\usepackage{algorithm}
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+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multirow}
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%
% paper title
% can use linebreaks \\ within to get better formatting as desired
-\title{A Krylov two-stage algorithm to solve large sparse linear systems}
+\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}
%où
%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
%où
% use a multiple column layout for up to two different
% affiliations
-\author{\IEEEauthorblockN{Rapha\"el Couturier}
-\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
-Universit\'e de Franche-Comt\'e, IUT de Belfort-Montb\'eliard\\
-19 avenue de Mar\'echal Juin, BP 527 \\
-90016 Belfort Cedex, France\\
-Email: raphael.couturier@univ-fcomte.fr}
-\and
-\IEEEauthorblockN{Lilia Ziane Khodja}
-\IEEEauthorblockA{Centre de Recherche INRIA Bordeaux Sud-Ouest\\
-200 avenue de la Vieille Tour\\
-33405 Talence Cedex, France\\
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2} and Christophe Guyeux\IEEEauthorrefmark{1}}
+\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
+Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
Email: lilia.ziane@inria.fr}
}
+
+
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% the acknowledgment of grants, issue a \IEEEoverridecommandlockouts
\begin{abstract}
-%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT.
+In this paper we propose a two stage iterative method which increases the
+convergence of Krylov iterative methods, typically those of GMRES variants. The
+principle of our approach is to build an external iteration over the Krylov
+method and to save the current residual frequently (for example, for each
+restart of GMRES). Then after a given number of outer iterations, a minimization
+step is applied on the matrix composed of the save residuals in order to compute
+a better solution and make a new iteration if necessary. We prove that our
+method has the same convergence property than the inner method used. Some
+experiments using up to 16,394 cores show that compared to GMRES our algorithm
+can be around 7 times faster.
\end{abstract}
\begin{IEEEkeywords}
-Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à voir...
+Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir...
\end{IEEEkeywords}
% no \IEEEPARstart
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
+Iterative methods are become more attractive than direct ones to solve very
+large sparse linear systems. They are more effective in a parallel context and
+require less memory and arithmetic operations than direct methods. A number of
+iterative methods are proposed and adapted by many researchers and the increased
+need for solving very large sparse linear systems triggered the development of
+efficient iterative techniques suitable for the parallel processing.
+
+Most of the successful iterative methods currently available are based on Krylov
+subspaces which consist in forming a basis of a sequence of successive matrix
+powers times an initial vector for example the residual. These methods are based
+on orthogonality of vectors of the Krylov subspace basis to solve linear
+systems. The most well-known iterative Krylov subspace methods are Conjugate
+Gradient method and GMRES method (generalized minimal residual).
+
+However, iterative methods suffer from scalability problems on parallel
+computing platforms with many processors due to their need for reduction
+operations and collective communications to perform matrix-vector
+multiplications. The communications on large clusters with thousands of cores
+and large sizes of messages can significantly affect the performances of
+iterative methods. In practice, Krylov subspace iteration methods are often used
+with preconditioners in order to increase their convergence and accelerate their
+performances. However, most of the good preconditioners are not scalable on
+large clusters.
+
+In this paper we propose a two-stage algorithm based on two nested iterations
+called inner-outer iterations. This algorithm consists in solving the sparse
+linear system iteratively with a small number of inner iterations and restarts
+the outer step with a new solution minimizing some error functions over a Krylov
+subspace. This algorithm is iterative and easy to parallelize on large clusters
+and the minimization technique improves its convergence and performances.
+
+The present paper is organized as follows. In Section~\ref{sec:02} some related
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
+on Krylov subspace iteration methods. Section~\ref{sec:04} shows some
+experimental results obtained on large clusters of our algorithm using routines
+of PETSc toolkit.
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\section{Related works}
+\label{sec:02}
%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
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\section{A Krylov two-stage algorithm}
-
-
-\begin{algorithm}[!t]
+\label{sec:03}
+A two-stage algorithm is proposed to solve large sparse linear systems of the
+form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
+nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector and
+$b\in\mathbb{R}^n$ is the right-hand side. The algorithm is implemented as an
+inner-outer iteration solver based on iterative Krylov methods. The main key
+points of our solver are given in Algorithm~\ref{algo:01}.
+
+In order to accelerate the convergence, the outer iteration is implemented as an
+iterative Krylov method which minimizes some error functions over a Krylov
+subspace~\cite{saad96}. At each iteration, the sparse linear system $Ax=b$ is
+solved iteratively with an iterative method, for example GMRES
+method~\cite{saad86} or some of its variants, and the Krylov subspace that we
+used is spanned by a basis $S$ composed of successive solutions issued from the
+inner iteration
+\begin{equation}
+ S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
+\end{equation}
+The advantage of such a Krylov subspace is that we neither need an orthogonal
+basis nor any synchronization between processors to generate this basis. The
+algorithm is periodically restarted every $s$ iterations with a new initial
+guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
+subspace spanned by vectors of $S$, where $\alpha$ is a solution of the normal
+equations
+\begin{equation}
+ R^TR\alpha = R^Tb,
+\end{equation}
+which is associated with the least-squares problem
+\begin{equation}
+ \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
+\label{eq:01}
+\end{equation}
+such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$,
+$s\ll n$, and $R^T$ denotes the transpose of matrix $R$. We use an iterative
+method to solve the least-squares problem~(\ref{eq:01}) such as CGLS
+~\cite{hestenes52} or LSQR~\cite{paige82} which are more appropriate than a
+direct method in the parallel context.
