%
% paper title
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-\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
%\title{???}
+
+
+
% author names and affiliations
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% affiliations
\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}
% 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.
+
+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
+In order to accelerate the convergence, the outer iteration applies a least-square minimization on the residuals computed by the inner some error functions over a Krylov
+subspace~\cite{Saad2003}. 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
+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}
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
+~\cite{Hestenes52} or LSQR~\cite{Paige82} which are more appropriate than a
direct method in the parallel context.
\begin{algorithm}[t]
%%%*********************************************************
%%%*********************************************************
-
+\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
\end{table*}
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ 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 \\
+\hline
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
+\label{tab:04}
+\end{center}
+\end{table*}
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:05}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************
+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- adaptative number of outer iterations to minimize\\
+- other methods to minimize the residuals?\\
+- implement our solver inside PETSc
+
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-%\bibliographystyle{IEEEtran}
+\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
+\bibliography{biblio}
%
% <OR> manually copy in the resultant .bbl file
% 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}
