[GMRES2stage.git] / paper.tex

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\hyphenation{op-tical net-works semi-conduc-tor}



\usepackage{algorithm}
\usepackage{algpseudocode}

\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}

\algnewcommand\algorithmicoutput{\textbf{Output:}}
\algnewcommand\Output{\item[\algorithmicoutput]}



\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{A Krylov two-stage algorithm 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
% use a multiple column layout for up to two different
% affiliations

\author{\IEEEauthorblockN{Rapha\"el Couturier}
\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
Universit\'e de Franche-Comt\'e, IUT de Belfort-Montb\'eliard\\
19 avenue de Mar\'echal Juin, BP 527 \\
90016 Belfort Cedex, France\\
Email: raphael.couturier@univ-fcomte.fr}
\and
\IEEEauthorblockN{Lilia Ziane Khodja}
\IEEEauthorblockA{Centre de Recherche INRIA Bordeaux Sud-Ouest\\
200 avenue de la Vieille Tour\\
33405 Talence Cedex, France\\
Email: lilia.ziane@inria.fr}
}

% conference papers do not typically use \thanks and this command
% is locked out in conference mode. 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




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%\caption{Simulation Results}
%\label{fig_sim}
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%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
%\label{fig_first_case}}
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%\label{fig_second_case}}}
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%\label{fig_sim}
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% if using array.sty, it might be a good idea to tweak the value of
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%One & Two\\
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%Three & Four\\
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% positioning is not used. Most IEEE journals/conferences use top floats
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% 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)
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Related works}
%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}


\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
\end{algorithmic}
\label{algo:01}
\end{algorithm}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
%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
%%%*********************************************************
%%%*********************************************************


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% 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
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% The "triggered" command can be changed if desired:
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% references section

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%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <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}

\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.

\end{thebibliography}




% that's all folks
\end{document}