+
+%% 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
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+%% any damages or losses, including, but not limited to, incidental,
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+%% of any information contained here.
+%%
+%% All comments are the opinions of their respective authors and are not
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+%%
+%% 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. A copy of the LPPL, version 1.3, is included
+%% in the base LaTeX documentation of all distributions of LaTeX released
+%% 2003/12/01 or later.
+%% Retain all contribution notices and credits.
+%% ** Modified files should be clearly indicated as such, including **
+%% ** renaming them and changing author support contact information. **
+%%
+%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
+%% bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
+%%*************************************************************************
+
+% *** Authors should verify (and, if needed, correct) their LaTeX system ***
+% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
+% *** with production work. IEEE's font choices can trigger bugs that do ***
+% *** not appear when using other class files. ***
+% The testflow support page is at:
+% http://www.michaelshell.org/tex/testflow/
+
+
+
+% Note that the a4paper option is mainly intended so that authors in
+% countries using A4 can easily print to A4 and see how their papers will
+% look in print - the typesetting of the document will not typically be
+% affected with changes in paper size (but the bottom and side margins will).
+% Use the testflow package mentioned above to verify correct handling of
+% both paper sizes by the user's LaTeX system.
+%
+% Also note that the "draftcls" or "draftclsnofoot", not "draft", option
+% should be used if it is desired that the figures are to be displayed in
+% draft mode.
+%
+\documentclass[10pt, conference, compsocconf]{IEEEtran}
+% Add the compsocconf option for Computer Society conferences.
+%
+% If IEEEtran.cls has not been installed into the LaTeX system files,
+% manually specify the path to it like:
+% \documentclass[conference]{../sty/IEEEtran}
+
+
+
+
+
+% Some very useful LaTeX packages include:
+% (uncomment the ones you want to load)
+
+
+% *** MISC UTILITY PACKAGES ***
+%
+%\usepackage{ifpdf}
+% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
+% compilation based on whether the output is pdf or dvi.
+% usage:
+% \ifpdf
+% % pdf code
+% \else
+% % dvi code
+% \fi
+% The latest version of ifpdf.sty can be obtained from:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/oberdiek/
+% Also, note that IEEEtran.cls V1.7 and later provides a builtin
+% \ifCLASSINFOpdf conditional that works the same way.
+% When switching from latex to pdflatex and vice-versa, the compiler may
+% have to be run twice to clear warning/error messages.
+
+
+
+
+
+
+% *** CITATION PACKAGES ***
+%
+%\usepackage{cite}
+% cite.sty was written by Donald Arseneau
+% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
+% \cite{} output to follow that of IEEE. Loading the cite package will
+% result in citation numbers being automatically sorted and properly
+% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
+% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
+% \cite will automatically add leading space, if needed. Use cite.sty's
+% noadjust option (cite.sty V3.8 and later) if you want to turn this off.
+% cite.sty is already installed on most LaTeX systems. Be sure and use
+% version 4.0 (2003-05-27) and later if using hyperref.sty. cite.sty does
+% not currently provide for hyperlinked citations.
+% The latest version can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/cite/
+% The documentation is contained in the cite.sty file itself.
+
+
+
+
+
+
+% *** GRAPHICS RELATED PACKAGES ***
+%
+\ifCLASSINFOpdf
+ % \usepackage[pdftex]{graphicx}
+ % declare the path(s) where your graphic files are
+ % \graphicspath{{../pdf/}{../jpeg/}}
+ % and their extensions so you won't have to specify these with
+ % every instance of \includegraphics
+ % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
+\else
+ % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
+ % will default to the driver specified in the system graphics.cfg if no
+ % driver is specified.
