\usepackage{amsfonts,amssymb}
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-%\usepackage{algorithmic}
-%\usepackage[ruled,english,boxed,linesnumbered]{algorithm2e}
-%\usepackage[english]{algorithme}
\usepackage{algorithm}
\usepackage{algpseudocode}
-
-
-\title{A scalable multisplitting algorithm for solving large sparse linear systems}
-
-
-
+\usepackage{multirow}
+\usepackage{authblk}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
+\newcommand{\Prec}{\mathit{prec}}
+\newcommand{\Ratio}{\mathit{Ratio}}
+
+\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
+\let\endchangemargin=\endlist
+
+\title{A scalable multisplitting algorithm to solve large sparse linear systems}
+\date{}
+
+\author[1]{Raphaël Couturier}
+\author[2]{ Lilia Ziane Khodja}
+\affil[1]{ Femto-ST Institute\\
+ University of Franche Comte\\
+ France\\
+ email: raphael.couturier@univ-fcomte.fr}
+\affil[2]{Inria Bordeaux Sud-Ouest\\
+ France\\
+ email: lilia.ziane@inria.fr}
\begin{document}
-\author{Raphaël Couturier \and Lilia Ziane Khodja}
+
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{abstract}
-In this paper we revist the krylov multisplitting algorithm presented in
-\cite{huang1993krylov} which uses a scalar method to minimize the krylov
+In this paper we revisit the Krylov multisplitting algorithm presented in
+\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
+parallel GMRES method inside each block and on a parallel Krylov minimization in
order to improve the convergence. Some large scale experiments with a 3D Poisson
-problem are presented. They show the obtained improvements compared to a
-classical GMRES both in terms of number of iterations and execution times.
+problem are presented with up to 8,192 cores. They show the obtained
+improvements compared to a classical GMRES both in terms of number of iterations
+and in terms of execution times.
\end{abstract}
-
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-
\section{Introduction}
-
Iterative methods are used to solve large sparse linear systems of equations of
the form $Ax=b$ because they are easier to parallelize than direct ones. Many
-iterative methods have been proposed and adapted by many researchers. For
+iterative methods have been proposed and adapted by different researchers. For
example, the GMRES method and the Conjugate Gradient method are very well known
-and used by many researchers ~\cite{S96}. Both the method are based on the
-Krylov subspace which consists in forming a basis of the sequence of successive
+and used~\cite{S96}. Both methods are based on the
+Krylov subspace which consists in forming a basis of a sequence of successive
matrix powers times the initial residual.
When solving large linear systems with many cores, iterative methods often
suffer from scalability problems. This is due to their need for collective
communications to perform matrix-vector products and reduction operations.
-Preconditionners can be used in order to increase the convergence of iterative
-solvers. However, most of the good preconditionners are not sclalable when
+Preconditioners can be used in order to increase the convergence of iterative
+solvers. However, most of the good preconditioners are not scalable when
thousands of cores are used.
+%Traditional iterative solvers have global synchronizations that penalize the
+%scalability. Two possible solutions consists either in using asynchronous
+%iterative methods~\cite{ref18} or to use 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.
+
+Traditional parallel iterative solvers are based on fine-grain computations that
+frequently require data exchanges between computing nodes and have global
+synchronizations that penalize the scalability. Particularly, they are more
+penalized on large scale architectures or on distributed platforms composed of
+distant clusters interconnected by a high-latency network. It is therefore
+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
+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.
+
+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~\ref{sec:03} the algorithm of our Krylov multisplitting
+method, based on inner-outer iterations, is presented. 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.
-A completer...
-On ne peut pas parler de tout...\\
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}
+A general framework to study parallel multisplitting methods has been presented in~\cite{o1985multi}
+by O'Leary and White. Convergence conditions are given for the
+most general cases. Many authors have improved multisplitting algorithms by proposing,
+for example, an asynchronous version~\cite{bru1995parallel} or convergence
+conditions~\cite{bai1999block,bahi2000asynchronous} or other
+two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
+
+In~\cite{huang1993krylov}, the authors have proposed a parallel multisplitting
+algorithm in which all the tasks except one are devoted to solve a sub-block of
+the splitting and to send their local solutions to the first task which is in
+charge of combining the vectors at each iteration. These vectors form a Krylov
+basis for which the first task minimizes the error function over the basis to
+increase the convergence, then the other tasks receive the updated solution until the
+convergence of the global system.