+
+\begin{algorithm}[t]
\caption{A Krylov two-stage algorithm}
\begin{algorithmic}[1]
-\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 $x^0=\tilde{x}^{k-1}$:
-\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 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}{$S_\ell$, $R$, $b$, $k$}
-\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
+ \Input $A$ (sparse matrix), $b$ (right-hand side)
+ \Output $x$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+ \State Solve iteratively $Ax^k=b$ \label{algo:solve}
+ \State $S_{k~mod~s}=x^k$
+ \If {$k$ mod $s=0$ {\bf and} not convergence}
+ \State Compute dense matrix $R=AS$
+ \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+ \State Compute minimizer $x^k=S\alpha$
+ \EndIf
+ \EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}
+
+Operation $S_{k~ mod~ s}=x^k$ consists in copying the residual $x_k$ into the
+column $k~ mod~ s$ of the matrix $S$. After the minimization, the matrix $S$ is
+reused with the new values of the residuals.
+
%%%*********************************************************
%%%*********************************************************
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\section{Experiments using petsc}
+\label{sec:04}
+
+
+In order to see the influence of our algorithm with only one processor, we first
+show a comparison with the standard version of GMRES and our algorithm. In
+table~\ref{tab:01}, we show the matrices we have used and some of them
+characteristics. For all the matrices, the name, the field, the number of rows
+and the number of nonzero elements is given.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|}
+\hline
+Matrix name & Field &\# Rows & \# Nonzeros \\\hline \hline
+crashbasis & Optimization & 160,000 & 1,750,416 \\
+parabolic\_fem & Computational fluid dynamics & 525,825 & 2,100,225 \\
+epb3 & Thermal problem & 84,617 & 463,625 \\
+atmosmodj & Computational fluid dynamics & 1,270,432 & 8,814,880 \\
+bfwa398 & Electromagnetics problem & 398 & 3,678 \\
+torso3 & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\end{center}
+\end{table}
+
+The following parameters have been chosen for our experiments. As by default
+the restart of GMRES is performed every 30 iterations, we have chosen to stop
+the GMRES every 30 iterations (line \ref{algo:solve} in
+Algorithm~\ref{algo:01}). $s$ is set to 8. CGLS is chosen to minimize the
+least-squares problem. Two conditions are used to stop CGLS, either the
+precision is under $1e-40$ or the number of iterations is greater to $20$. The
+external precision is set to $1e-10$ (line \ref{algo:conv} in
+Algorithm~\ref{algo:01}). Those experiments have been performed on a Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+
+
+In Table~\ref{tab:02}, some experiments comparing the solving of the linear
+systems obtained with the previous matrices with a GMRES variant and with out 2
+stage algorithm are given. In the second column, it can be noticed that either
+gmres or fgmres is used to solve the linear system. According to the matrices,
+different preconditioner is used. With the 2 stage algorithm, the same solver
+and the same preconditionner is used. This Table shows that the 2 stage
+algorithm can drastically reduce the number of iterations to reach the
+convergence when the number of iterations for the normal GMRES is more or less
+greater than 500. In fact this also depends on tow parameters: the number of
+iterations to stop GMRES and the number of iterations to perform the
+minimization.
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|}
+\hline
+
+ \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\
+\cline{3-6}
+ & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
+
+crashbasis & gmres / none & 15.65 & 518 & 14.12 & 450 \\
+parabolic\_fem & gmres / ilu & 1009.94 & 7573 & 401.52 & 2970 \\
+epb3 & fgmres / sor & 8.67 & 600 & 8.21 & 540 \\
+atmosmodj & fgmres / sor & 104.23 & 451 & 88.97 & 366 \\
+bfwa398 & gmres / none & 1.42 & 9612 & 0.28 & 1650 \\
+torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+
+
+Larger experiments ....
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ nb. cores & precond & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\
+\cline{3-8}
+ & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
+ 2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\
+ 2,048 & sor & 745.37 & 57,060 & 87.31 & 6,150 & 104.21 & 7,230 & 8.53 \\
+ 4,096 & mg & 562.25 & 25,170 & 97.23 & 3,990 & 89.71 & 3,630 & 6.27 \\
+ 4,096 & sor & 912.12 & 70,194 & 145.57 & 9,750 & 168.97 & 10,980 & 6.26 \\
+ 8,192 & mg & 917.02 & 40,290 & 148.81 & 5,730 & 143.03 & 5,280 & 6.41 \\
+ 8,192 & sor & 1,404.53 & 106,530 & 212.55 & 12,990 & 180.97 & 10,470 & 7.76 \\
+ 16,384 & mg & 1,430.56 & 63,930 & 237.17 & 8,310 & 244.26 & 7,950 & 6.03 \\
+ 16,384 & sor & 2,852.14 & 216,240 & 418.46 & 21,690 & 505.26 & 23,970 & 6.82 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
+
+
%%%*********************************************************
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\section{Conclusion}
+\label{sec:05}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
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+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- adaptative number of outer iterations to minimize\\
+- implement our solver inside PETSc
+
% conference papers do not normally have an appendix
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\section*{Acknowledgment}
-%The authors would like to thank...
-%more thanks here
-%%%*********************************************************
-%%%*********************************************************
+This paper is partially funded by the Labex ACTION program (contract
+ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource
+Curie and Juqueen respectively based in France and Germany.
+
% trigger a \newpage just before the given reference
% (used to reserve space for the reference number labels box)
\begin{thebibliography}{1}
-\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
-% 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+\bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
+
+\bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+
+\bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
+\bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
\end{thebibliography}
