[GMRES2stage.git] / paper.tex

%% 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. 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-Comt\'e, 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  achieve   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 
last obtained approximation. 
GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
$S$, which is composed by the $s$ last solutions that have been computed during 
the inner iterations phase.
In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. 

At each $s$ iterations, another kind of 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 $R=AS$. The new solution $x$ is then 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  the proposed  method.  The  outer
iteration is  inside the \emph{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
equal to  the restart  number in 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 $S$.
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  parallelization  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
columns 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} for further readings.
\begin{proposition}
\label{prop:saad}
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}


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. Furthermore, 
let $r_k$ be the
$k$-th residue of TSIRM, then
we still have:
\begin{equation}
||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
\end{equation}
where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.
\end{proposition}

\begin{proof}
We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$

The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.

Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
We will show that the statement holds too for $r_k$. Two situations can occur:
\begin{itemize}
\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis.
\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\
$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$

$\begin{array}{ll}
& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\
& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
& \leqslant ||b-Ax_{k}||_2\\
& = ||r_k||_2\\
& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||,
\end{array}$
\end{itemize}
which concludes the induction and the proof.
\end{proof}

We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
than the one of the GMRES method.

%%%*********************************************************
%%%*********************************************************
\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 TSIRM with (F)GMRES 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 core  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:03}  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|}{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      & 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 TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 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|}{FGMRES} & \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. &   & FGMRES & 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 TSIRM with FGMRES 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, threshold 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
%%%*********************************************************
%%%*********************************************************

A novel two-stage iterative  algorithm has been proposed in this article,
in order to accelerate the convergence Krylov iterative  methods.
Our TSIRM proposal acts as a merger between Krylov based solvers and
a least-squares minimization step.
The convergence of the method has been proven in some situations, while 
experiments up to 16,394 cores have been led to verify that TSIRM runs
5 or  7 times  faster than GMRES.


For future work, the authors' intention is to investigate 
other kinds of matrices, problems, and inner solvers. The 
influence of all parameters must be tested too, while 
other methods to minimize the residuals must be regarded.
The number of outer iterations to minimize should become 
adaptative to improve the overall performances of the proposal.
Finally, this solver will be implemented 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 resources
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}