\usepackage{algorithm}
\usepackage{algpseudocode}
\usepackage{multirow}
+\usepackage{authblk}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\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}
+\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
+\let\endchangemargin=\endlist
-\title{A scalable multisplitting algorithm for solving large sparse linear systems}
+\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 revisit the Krylov multisplitting algorithm presented in
-\cite{huang1993krylov} which uses a scalar method to minimize the Krylov
+\cite{huang1993krylov} which uses a sequential method to minimize the Krylov
iterations computed by a multisplitting algorithm. Our new algorithm is based on
a parallel multisplitting algorithm with few blocks of large size using a
parallel GMRES method inside each block and on a parallel Krylov minimization in
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 methods are based on the
+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.
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 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.
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Related works}
-A general framework for studying parallel multisplitting has been presented in~\cite{o1985multi}
+\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 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
+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 proposed a parallel multisplitting
+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 to combine the vectors at each iteration. These vectors form a Krylov
+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
+increase the convergence, then the other tasks receive the updated solution until the
convergence of the global system.
-In~\cite{couturier2008gremlins}, the authors proposed practical implementations
-of multisplitting algorithms that take benefit from multisplitting algorithms {\bf ???} 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{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 proposed a parallel multisplitting
+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
-asynchronous multisplitting algorithm could be more efficient than traditional
+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.
+\\
+
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_\ell - N_\ell,
\label{eq01}
\end{equation}
-where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by iteration based on the obtained splittings 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_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
\label{eq02}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{A two-stage method with a minimization}
+\label{sec:03}
Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. The multisplitting of this linear system is defined as follows
\begin{equation}
\left\{
\end{equation}
where $A_{\ell m}$ is a sub-block of size $n_\ell\times n_m$ of the rectangular matrix $A_\ell$, $X_m\neq X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
-Our multisplitting method proceeds by iteration for solving the linear system in such a way each sub-system
+Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system
\begin{equation}
\left\{
\begin{array}{l}
\end{equation}
is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
-It should be noted that the convergence of the inner iterative solver for the different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the multisplitting method. It strongly depends on the properties of the global sparse linear system to be solved 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.
+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 of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations between clusters to generate this basis.
+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}
\label{algo:01}
\end{algorithm}
-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: 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~\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.
+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}
-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
+\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.
preconditioner it is possible to reduce the number of iterations but
preconditioners are not scalable when using many cores.
-Doing many experiments with many cores is not easy and requires to access to a
-supercomputer with several hours for developing a code and then improving
-it. In the following we presented some experiments we could achieved out on the
-Hector architecture, the previous UK's high-end computing resource, funded by
-the UK Research Councils, which has been stopped in the early 2014.
+%Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.
+In the following we present some experiments we could 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 parenthesis, there is the decomposition used for 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 describes the number of iterations. For the
+and the seventh column describe the number of iterations. For the
multisplitting code, the total number of inner iterations is represented in
-parenthesis. For the GMRES code (alone and in the multisplitting version) the
+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
\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
-$468^3$ & 2048 (2x1024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
+$468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
\hline
-$590^3$ & 4096 (2x2048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
+$590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
\hline
-$743^3$ & 8192 (2x4096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
+$743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
\hline
-$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
+$743^3$ & 8,192 (4x2,048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
\hline
\end{tabular}
\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 is between 4 and 6. It can be noticed that the number of
-iterations is drastically reduced with the multisplitting version even it is not
-neglectable.
+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 multisplitting and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a multi-cluster environment based on processors of the computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. Second, we have implemented the outer iteration of the multisplitting method using Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
-
-We have tested our multisplitting method for solving the sparse linear system issued from the discretization of a 3D Poisson problem. We have compared its performances to those of classical GMRES method on a supercomputer composed of 2048 to 8192 cores. The experimental results showed that the multisplitting method is about 4 to 6 times faster than the GMRES method for different sizes of the problem split into 2 or 4 blocks when using multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
-
-In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.
-
+We have implemented a Krylov multisplitting method to solve sparse linear
+systems on large-scale computing platforms. We have developed a synchronous
+two-stage method based on the block Jacobi multisaplitting which uses GMRES
+iterative method as an inner iteration. Our contribution in this paper is
+twofold. First we provide a multi cluster decomposition that allows us to choose
+the appropriate size of the clusters according to the architecures of the
+supercomputer. Second, we have implemented the outer iteration of the
+multisplitting method as a Krylov subspace method which minimizes some error
+function. This increases the convergence and improves the scalability of the
+multisplitting method.
+
+We have tested our multisplitting method to solve the sparse linear system
+issued from the discretization of a 3D Poisson problem. We have compared its
+performances to the classical GMRES method on a supercomputer composed of 2,048
+to 8,192 cores. The experimental results showed that the multisplitting method is
+about 4 to 6 times faster than the GMRES method for different sizes of the
+problem split into 2 or 4 blocks when using 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\\