+ % \usepackage[dvips]{graphicx}
+ % declare the path(s) where your graphic files are
+ % \graphicspath{{../eps/}}
+ % and their extensions so you won't have to specify these with
+ % every instance of \includegraphics
+ % \DeclareGraphicsExtensions{.eps}
+\fi
+% graphicx was written by David Carlisle and Sebastian Rahtz. It is
+% required if you want graphics, photos, etc. graphicx.sty is already
+% installed on most LaTeX systems. The latest version and documentation can
+% be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/required/graphics/
+% Another good source of documentation is "Using Imported Graphics in
+% LaTeX2e" by Keith Reckdahl which can be found as epslatex.ps or
+% epslatex.pdf at: http://www.ctan.org/tex-archive/info/
+%
+% latex, and pdflatex in dvi mode, support graphics in encapsulated
+% postscript (.eps) format. pdflatex in pdf mode supports graphics
+% in .pdf, .jpeg, .png and .mps (metapost) formats. Users should ensure
+% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
+% not a bitmapped formats (.jpeg, .png). IEEE frowns on bitmapped formats
+% which can result in "jaggedy"/blurry rendering of lines and letters as
+% well as large increases in file sizes.
+%
+% You can find documentation about the pdfTeX application at:
+% http://www.tug.org/applications/pdftex
+
+
+
+
+
+% *** MATH PACKAGES ***
+%
+%\usepackage[cmex10]{amsmath}
+% A popular package from the American Mathematical Society that provides
+% many useful and powerful commands for dealing with mathematics. If using
+% it, be sure to load this package with the cmex10 option to ensure that
+% only type 1 fonts will utilized at all point sizes. Without this option,
+% it is possible that some math symbols, particularly those within
+% footnotes, will be rendered in bitmap form which will result in a
+% document that can not be IEEE Xplore compliant!
+%
+% Also, note that the amsmath package sets \interdisplaylinepenalty to 10000
+% thus preventing page breaks from occurring within multiline equations. Use:
+%\interdisplaylinepenalty=2500
+% after loading amsmath to restore such page breaks as IEEEtran.cls normally
+% does. amsmath.sty is already installed on most LaTeX systems. The latest
+% version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/
+
+
+
+
+
+% *** SPECIALIZED LIST PACKAGES ***
+%
+%\usepackage{algorithmic}
+% algorithmic.sty was written by Peter Williams and Rogerio Brito.
+% This package provides an algorithmic environment fo describing algorithms.
+% You can use the algorithmic environment in-text or within a figure
+% environment to provide for a floating algorithm. Do NOT use the algorithm
+% floating environment provided by algorithm.sty (by the same authors) or
+% algorithm2e.sty (by Christophe Fiorio) as IEEE does not use dedicated
+% algorithm float types and packages that provide these will not provide
+% correct IEEE style captions. The latest version and documentation of
+% algorithmic.sty can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithms/
+% There is also a support site at:
+% http://algorithms.berlios.de/index.html
+% Also of interest may be the (relatively newer and more customizable)
+% algorithmicx.sty package by Szasz Janos:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/
+
+
+
+
+% *** ALIGNMENT PACKAGES ***
+%
+%\usepackage{array}
+% Frank Mittelbach's and David Carlisle's array.sty patches and improves
+% the standard LaTeX2e array and tabular environments to provide better
+% appearance and additional user controls. As the default LaTeX2e table
+% generation code is lacking to the point of almost being broken with
+% respect to the quality of the end results, all users are strongly
+% advised to use an enhanced (at the very least that provided by array.sty)
+% set of table tools. array.sty is already installed on most systems. The
+% latest version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/required/tools/
+
+
+%\usepackage{mdwmath}
+%\usepackage{mdwtab}
+% Also highly recommended is Mark Wooding's extremely powerful MDW tools,
+% especially mdwmath.sty and mdwtab.sty which are used to format equations
+% and tables, respectively. The MDWtools set is already installed on most
+% LaTeX systems. The lastest version and documentation is available at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/mdwtools/
+
+
+% IEEEtran contains the IEEEeqnarray family of commands that can be used to
+% generate multiline equations as well as matrices, tables, etc., of high
+% quality.
+
+
+\usepackage{eqparbox}
+% Also of notable interest is Scott Pakin's eqparbox package for creating
+% (automatically sized) equal width boxes - aka "natural width parboxes".
