\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
\usepackage{graphicx}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
\title{A scalable multisplitting algorithm for solving large sparse linear systems}
+\date{}
\maketitle
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
\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 simply a
-parallel multisplitting algorithm with few blocks of large size and a parallel
-krylov minimization is used 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.
+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
+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.
\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 adpated 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
-communications to perform matrix-vector products and reduction operations.
+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.
+
+
+Traditionnal 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.
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% 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
+\begin{equation}
+A = M_l - N_l,~l\in\{1,\ldots,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
+\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
+\begin{equation}
+\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\label{eq03}
+\end{equation}
+
+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\},
+\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
+\begin{equation}
+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. $v_l^k$ 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.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\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:
+\begin{equation}
+\left\{
+\begin{array}{lll}
+A & = & [A_{1}, \ldots, A_{L}]\\
+x & = & [X_{1}, \ldots, X_{L}]\\
+b & = & [B_{1}, \ldots, B_{L}]
+\end{array}
+\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:
+\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,
+\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
+$i\in\{1,\ldots,l-1,l+1,\ldots,L\}$.
+
+The 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,
+\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 to solve the
+sub-systems~(\ref{sec03:eq03}). It is a well-known iterative method
+which gives good performances to solve 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})
+\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 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
+\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
+\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 $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$
+\State Initialize 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 $S_l^k$
+\State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters
+\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$
+\State\textbf{end for}
+\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
+\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 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 $A_{ll}X_l^j=Y_l$ using the 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 the normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using the parallel CGNR method
+\State Compute the 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 the 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 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 to solve each splitting on a
+cluster of processors, and the CGNR method executed in parallel by all
+clusters to minimize 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.
+
+
+\section{Experiments}
+
+In order to illustrate the interest of our algorithm. We have compared our
+algorithm with the GMRES method which 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.
+
+\section{Conclusion and perspectives}
+
+Other applications (=> other matrices)\\
+Larger experiments\\
+Async\\
+Overlapping
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{biblio}
