\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
\usepackage{graphicx}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{multirow}
+
+\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}}
+
+\usepackage{xspace}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\LZK}[2][inline]{%
+\todo[color=green!40,#1]{\sffamily\textbf{LZK:} #2}\xspace}
\title{A scalable multisplitting algorithm for solving large sparse linear systems}
+\date{}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{abstract}
-In this paper we revist the krylov multisplitting algorithm presented in
+In this paper we revisit the krylov multisplitting algorithm presented in
\cite{huang1993krylov} which uses a scalar 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
classical GMRES both in terms of number of iterations and 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. When
-solving large linear systems with many cores, iterative methods often suffer
-from scalability problems. This is due to their need for collective
+iterative methods have been proposed and adapted by many 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
+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
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
+traditionnal multisplitting method that suffer from slow convergence, as
+proposed in~\cite{huang1993krylov}, the use of a minimization process can
+drastically improve the convergence.
-A completer...
-On ne peut pas parler de tout...\\
+\LZK[]{Suite\dots}
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+\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 updated solution until
+convergence of the global system.
+
+In~\cite{couturier2008gremlins}, the authors proposed practical implementations
+of multisplitting algorithms that take benefit from multisplitting algorithms\LZK[]{répétition ???} 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.\LZK[]{lu et gmres par exemple}
+
+In~\cite{prace-multi}, the authors have proposed 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
+asynchronous multisplitting algorithm could be more efficient than traditional
+solvers on exascale architecture with hundreds of thousands of cores.
-%%%%%%%%%%%%%%%%%%%%%%%
-%% 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
+\LZK[]{Peut-être autres related works\ldots}\\
+
+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_l - N_l,
\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 $l\in\{1,\ldots,L\}$ $M_l$ are non-singular matrices. Then the linear system is solved by 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
\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 $E_l$ 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.
\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}
-y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
\label{eq04}
\end{equation}
-to be solved independently by a direct or an iterative method, where
-$y_l^k$ is the solution of the local sub-system. 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 $y_l^k$ 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 $v_l^k$ is the solution of the local sub-system. Thus the computations of $\{v_l\}_{1\leq l\leq L}$ 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 $v_l$ 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 y_l^k,
+x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
\label{eq05}
\end{equation}
-In the case where the diagonal weighting matrices $E_l$ have only zero
-and one factors (i.e. $y_l^k$ 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 a asynchronous version \cite{bru1995parallel} and convergence
-condition \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 tasks minimize 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 the case where the diagonal weighting matrices $E_l$ have only zero and one factors (i.e. $v_l$ 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:
+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 all $l\in\{1,\ldots,L\}$ $A_l$ 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 $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size $n_l\times n$ and $X_l$ and $B_l$ are sub-vectors of size $n_l$ each, such that $\sum_ln_l=n$. 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 l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{m=1}^{l-1}A_{lm}X_m + A_{ll}X_l + \displaystyle\sum_{m=l+1}^{L}A_{lm}X_m = B_l,
\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 $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
-the following spare sub-linear system:
+where $A_{lm}$ is a sub-block of size $n_l\times n_m$ of the rectangular matrix $A_l$, $X_m\neq X_l$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq l}n_m+n_l=n$, 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
\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,
+Y_l = B_l - \displaystyle\sum_{\substack{m=1\\m\neq l}}^{L}A_{lm}X_m,
\end{array}
\right.
\label{sec03:eq03}
\end{equation}
-where the sub-vectors $X_i$ define the data dependencies between the cluster $l$ and other clusters.
+is solved independently by a {\it cluster of processors} and communication are required to update the right-hand side vectors $Y_l$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel 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 parallel on clusters of processors. 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 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 work, 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 subspace 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 need neither an orthogonal basis nor synchronizations 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 a system of normal equations
+\begin{equation}
+R^TR\alpha=R^Tb,
+\label{sec03:eq06}
+\end{equation}
+which is associated with the least squares problem
+\begin{equation}
+\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 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]
+\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
+\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$
+\State Initialize 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 $x^0=\tilde{x}^{k-1}$:
+\For {$j=1,2,\ldots,s$}
+\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
+\State Construct basis $S$: add column vector $X_l^j$ to the matrix $S_l^k$
+\State Exchange local values of $X_l^j$ with the neighboring clusters
+\State Compute dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$
+\EndFor
+\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
+\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
+\EndFor
+
+\Statex
+
+\Function {InnerSolver}{$x^0$, $j$}
+\State Compute local right-hand side $Y_l = B_l - \sum^L_{\substack{m=1\\m\neq l}}A_{lm}X_m^0$
+\State Solving local splitting $A_{ll}X_l^j=Y_l$ using parallel GMRES method, such that $X_l^0$ is the initial guess
+\State \Return $X_l^j$
+\EndFunction
+
+\Statex
+
+\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
+\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 $\tilde{X}_l^k=S_l^k\alpha^k$
+\State \Return $\tilde{X}_l^k$
+\EndFunction
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+
+The main key points of our 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 GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $l$ represent the local data for cluster $l$, where $l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and 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 at line~$12$ 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 $R$ of the Krylov subspace. We chose 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 sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Experiments}
+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
+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 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.
+
+In the experiments we report the size of the 3D poisson considered\LZK[]{Suite\dots ?}
+
+
+The first column shows the size of the 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 parenthesis, there is the decomposition used for the
+Krylov multisplitting. The third column and the sixth column respectively show
+the execution time for the GMRES and the Kyrlow multisplitting code. The fourth
+and the seventh column describes the number of iterations. For the
+multisplitting code, the total number of inner iterations is represented in
+parenthesis.
+
+ We also give the other parameters: the restart for the GRMES method....
+
+\begin{table}[p]
+\begin{center}
+\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
+
+$590^3$ & 4096 (2x2048) & 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 \\
+\hline
+$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
+\hline
+
+\end{tabular}
+\caption{Results without preconditioner}
+\label{tab1}
+\end{center}
+\end{table}
+
+
+\begin{table}[p]
+\begin{center}
+\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
+
+$590^3$ & 4096 (2x2048) & 433.0 & 55,494 & 4.92e-7 & 80.4 & 1,091(9,545) & 7.64e-08 & 5.39 \\
+\hline
+$743^3$ & 8192 (2x4096) & 704.4 & 87,822 & 4.80e-07 & 110.2 & 1,401(12,379) & 1.11e-07 & 6.39 \\
+\hline
+$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 139.8 & 1,891(15,960) & 1.60e-07& 5.03 \\
+\hline
+
+\end{tabular}
+\caption{Results with preconditioner}
+\label{tab2}
+\end{center}
+\end{table}
+
+\section{Conclusion and perspectives}
+
+Other applications (=> other matrices)\\
+Larger experiments\\
+Async\\
+Overlapping
%%%%%%%%%%%%%%%%%%%%%%%%