+% Available at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/
+
+
+
+
+
+% *** SUBFIGURE PACKAGES ***
+%\usepackage[tight,footnotesize]{subfigure}
+% subfigure.sty was written by Steven Douglas Cochran. This package makes it
+% easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
+% work, it is a good idea to load it with the tight package option to reduce
+% the amount of white space around the subfigures. subfigure.sty is already
+% installed on most LaTeX systems. The latest version and documentation can
+% be obtained at:
+% http://www.ctan.org/tex-archive/obsolete/macros/latex/contrib/subfigure/
+% subfigure.sty has been superceeded by subfig.sty.
+
+
+
+%\usepackage[caption=false]{caption}
+%\usepackage[font=footnotesize]{subfig}
+% subfig.sty, also written by Steven Douglas Cochran, is the modern
+% replacement for subfigure.sty. However, subfig.sty requires and
+% automatically loads Axel Sommerfeldt's caption.sty which will override
+% IEEEtran.cls handling of captions and this will result in nonIEEE style
+% figure/table captions. To prevent this problem, be sure and preload
+% caption.sty with its "caption=false" package option. This is will preserve
+% IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later
+% (recommended due to many improvements over 1.2) of subfig.sty supports
+% the caption=false option directly:
+%\usepackage[caption=false,font=footnotesize]{subfig}
+%
+% The latest version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
+% The latest version and documentation of caption.sty can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/caption/
+
+
+
+
+% *** FLOAT PACKAGES ***
+%
+%\usepackage{fixltx2e}
+% fixltx2e, the successor to the earlier fix2col.sty, was written by
+% Frank Mittelbach and David Carlisle. This package corrects a few problems
+% in the LaTeX2e kernel, the most notable of which is that in current
+% LaTeX2e releases, the ordering of single and double column floats is not
+% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
+% single column figure to be placed prior to an earlier double column
+% figure. The latest version and documentation can be found at:
+% http://www.ctan.org/tex-archive/macros/latex/base/
+
+
+
+%\usepackage{stfloats}
+% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
+% the ability to do double column floats at the bottom of the page as well
+% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
+% LaTeX2e). It also provides a command:
+%\fnbelowfloat
+% to enable the placement of footnotes below bottom floats (the standard
+% LaTeX2e kernel puts them above bottom floats). This is an invasive package
+% which rewrites many portions of the LaTeX2e float routines. It may not work
+% with other packages that modify the LaTeX2e float routines. The latest
+% version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/sttools/
+% Documentation is contained in the stfloats.sty comments as well as in the
+% presfull.pdf file. Do not use the stfloats baselinefloat ability as IEEE
+% does not allow \baselineskip to stretch. Authors submitting work to the
+% IEEE should note that IEEE rarely uses double column equations and
+% that authors should try to avoid such use. Do not be tempted to use the
+% cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does
+% not format its papers in such ways.
+
+
+
+
+
+% *** PDF, URL AND HYPERLINK PACKAGES ***
+%
+%\usepackage{url}
+% url.sty was written by Donald Arseneau. It provides better support for
+% handling and breaking URLs. url.sty is already installed on most LaTeX
+% systems. The latest version can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/misc/
+% Read the url.sty source comments for usage information. Basically,
+% \url{my_url_here}.
+
+
+
+
+
+% *** Do not adjust lengths that control margins, column widths, etc. ***
+% *** Do not use packages that alter fonts (such as pslatex). ***
+% There should be no need to do such things with IEEEtran.cls V1.6 and later.
+% (Unless specifically asked to do so by the journal or conference you plan
+% to submit to, of course. )
+
+
+% correct bad hyphenation here
+\hyphenation{op-tical net-works semi-conduc-tor}
+
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multirow}
+\usepackage{graphicx}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+\newtheorem{proposition}{Proposition}
+
+\begin{document}
+%
+% paper title
+% can use linebreaks \\ within to get better formatting as desired
+\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}
+
+
+
+
+
+
+% author names and affiliations
+% use a multiple column layout for up to two different
+% affiliations
+
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
+\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
+Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, 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}
+In this article, a two-stage iterative algorithm is proposed to improve the
+convergence of Krylov based iterative methods, typically those of GMRES
+variants. The principle of the proposed approach is to build an external
+iteration over the Krylov method, and to frequently store its current residual
+(at each GMRES restart for instance). After a given number of outer iterations,
+a least-squares minimization step is applied on the matrix composed by the saved
+residuals, in order to compute a better solution and to make new iterations if
+required. It is proven that the proposal has the same convergence properties
+than the inner embedded method itself. Experiments using up to 16,394 cores
+also show that the proposed algorithm runs around 5 or 7 times faster than
+GMRES.
