}
by O'Leary and White. Convergence conditions are given for the
most general case. Many authors improved multisplitting algorithms by proposing
In~\cite{couturier2008gremlins}, the authors proposed practical implementations
of multisplitting algorithms to solve large scale linear systems. Inner solvers
-could be based on scalar direct method with the LU method or scalar iterative
+could be based on sequential direct method with the LU method or sequential iterative
one with GMRES.
In~\cite{prace-multi}, the authors have proposed a parallel multisplitting
asynchronous multisplitting algorithm could be more efficient than traditional
solvers on an exascale architecture with hundreds of thousands of cores.
-
-So compared to these works, we propose in this paper a practical multisplitting
-which is based on parallel iterative blocks and which give better result than
-GMRES for the 3D Poisson problem we considered.
+So compared to these works, we propose in this paper a practical multisplitting method based on parallel iterative blocks and gives better results than classical GMRES method for the 3D Poisson problem we considered.
\\
The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways
%%%%%%%%%%%%%%%%%%%%%%%%
\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 to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
\begin{equation}
\left\{
\end{equation}
where $A_{\ell m}$ is a sub-block of size $n_\ell\times n_m$ of the rectangular matrix $A_\ell$, $X_m\neq X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
-Our multisplitting method proceeds by iteration for solving the linear system in such a way each sub-system
+Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system
\begin{equation}
\left\{
\begin{array}{l}
\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. %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 convergence of the multisplitting method. It strongly depends on the properties of the global sparse linear system to be solved~\cite{o1985multi,ref18}. Furthermore, the multisplitting of the linear system among several clusters of processors increases the spectral radius of the iteration matrix, thereby slowing the convergence. In this paper, we based on the work presented in~\cite{huang1993krylov} to increase the convergence and improve the scalability of the multisplitting methods.
+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
+convergence of the multisplitting method. It strongly depends on the properties
+of the global sparse linear system to be
+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, we
+based on the work presented in~\cite{huang1993krylov} to increase the
+convergence and improve the scalability of the multisplitting methods.
In order to accelerate the convergence, we implemented the outer iteration of the 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}
%%%%%%%%%%%%%%%%%%%%%%%%
\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 very well used method in many
situations. We have chosen to focus on only one problem which is very simple to
preconditioner it is possible to reduce the number of iterations but
preconditioners are not scalable when using many cores.
-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 presented some experiments we could achieved out on the
-Hector architecture, the previous UK's high-end computing resource, funded by
-the UK Research Councils, which has been stopped in the early 2014.
+%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 achieved 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.
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
\cline{3-8}
& & Time (s) & nb Iter. & $\Delta$ & Time (s)& nb Iter. & $\Delta$ & \\
\hline
-$468^3$ & 2048 (2x1024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
+$468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
\hline
-$590^3$ & 4096 (2x2048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
+$590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
\hline
-$743^3$ & 8192 (2x4096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
+$743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
\hline
-$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
+$743^3$ & 8,192 (4x2,048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
\hline
\end{tabular}
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 is between 4 and 6. It can be noticed that the number of
-iterations is drastically reduced with the multisplitting version even it is not
-neglectable.
+iterations is drastically reduced with the multisplitting version even it is not
+neglectable. Moreover, with 8,192 cores, we can see that using 4 clusters gives
+better performance than simply using 2 clusters. In fact, we can remark that the
+precision with 2 clusters is slightly better but in both cases the precision is
+under the specified threshold.
\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 two-stage method based on the block Jacobi multisplitting and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a multi-cluster environment based on processors of the computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. Second, we have implemented the outer iteration of the multisplitting method using Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
-
-We have tested our multisplitting method for solving the sparse linear system issued from the discretization of a 3D Poisson problem. We have compared its performances to those of classical GMRES method on a supercomputer composed of 2048 to 8192 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 multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
-
-In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.
-
+We have implemented a Krylov multisplitting method to solve sparse linear
+systems on large-scale computing platforms. We have developed a synchronous
+two-stage method based on the block Jacobi multisaplitting which uses GMRES
+iterative method as an inner iteration. Our contribution in this paper is
+twofold. First we provide a multi cluster decomposition that allows us to choose
+the appropriate size of the clusters according to the architecures of the
+supercomputer. Second, we have implemented the outer iteration of the
+multisplitting method as a Krylov subspace method which minimizes some error
+function. This increases the convergence and improves the scalability of the
+multisplitting method.
+
+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
+about 4 to 6 times faster than the GMRES method for different sizes of the
+problem split into 2 or 4 blocks when using multisplitting method. Indeed, the
+GMRES method has difficulties to scale with many cores while the Krylov
+multisplitting method allows to hide latency and reduce the inter-cluster
+communications.
+
+In future works, we plan to conduct experiments on larger number of cores and
+test the scalability of our Krylov multisplitting method. It would be
+interesting to validate its performances to solve other linear/nonlinear and
+symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting
+methods based on asynchronous iteration in which communications are overlapped
+by computations. These methods would be interesting for platforms composed of
+distant clusters interconnected by a high-latency network. In addition, we
+intend to investigate the convergence improvements of our method by using
+preconditioning techniques for Krylov iterative methods and multisplitting
+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.
%Other applications (=> other matrices)\\
%Larger experiments\\
