\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}}
\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
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.
-A completer...
-On ne peut pas parler de tout...\\
+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
+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
+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
+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
-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 system.
-A multisplitting method using an iterative method for solving the $L$ linear systems is called an inner-outer
-iterative method or a two-stage method. The solution of the global linear system at the iteration $k$ is computed
-as follows
+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 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.
+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
%%%%%%%%%%%%%%%%%%%%%%%
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}
+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 tasks minimize the error function over the basis to
+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.
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:
+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}
\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 $n_l$, 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 $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
-the following spare sub-linear system:
+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}
\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 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.
+
+Doing many experiments with many cores is not easy and require to access to a
+supercomputer with several hours for developping 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
+
+
+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
%%%%%%%%%%%%%%%%%%%%%%%%