+\end{abstract}
+
+\begin{IEEEkeywords}
+Iterative Krylov methods; sparse linear systems; residual 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 have recently become more attractive than direct ones to solve very large
+sparse linear systems. They are more efficient in a parallel
+context, supporting thousands of cores, and they require less memory and arithmetic
+operations than direct methods. This is why new iterative methods are frequently
+proposed or adapted by researchers, and the increasing need to solve very large sparse
+linear systems has triggered the development of such efficient iterative techniques
+suitable for parallel processing.
+
+Most of the successful iterative methods currently available are based on so-called ``Krylov
+subspaces''. They consist in forming a basis of successive matrix
+powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov subspace basis in order to solve linear
+systems. The most known iterative Krylov subspace methods are conjugate
+gradient and GMRES ones (Generalized Minimal RESidual).
+
+
+However, iterative methods suffer from scalability problems on parallel
+computing platforms with many processors, due to their need of reduction
+operations, and to collective communications to achive matrix-vector
+multiplications. The communications on large clusters with thousands of cores
+and large sizes of messages can significantly affect the performances of these
+iterative methods. As a consequence, Krylov subspace iteration methods are often used
+with preconditioners in practice, to increase their convergence and accelerate their
+performances. However, most of the good preconditioners are not scalable on
+large clusters.
+
+In this research work, a two-stage algorithm based on two nested iterations
+called inner-outer iterations is proposed. This algorithm consists in solving the sparse
+linear system iteratively with a small number of inner iterations, and restarting
+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. Furthermore, the minimization technique improves its convergence and
+performances.
+
+The present article is organized as follows. Related works are presented in
+Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
+a least-squares residual minimization, while Section~\ref{sec:04} provides
+convergence results regarding this method. Section~\ref{sec:05} shows some
+experimental results obtained on large clusters using routines of PETSc
+toolkit. This research work ends by a conclusion section, in which the proposal
+is summarized while intended perspectives are provided.
+
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\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{Two-stage iteration with least-squares residuals minimization 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. As explained previously,
+the algorithm is implemented as an
+inner-outer iteration solver based on iterative Krylov methods. The main
+key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
+It can be summarized as follows: the
+inner solver is a Krylov based one. In order to accelerate its convergence, the
+outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed.
+
+At each outer iteration, the sparse linear system $Ax=b$ is partially
+solved using only $m$
+iterations of an iterative method, this latter being initialized with the
+best known approximation previously obtained.
+GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
+inner solver. The current approximation of the Krylov method is then stored inside a matrix
+$S$ composed by the successive solutions that are computed during inner iterations.
+
+At each $s$ iterations, the minimization step is applied in order to
+compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by
+the inner iterations 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}
+with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+
+
+In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$,
+with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-squares method such as
+CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more
+appropriate than a single direct method in a parallel context.
+
+
+
+\begin{algorithm}[t]
+\caption{TSIRM}
+\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 (error$<\epsilon_{tsirm}$)} \label{algo:conv}
+ \State $x_k=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
+ \State retrieve error
+ \State $S_{k \mod s}=x_k$ \label{algo:store}
+ \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
+ \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
+ \State $x_k=S\alpha$ \Comment{compute new solution}
+ \EndIf
+ \EndFor
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+
+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 $max\_iter_{kryl}$ iterations. In practice, we suggest to set this parameter
+equals to the restart number of the GMRES-like method. Moreover, a tolerance
+threshold must be specified for the solver. In practice, this threshold must be
+much smaller than the convergence threshold of the TSIRM algorithm (\emph{i.e.}
+$\epsilon_{tsirm}$). 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. To
+solve the minimization problem, an iterative method is used. Two parameters are
+required for that: the maximum number of iterations and the threshold to stop the
+method.
