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-{\bf RAPH : EST ce qu'on parle de Krylov pour dire que les résidus constituent une base de Krylov... J'hésite... Tof t'en penses quoi?}
-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.
+
+Iterative methods became more attractive than direct ones to solve very large
+sparse linear systems. Iterative methods are more effecient in a parallel
+context, with thousands of cores, 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
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 outer step with a new solution minimizing some error functions over some
+previous residuals. 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
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
+a least-square residual minimization. Section~\ref{sec:04} describes some
+convergence results on this method. Section~\ref{sec:05} shows some
experimental results obtained on large clusters of our algorithm using routines
-of PETSc toolkit.
+of PETSc toolkit. Finally Section~\ref{sec:06} concludes and gives some
+perspectives.
%%%*********************************************************
%%%*********************************************************
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
+In order to accelerate the convergence, the outer iteration periodically applies
+a least-square minimization on the residuals computed by the inner solver. The
+inner solver is a Krylov based solver which does not required to be changed.
+
+At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
+iterations, using an iterative method restarting with the previous solution. For
+example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
+inner solver. The current solution of the Krylov method is saved inside a matrix
+$S$ composed of successive solutions computed by the inner iteration.
+
+Periodically, every $s$ iterations, the minimization step is applied in order to
+compute a new solution $x$. For that, the previous residuals are computed with
+$(b-AS)$. The minimization of the residuals is obtained by
\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.
+with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+
+
+In practice, $R$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$,
+$s\ll n$. In order to minimize~(\ref{eq:01}), a least-square method such as
+CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Those methods are more
+appropriate than a direct method in a parallel context.
\begin{algorithm}[t]
-\caption{A Krylov two-stage algorithm}
+\caption{TSARM}
\begin{algorithmic}[1]
\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 $S_{k~mod~s}=x^k$
+ \State $x^k=Solve(A,b,x^{k-1},m)$ \label{algo:solve}
+ \State $S_{k~mod~s}=x^k$ \label{algo:store}
\If {$k$ mod $s=0$ {\bf and} not convergence}
- \State Compute dense matrix $R=AS$
+ \State $R=AS$ \Comment{compute dense matrix}
\State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
- \State Compute minimizer $x^k=S\alpha$
+ \State $x^k=S\alpha$ \Comment{compute new solution}
\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.
+Algorithm~\ref{algo:01} summarizes the principle of our method. The outer
+iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is
+called for a maximum of $m$ iterations. In practice, we suggest to choose $m$
+equals to the restart number of the GMRES like method. Line~\ref{algo:store},
+$S_{k~ mod~ s}=x^k$ consists in copying the solution $x_k$ 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.
%%%*********************************************************
%%%*********************************************************
-
+\section{Convergence results}
+\label{sec:04}
%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
-\label{sec:04}
+\label{sec:05}
In order to see the influence of our algorithm with only one processor, we first
characteristics. For all the matrices, the name, the field, the number of rows
and the number of nonzero elements is given.
-\begin{table}
+\begin{table*}
\begin{center}
\begin{tabular}{|c|c|r|r|r|}
\hline
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
-\end{table}
+\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
nb. cores & threshold & \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 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
+ 2,048 & 6e-5 & 194.01 & 30,270 & 35.50 & 5,430 & 27.74 & 4,350 & 6.99 \\
+ 4,096 & 7e-5 & 160.59 & 22,530 & 35.15 & 5,130 & 29.21 & 4,350 & 5.49 \\
+ 4,096 & 6e-5 & 249.27 & 35,520 & 52.13 & 7,950 & 39.24 & 5,790 & 6.35 \\
8,192 & 6e-5 & 149.54 & 17,280 & 28.68 & 3,810 & 29.05 & 3,990 & 5.21 \\
8,192 & 5e-5 & 792.11 & 109,590 & 76.83 & 10,470 & 65.20 & 9,030 & 12.14 \\
16,384 & 4e-5 & 718.61 & 86,400 & 98.98 & 10,830 & 131.86 & 14,790 & 7.26 \\
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:05}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
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+\bibliographystyle{IEEEtran}
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-%\bibliography{IEEEabrv,../bib/paper}
+\bibliography{biblio}
%
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% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
-\begin{thebibliography}{1}
+%% \begin{thebibliography}{1}
-\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{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{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{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}
+%% \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}
