X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/4f3541e3efca5ecdede512ef9e890b58c3c4acad..16f0ca498711e0af6a2de8200c2fbee0e5f8810e:/paper.tex diff --git a/paper.tex b/paper.tex index fbcf12d..35d9ed7 100644 --- a/paper.tex +++ b/paper.tex @@ -1 +1,810 @@ -toto + +%% bare_conf.tex +%% V1.3 +%% 2007/01/11 +%% by Michael Shell +%% See: +%% http://www.michaelshell.org/ +%% for current contact information. +%% +%% This is a skeleton file demonstrating the use of IEEEtran.cls +%% (requires IEEEtran.cls version 1.7 or later) with an IEEE conference paper. +%% +%% Support sites: +%% http://www.michaelshell.org/tex/ieeetran/ +%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/ +%% and +%% http://www.ieee.org/ + +%%************************************************************************* +%% Legal Notice: +%% This code is offered as-is without any warranty either expressed or +%% implied; without even the implied warranty of MERCHANTABILITY or +%% FITNESS FOR A PARTICULAR PURPOSE! +%% User assumes all risk. +%% In no event shall IEEE or any contributor to this code be liable for +%% any damages or losses, including, but not limited to, incidental, +%% consequential, or any other damages, resulting from the use or misuse +%% of any information contained here. +%% +%% All comments are the opinions of their respective authors and are not +%% necessarily endorsed by the IEEE. +%% +%% This work is distributed under the LaTeX Project Public License (LPPL) +%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used, +%% distributed and modified. 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If really needed, such as for +% the acknowledgment of grants, issue a \IEEEoverridecommandlockouts +% after \documentclass + +% for over three affiliations, or if they all won't fit within the width +% of the page, use this alternative format: +% +%\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1}, +%Homer Simpson\IEEEauthorrefmark{2}, +%James Kirk\IEEEauthorrefmark{3}, +%Montgomery Scott\IEEEauthorrefmark{3} and +%Eldon Tyrell\IEEEauthorrefmark{4}} +%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\ +%Georgia Institute of Technology, +%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html} +%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\ +%Email: homer@thesimpsons.com} +%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\ +%Telephone: (800) 555--1212, Fax: (888) 555--1212} +%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}} + + + + +% use for special paper notices +%\IEEEspecialpapernotice{(Invited Paper)} + + + + +% make the title area +\maketitle + + +\begin{abstract} +%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT. +\end{abstract} + +\begin{IEEEkeywords} +Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir... +\end{IEEEkeywords} + + +% For peer review papers, you can put extra information on the cover +% page as needed: +% \ifCLASSOPTIONpeerreview +% \begin{center} \bfseries EDICS Category: 3-BBND \end{center} +% \fi +% +% For peerreview papers, this IEEEtran command inserts a page break and +% creates the second title. It will be ignored for other modes. +\IEEEpeerreviewmaketitle + + + + +% An example of a floating figure using the graphicx package. +% Note that \label must occur AFTER (or within) \caption. +% For figures, \caption should occur after the \includegraphics. +% Note that IEEEtran v1.7 and later has special internal code that +% is designed to preserve the operation of \label within \caption +% even when the captionsoff option is in effect. However, because +% of issues like this, it may be the safest practice to put all your +% \label just after \caption rather than within \caption{}. +% +% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class +% option should be used if it is desired that the figures are to be +% displayed while in draft mode. +% +%\begin{figure}[!t] +%\centering +%\includegraphics[width=2.5in]{myfigure} +% where an .eps filename suffix will be assumed under latex, +% and a .pdf suffix will be assumed for pdflatex; or what has been declared +% via \DeclareGraphicsExtensions. +%\caption{Simulation Results} +%\label{fig_sim} +%\end{figure} + +% Note that IEEE typically puts floats only at the top, even when this +% results in a large percentage of a column being occupied by floats. + + +% An example of a double column floating figure using two subfigures. +% (The subfig.sty package must be loaded for this to work.) +% The subfigure \label commands are set within each subfloat command, the +% \label for the overall figure must come after \caption. +% \hfil must be used as a separator to get equal spacing. +% The subfigure.sty package works much the same way, except \subfigure is +% used instead of \subfloat. +% +%\begin{figure*}[!t] +%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}% +%\label{fig_first_case}} +%\hfil +%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}% +%\label{fig_second_case}}} +%\caption{Simulation results} +%\label{fig_sim} +%\end{figure*} +% +% Note that often IEEE papers with subfigures do not employ subfigure +% captions (using the optional argument to \subfloat), but instead will +% reference/describe all of them (a), (b), etc., within the main caption. + + +% An example of a floating table. Note that, for IEEE style tables, the +% \caption command should come BEFORE the table. Table text will default to +% \footnotesize as IEEE normally uses this smaller font for tables. +% The \label must come after \caption as always. +% +%\begin{table}[!t] +%% increase table row spacing, adjust to taste +%\renewcommand{\arraystretch}{1.3} +% if using array.sty, it might be a good idea to tweak the value of +% \extrarowheight as needed to properly center the text within the cells +%\caption{An Example of a Table} +%\label{table_example} +%\centering +%% Some packages, such as MDW tools, offer better commands for making tables +%% than the plain LaTeX2e tabular which is used here. +%\begin{tabular}{|c||c|} +%\hline +%One & Two\\ +%\hline +%Three & Four\\ +%\hline +%\end{tabular} +%\end{table} + + +% Note that IEEE does not put floats in the very first column - or typically +% anywhere on the first page for that matter. Also, in-text middle ("here") +% positioning is not used. Most IEEE journals/conferences use top floats +% exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places +% footnotes above bottom floats. This can be corrected via the \fnbelowfloat +% command of the stfloats package. + + + +%%%********************************************************* +%%%********************************************************* +\section{Introduction} +% 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. +%%%********************************************************* +%%%********************************************************* + + + +%%%********************************************************* +%%%********************************************************* +\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. +%%%********************************************************* +%%%********************************************************* + + + +%%%********************************************************* +%%%********************************************************* +\section{A Krylov two-stage algorithm} +\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 $A$ (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. + +%%%********************************************************* +%%%********************************************************* + + + +%%%********************************************************* +%%%********************************************************* +\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} + + + +%%%********************************************************* +%%%********************************************************* + + + +%%%********************************************************* +%%%********************************************************* +\section{Conclusion} +\label{sec:05} +%The conclusion goes here. this is more of the conclusion +%%%********************************************************* +%%%********************************************************* + + + +% conference papers do not normally have an appendix + + + +% use section* for acknowledgement +%%%********************************************************* +%%%********************************************************* +\section*{Acknowledgment} +%The authors would like to thank... +%more thanks here +%%%********************************************************* +%%%********************************************************* + + +% trigger a \newpage just before the given reference +% number - used to balance the columns on the last page +% adjust value as needed - may need to be readjusted if +% the document is modified later +%\IEEEtriggeratref{8} +% The "triggered" command can be changed if desired: +%\IEEEtriggercmd{\enlargethispage{-5in}} + +% references section + +% can use a bibliography generated by BibTeX as a .bbl file +% BibTeX documentation can be easily obtained at: +% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/ +% The IEEEtran BibTeX style support page is at: +% http://www.michaelshell.org/tex/ieeetran/bibtex/ +%\bibliographystyle{IEEEtran} +% argument is your BibTeX string definitions and bibliography database(s) +%\bibliography{IEEEabrv,../bib/paper} +% +% 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} + +\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} + + + + +% that's all folks +\end{document} + +