+
+Let us summarize the most important parameters of TSIRM:
+\begin{itemize}
+\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
+\item $s$: the number of outer iterations before applying the minimization step;
+\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
+\item $\epsilon_{ls}$: the threshold used to stop the least-squares method.
+\end{itemize}
+
+
+The parallelisation of TSIRM relies on the parallelization of all its
+parts. More precisely, except the least-squares step, all the other parts are
+obvious to achieve out in parallel. In order to develop a parallel version of
+our code, we have chosen to use PETSc~\cite{petsc-web-page}. For
+line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and
+efficient since the matrix $A$ is sparse and since the matrix $S$ contains few
+colums in practice. As explained previously, at least two methods seem to be
+interesting to solve the least-squares minimization, CGLS and LSQR.
+
+In the following we remind the CGLS algorithm. The LSQR method follows more or
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+
+\begin{algorithm}[t]
+\caption{CGLS}
+\begin{algorithmic}[1]
+ \Input $A$ (matrix), $b$ (right-hand side)
+ \Output $x$ (solution vector)\vspace{0.2cm}
+ \State Let $x_0$ be an initial approximation
+ \State $r_0=b-Ax_0$
+ \State $p_1=A^Tr_0$
+ \State $s_0=p_1$
+ \State $\gamma=||s_0||^2_2$
+ \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
+ \State $q_k=Ap_k$
+ \State $\alpha_k=\gamma/||q_k||^2_2$
+ \State $x_k=x_{k-1}+\alpha_kp_k$
+ \State $r_k=r_{k-1}-\alpha_kq_k$
+ \State $s_k=A^Tr_k$
+ \State $\gamma_{old}=\gamma$
+ \State $\gamma=||s_k||^2_2$
+ \State $\beta_k=\gamma/\gamma_{old}$
+ \State $p_{k+1}=s_k+\beta_kp_k$
+ \EndFor
+\end{algorithmic}
+\label{algo:02}
+\end{algorithm}
+
+
+In each iteration of CGLS, there is two matrix-vector multiplications and some
+classical operations: dot product, norm, multiplication and addition on vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+
+\section{Convergence results}
+\label{sec:04}
+Let us recall the following result, see~\cite{Saad86}.
+\begin{proposition}
+Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
+\begin{equation}
+||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+\end{equation}
+where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves
+the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
+\end{proposition}
+
+<<<<<<< HEAD
+
+=======
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+\end{proposition}
+
+\begin{proof}
+Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
+$k$-th iterate of TSIRM.
+We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+
+Each step of the TSIRM algorithm
+\end{proof}
+>>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Experiments using PETSc}
+\label{sec:05}
+
+
+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 are given.
+
+\begin{table}[htbp]
+\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 & Comput. fluid dynamics & 525,825 & 2,100,225 \\
+epb3 & Thermal problem & 84,617 & 463,625 \\
+atmosmodj & Comput. fluid dynamics & 1,270,432 & 8,814,880 \\
+bfwa398 & Electromagnetics pb & 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 (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
+chosen to minimize the least-squares problem with the following parameters:
+$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
+$\epsilon_{tsirm}=1e-10$. 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 TSIRM, the same solver and the same
+preconditionner are used. This Table shows that TSIRM 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}[htbp]
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|}
+\hline
+
+ \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\
+\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 (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+
+
+
+In order to perform larger experiments, we have tested some example applications
+of PETSc. Those applications are available in the ksp part which is suited for
+scalable linear equations solvers:
+\begin{itemize}
+\item ex15 is an example which solves in parallel an operator using a finite
+ difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
+ representing the neighbors in each directions are equal to -1. This example is
+ used in many physical phenomena, for example, heat and fluid flow, wave
+ propagation, etc.
+\item ex54 is another example based on 2D problem discretized with quadrilateral
+ finite elements. For this example, the user can define the scaling of material
+ coefficient in embedded circle called $\alpha$.
+\end{itemize}
+For more technical details on these applications, interested readers are invited
+to read the codes available in the PETSc sources. Those problems have been
+chosen because they are scalable with many cores which is not the case of other problems that we have tested.
