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\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}
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+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}
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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} \\
+\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 GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\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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% (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}
