[GMRES2stage.git] / paper.tex

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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{A Krylov two-stage algorithm to solve large sparse linear systems}
%où
%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
%où
%\title{???}


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

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%Montgomery Scott\IEEEauthorrefmark{3} and
%Eldon Tyrell\IEEEauthorrefmark{4}}
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%Georgia Institute of Technology,
%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
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\begin{abstract}
%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT.
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\begin{IEEEkeywords}
Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir... 
\end{IEEEkeywords}


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%%%*********************************************************
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\section{Introduction}
% no \IEEEPARstart
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
Iterative  methods are become  more attractive  than direct  ones to  solve very
large sparse linear  systems. They are more effective in  a parallel context and
require less memory  and arithmetic operations than direct  methods. A number of
iterative methods are proposed and adapted by many researchers and the increased
need for solving  very large sparse linear systems  triggered the development of
efficient iterative techniques suitable for the parallel processing.

Most of the successful iterative methods currently available are based on Krylov
subspaces which  consist in forming a  basis of a sequence  of successive matrix
powers times an initial vector for example the residual. These methods are based
on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
Gradient method and GMRES method (generalized minimal residual).

However,  iterative  methods suffer  from scalability  problems  on parallel
computing  platforms  with many  processors  due  to  their need  for  reduction
operations    and   collective    communications   to    perform   matrix-vector
multiplications. The  communications on large  clusters with thousands  of cores
and  large  sizes of  messages  can  significantly  affect the  performances  of
iterative methods. In practice, Krylov subspace iteration methods are often used
with preconditioners in order to increase their convergence and accelerate their
performances.  However, most  of the  good preconditioners  are not  scalable on
large clusters.

In this  paper we propose a  two-stage algorithm based on  two nested iterations
called inner-outer  iterations.  This algorithm  consists in solving  the sparse
linear system iteratively  with a small number of  inner iterations and restarts
the outer step with a new solution minimizing some error functions over a Krylov
subspace. This algorithm is iterative  and easy to parallelize on large clusters
and the minimization technique improves its convergence and performances.

The present paper is organized  as follows. In Section~\ref{sec:02} some related
works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
on   Krylov  subspace   iteration  methods.   Section~\ref{sec:04}   shows  some
experimental results obtained on large  clusters of our algorithm using routines
of PETSc toolkit.
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\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.
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\section{A Krylov two-stage algorithm}
\label{sec:03}
A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
points of our solver are given in Algorithm~\ref{algo:01}.

In order to accelerate the convergence, the outer iteration is implemented as an
iterative  Krylov method  which minimizes  some  error functions  over a  Krylov
subspace~\cite{saad96}. At  each iteration, the  sparse linear system  $Ax=b$ is
solved   iteratively    with   an   iterative   method,    for   example   GMRES
method~\cite{saad86} or  some of its variants,  and the Krylov  subspace that we
used is spanned by a basis  $S$ composed of successive solutions issued from the
inner iteration
\begin{equation}
  S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
\end{equation} 
The advantage  of such a Krylov subspace  is that we neither  need an orthogonal
basis nor  any synchronization  between processors to  generate this  basis. The
algorithm  is periodically  restarted every  $s$ iterations  with a  new initial
guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
subspace spanned by  vectors of $S$, where $\alpha$ is a  solution of the normal
equations
\begin{equation}
  R^TR\alpha = R^Tb,
\end{equation}
which is associated with the least-squares problem
\begin{equation}
   \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\end{equation}
such  that $R=AS$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
$s\ll n$,  and $R^T$ denotes  the transpose of  matrix $R$. We use  an iterative
method   to  solve   the  least-squares   problem~(\ref{eq:01})  such   as  CGLS
~\cite{hestenes52}  or LSQR~\cite{paige82}  which  are more  appropriate than  a
direct method in the parallel context.

\begin{algorithm}[t]
\caption{A Krylov two-stage algorithm}
\begin{algorithmic}[1]
  \Input $A$ (sparse matrix), $b$ (right-hand side)
  \Output $x$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence}
    \State Solve iteratively $Ax^k=b$
    \State $S_{k~mod~s}=x^k$ 
    \If {$k$ mod $s=0$ {\bf and} not convergence}
      \State Compute dense matrix $R=AS$
      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
      \State Compute minimizer $x^k=S\alpha$
    \EndIf
  \EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}

Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ into the
column $k~ mod~ s$ of the matrix  $S$. After the minimization, the matrix $S$ is
reused with the new values of the residuals.

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\section{Experiments using petsc}
\label{sec:04}

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\section{Conclusion}
\label{sec:05}
%The conclusion goes here. this is more of the conclusion
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\section*{Acknowledgment}
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% references section

% can use a bibliography generated by BibTeX as a .bbl file
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\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.
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% that's all folks
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