+
+In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
+
+
+{\bf Description of preconditioners}\\
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
+\cline{3-8}
+ & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
+ 2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\
+ 2,048 & sor & 745.37 & 57,060 & 87.31 & 6,150 & 104.21 & 7,230 & 8.53 \\
+ 4,096 & mg & 562.25 & 25,170 & 97.23 & 3,990 & 89.71 & 3,630 & 6.27 \\
+ 4,096 & sor & 912.12 & 70,194 & 145.57 & 9,750 & 168.97 & 10,980 & 6.26 \\
+ 8,192 & mg & 917.02 & 40,290 & 148.81 & 5,730 & 143.03 & 5,280 & 6.41 \\
+ 8,192 & sor & 1,404.53 & 106,530 & 212.55 & 12,990 & 180.97 & 10,470 & 7.76 \\
+ 16,384 & mg & 1,430.56 & 63,930 & 237.17 & 8,310 & 244.26 & 7,950 & 6.03 \\
+ 16,384 & sor & 2,852.14 & 216,240 & 418.46 & 21,690 & 505.26 & 23,970 & 6.82 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
+Table~\ref{tab:03} shows the execution times and the number of iterations of
+example ex15 of PETSc on the Juqueen architecture. Different numbers of cores
+are studied ranging from 2,048 up-to 16,383. Two preconditioners have been
+tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the
+problems) per processor is fixed to 25,000, also called weak scaling. This
+number can seem relatively small. In fact, for some applications that need a lot
+of memory, the number of components per processor requires sometimes to be
+small.
+
+
+
+In Table~\ref{tab:03}, we can notice that TSIRM is always faster than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSIRM according to
+the minimization procedure: CGLS or LSQR. Even if we have computed the worst
+case between CGLS and LSQR, it is clear that TSIRM is always faster than
+FGMRES. For this example, the multigrid preconditioner is faster than SOR. The
+gain between TSIRM and FGMRES is more or less similar for the two
+preconditioners. Looking at the number of iterations to reach the convergence,
+it is obvious that TSIRM allows the reduction of the number of iterations. It
+should be noticed that for TSIRM, in those experiments, only the iterations of
+the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
+corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
+\label{fig:01}
+\end{figure}
+
+
+In Figure~\ref{fig:01}, the number of iterations per second corresponding to
+Table~\ref{tab:01} is displayed. It can be noticed that the number of
+iterations per second of FMGRES is constant whereas it decreases with TSIRM with
+both preconditioners. This can be explained by the fact that when the number of
+cores increases the time for the least-squares minimization step also increases but, generally,
+when the number of cores increases, the number of iterations to reach the
+threshold also increases, and, in that case, TSIRM is more efficient to reduce
+the number of iterations. So, the overall benefit of using TSIRM is interesting.
+
+
+
+
+
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 \\
+ 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*}
+
+
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
+
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\
+\cline{2-7} \cline{9-11}
+ & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline
+ 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\
+ 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\
+ 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\
+ 4096 & 405.60 & 28,380 & 111.67 & 7,590 & 91.72 & 6,510 & 4.42 & 1.22 & .79 & .84 \\
+ 8192 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 & .32 & .58 & .56 \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.}
+\label{tab:05}
+\end{center}
+\end{table*}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
+\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
+\label{fig:02}
+\end{figure}
+
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Conclusion}
+\label{sec:06}
+%The conclusion goes here. this is more of the conclusion
+%%%*********************************************************
+%%%*********************************************************
+
+
+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- test the influence of all the parameters\\
+- adaptative number of outer iterations to minimize\\
+- other methods to minimize the residuals?\\
+- implement our solver inside PETSc
+
+
+% conference papers do not normally have an appendix
+
+
+
+% use section* for acknowledgement
+%%%*********************************************************
+%%%*********************************************************
+\section*{Acknowledgment}
+This paper is partially funded by the Labex ACTION program (contract
+ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource
+Curie and Juqueen respectively based in France and Germany.
+
+
+
+% 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{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}
+
+%% \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}
+
+