+
+In~\cite{couturier2008gremlins}, the authors have developed practical implementations
+of multisplitting algorithms to solve large scale linear systems. Inner solvers
+could be based on sequential direct method with the LU method or sequential iterative
+one with GMRES.
+In~\cite{prace-multi}, the authors have designed a parallel multisplitting
+algorithm in which large blocks are solved using a GMRES solver. The authors have
+performed large scale experiments up-to 32,768 cores and they conclude that
+an 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 method based on parallel iterative blocks which gives better results than classical GMRES method for the 3D Poisson problem we considered.
+\\
-%%%%%%%%%%%%%%%%%%%%%%%
-%% BEGIN
-%%%%%%%%%%%%%%%%%%%%%%%
-The key idea of the multisplitting method for solving a large system
-of linear equations $Ax=b$ consists in partitioning the matrix $A$ in
-$L$ several ways
+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
\begin{equation}
-A = M_l - N_l,~l\in\{1,\ldots,L\},
+A = M_\ell - N_\ell,
\label{eq01}
\end{equation}
-where $M_l$ are nonsingular matrices. Then the linear system is solved
-by iteration based on the multisplittings as follows
+where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by an iteration based on the obtained splittings as follows
\begin{equation}
-x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
\label{eq02}
\end{equation}
-where $E_l$ are non-negative and diagonal weighting matrices such that
-$\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix). Thus the convergence
-of such a method is dependent on the condition
+where $E_\ell$ are non-negative and diagonal weighting matrices and their sum is an identity matrix $I$. The convergence of such a method is dependent on the condition
\begin{equation}
-\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\rho(\displaystyle\sum^L_{\ell=1}E_\ell M^{-1}_\ell N_\ell)<1.
\label{eq03}
\end{equation}
+where $\rho$ is the spectral radius of the square matrix.
-The advantage of the multisplitting method is that at each iteration
-$k$ there are $L$ different linear sub-systems
+The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems
\begin{equation}
-v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+v_\ell^k=M^{-1}_\ell N_\ell x_\ell^{k-1} + M^{-1}_\ell b,~\ell\in\{1,\ldots,L\},
\label{eq04}
\end{equation}
-to be solved independently by a direct or an iterative method, where
-$v_l^k$ is the solution of the local sub-system. Thus, the
-calculations of $v_l^k$ may be performed in parallel by a set of
-processors. A multisplitting method using an iterative method for
-solving the $L$ linear sub-systems is called an inner-outer iterative
-method or a two-stage method. The results $v_l^k$ obtained from the
-different splittings~(\ref{eq04}) are combined to compute the solution
-$x^k$ of the linear system by using the diagonal weighting matrices
+to be solved independently by a direct or an iterative method, where $v_\ell$ is the solution of the local sub-system. Thus the computations of $\{v_\ell\}_{1\leq \ell\leq L}$ may be performed in parallel by a set of processors. A multisplitting method using an iterative method as an inner solver is called an inner-outer iterative method or a two-stage method. The results $v_\ell$ obtained from the different splittings~(\ref{eq04}) are combined to compute solution $x$ of the linear system by using the diagonal weighting matrices
\begin{equation}
-x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
+x^k = \displaystyle\sum^L_{\ell=1} E_\ell v_\ell^k,
\label{eq05}
\end{equation}
-In the case where the diagonal weighting matrices $E_l$ have only zero
-and one factors (i.e. $v_l^k$ are disjoint vectors), the
-multisplitting method is non-overlapping and corresponds to the block
-Jacobi method.
-%%%%%%%%%%%%%%%%%%%%%%%
-%% END
-%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Related works}
-
-
-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
-for example an asynchronous version \cite{bru1995parallel} and convergence
-conditions \cite{bai1999block,bahi2000asynchronous} in this case or other
-two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
-
-In \cite{huang1993krylov}, the authors proposed a parallel multisplitting
-algorithm in which all the tasks except one are devoted to solve a sub-block of
-the splitting and to send their local solution to the first task which is in
-charge to combine the vectors at each iteration. These vectors form a Krylov
-basis for which the first task minimizes the error function over the basis to
-increase the convergence, then the other tasks receive the update solution until
-convergence of the global system.
