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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}
-We propose a two-stage algorithm 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}.
-
-The outer iteration is implemented as an iterative Krylov method which minimizes some error function over a Krylov sub-space~\cite{saad96}. At every iteration, the sparse linear system $Ax=b$ is solved iteratively with an iterative method as GMRES method~\cite{saad86} and the Krylov sub-space that we used is spanned by a basis $S$ composed of successive solutions issued from the inner iteration
+\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 sub-space 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 sub-space spanned by vectors of $S$, where $\alpha$ is a solution of the normal equations
+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}
\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}) as CGLS~\cite{hestenes52} or LSQR~\cite{paige82} which is more appropriate that a direct method in the parallel context.
+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}
\State Set the initial guess $x^0$
\For {$k=1,2,3,\ldots$ until convergence}
\State Solve iteratively $Ax^k=b$
- \State Add vector $x^k$ to Krylov basis $S$
+ \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 $x^k=S\alpha$
- \State Reinitialize Krylov basis $S$
\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.
+
+\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}
+
+
+
+
+\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} \\
+ & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
+
+crashbasis & gmres / none & 15.65 & 518 & 14.12 & 450 \\
+parabolic\_fem & gmres / ilu & 2152 & ?? & 724 & ?? \\
+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 & 565 & 37.70 & 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}
+
+
+Param : retart 30 iters
+cols = 8
+iter cgls = 20
+cgls prec = 1e-40
+prec = 1e-10
+Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
+
+
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\section{Conclusion}
+\label{sec:05}
%The conclusion goes here. this is more of the conclusion
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\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 and M.~H.~Schultz, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+\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.