-
-
-
-In \cite{couturier2008gremlins}, the authors proposed practical implementations
-of multisplitting algorithms that take benefit from multisplitting algorithms to
-solve large scale linear systems. Inner solvers could be based on scalar direct
-method with the LU method or scalar iterative one with GMRES.
-
-In~\cite{prace-multi}, the authors have proposed a parallel multisplitting
-algorithm in which large block are solved using a GMRES solver. The authors have
-performed large scale experimentations upto 32.768 cores and they conclude that
-asynchronous multisplitting algorithm could more efficient than traditionnal
-solvers on exascale architecture with hunders of thousands of cores.
-
+In the case where the diagonal weighting matrices $E_\ell$ have only zero and one factors (i.e. $v_\ell$ are disjoint vectors), the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-
\section{A two-stage method with a minimization}
-Let $Ax=b$ be a given sparse and large linear system of $n$ equations
-to solve in parallel on $L$ clusters, physically adjacent or
-geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square
-and nonsingular 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:
+\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\{
\begin{array}{lll}
\right.
\label{sec03:eq01}
\end{equation}
-where for $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size
-$n_l\times n$ and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such
-that $\sum_ln_l=n$. In this case, we use a row-by-row splitting
-without overlapping in such a way that successive rows of the 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 $\sum_\ell n_\ell=n$. In this work, we use a row-by-row splitting 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 l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l,
+\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}
\end{equation}
-where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular
-matrix $A_l$, $X_i\neq X_l$ is a sub-vector of size $n_i$ of the
-solution vector $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$, for all
-$i\in\{1,\ldots,l-1,l+1,\ldots,L\}$.
+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 $m\in\{1,\ldots,L\}$.
-The 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}
-A_{ll}X_l = Y_l \mbox{,~such that}\\
-Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+A_{\ell \ell}X_\ell = Y_\ell \mbox{,~such that}\\
+Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_m,
\end{array}
\right.
\label{sec03:eq03}
\end{equation}
-is solved independently by a cluster of processors and communication
-are required to update the right-hand side vectors $Y_l$, such that
-the vectors $X_i$ represent the data dependencies between the
-clusters. In this work, we use the parallel GMRES method~\cite{ref34}
-as an inner iteration method for solving the
-sub-systems~(\ref{sec03:eq03}). It is a well-known iterative method
-which gives good performances for solving sparse linear systems in
-parallel on a cluster of processors.
-
-It should be noted that the convergence of the inner iterative solver
-for the different linear sub-systems~(\ref{sec03:eq03}) does not
-necessarily involve the convergence of the multisplitting method. It
-strongly depends on the properties of the sparse linear system to be
-solved and the computing
-environment~\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.
-
-In order to accelerate the convergence, we implement the outer
-iteration of the multisplitting solver as a Krylov subspace method
-which minimizes some error function over a Krylov subspace~\cite{S96}.
-The Krylov space of the method that we used is spanned by a basis
-composed of successive solutions issued from solving the $L$
-splittings~(\ref{sec03:eq03})
+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 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
+work is 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}
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 such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis.
-The advantage of such a Krylov subspace is that we need neither an
-orthogonal basis nor synchronizations between the different clusters
-to generate this basis.
-
-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 the vectors of $S$. So, $\alpha$ is defined as the
-solution of the large overdetermined linear system
+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 orthogonal basis nor any synchronization between clusters to generate this basis.
+
+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}
\end{equation}
-where $R=AS$ is a dense rectangular matrix of size $n\times s$ and
-$s\ll n$. This leads us to solve the system of normal equations
+where $R=AS$ is a dense rectangular matrix of size $n\times s$ and $s\ll n$. This leads us to solve a system of normal equations
\begin{equation}
R^TR\alpha=R^Tb,
\label{sec03:eq06}
\text{minimize}~\|b-R\alpha\|_2,
\label{sec03:eq07}
\end{equation}
-where $R^T$ denotes the transpose of the matrix $R$. Since $R$
-(i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric
-positive definite system~(\ref{sec03:eq06}) is solved in
-parallel. Thus, an iterative method would be more appropriate than a
-direct one for solving this system. We use the parallel conjugate
-gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
+where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate Gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
\begin{algorithm}[!t]
\caption{A two-stage linear solver with inner iteration GMRES method}
\begin{algorithmic}[1]
-\State Load $A_l$, $B_l$, initial guess $x^0$
-\State Initialize the minimizer $\tilde{x}^0=x^0$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
+\State Load $A_\ell$, $B_\ell$
+\State Set the initial guess $x^0$
+\State Set the minimizer $\tilde{x}^0=x^0$
\For {$k=1,2,3,\ldots$ until the global convergence}
-\State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do}
-\State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l$
-\State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters
-\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^j=\sum^L_{i=1}A_{li}X_i^j$
-\State\textbf{end for}
-\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$}
-\State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$
-\State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters
+\State Restart with $x^0=\tilde{x}^{k-1}$:
+\For {$j=1,2,\ldots,s$}
+\State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
+\State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
+\State Exchange local values of $X_\ell^j$ with the neighboring clusters
+\State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$
+\EndFor
+\State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
+\State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
+\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $j$}
-\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
-\State Solving the local splitting $A_{ll}X_l^j=Y_l$ with the parallel GMRES method, such that $X_l^0$ is the initial guess.
-\State \Return $X_l^j$
+\State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
+\State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
+\State \Return $X_\ell^j$
\EndFunction
\Statex
-\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
-\State Solving the normal equations $R^TR\alpha=R^Tb$ in parallel by $L$ clusters using the parallel CGNR method
-\State Compute the local minimizer: $\tilde{X}_l^k=S_l\alpha$
-\State \Return $\tilde{X}_l^k$
+\Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
+\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
+\State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
+\State \Return $\tilde{X}_\ell^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}
-The main key points of the multisplitting method for solving large
-sparse linear systems are given in Algorithm~\ref{algo:01}. This
-algorithm is based on a two-stage method with a minimization using the
-GMRES iterative method as an inner solver. It is executed in parallel
-by each cluster of processors. The matrices and vectors with the
-subscript $l$ represent the local data for the cluster $l$, where
-$l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel
-iterative algorithms: the GMRES method for solving each splitting on a
-cluster of processors, and the CGNR method executed in parallel by all
-clusters for minimizing the function error over the Krylov subspace
-spanned by $S$. The algorithm requires two global synchronizations
-between the $L$ clusters. The first one is performed at line~$12$ in
-Algorithm~\ref{algo:01} to exchange the local values of the vector
-solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the
-multisplitting solver. The second one is needed to construct the
-matrix $R$ of the Krylov subspace. We choose to perform this latter
-synchronization $s$ times in every outer iteration $k$ (line~$7$ in
-Algorithm~\ref{algo:01}). This is a straightforward way to compute the
-matrix-matrix multiplication $R=AS$. We implement all
-synchronizations by using the MPI collective communication
-subroutines.
+The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $\ell$ represent the local data for cluster $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: the GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and the CGNR method, executed in parallel by all clusters, to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. The first one is performed line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+\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.
+
+\begin{equation}
+\left\{
+ \begin{array}{ll}
+ \nabla u&=f \mbox{~in~} \omega\\
+ u &=0 \mbox{~on~} \Gamma=\partial \omega
+ \end{array}
+ \right.
+\end{equation}
+After discretization, with a finite difference scheme, a seven point stencil is
+used. It is well-known that the spectral radius of matrices representing such
+problems are very close to 1. Moreover, the larger the number of discretization
+points is, the closer to 1 the spectral radius is. Hence, to solve a matrix
+obtained for a 3D Poisson problem, the number of iterations is high. Using a
+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 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.
+
+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
+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
+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.
+
+\begin{table}[htbp]
+\begin{center}
+\begin{changemargin}{-1.8cm}{0cm}
+\begin{small}
+\begin{tabular}{|c|c||c|c|c||c|c|c||c|}
+\hline
+\multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} & \multicolumn{3}{c||}{GMRES} & \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
+ \cline{3-8}
+ & & Time (s) & nb Iter. & $\Delta$ & Time (s)& nb Iter. & $\Delta$ & \\
+\hline
+$468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
+\hline
+$590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
+\hline
+$743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
+\hline
+$743^3$ & 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}
+\caption{Results}
+\label{tab1}
+\end{small}
+\end{changemargin}
+\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
+iterations is drastically reduced with the multisplitting version even it is not
+negligible. Moreover, with 8,192 cores, we can see that using 4 clusters gives a
+better performance than simply using 2 clusters. In fact, we can notice 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 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 the 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 numbers 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 iterations 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\\
+%Async\\
+%Overlapping\\
+%preconditioning
%%%%%%%%%%%%%%%%%%%%%%%%
