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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization 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\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}
}



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%Homer Simpson\IEEEauthorrefmark{2},
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%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\\
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%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
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% make the title area
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\begin{abstract}
In  this paper  we propose  a  two stage  iterative method  which increases  the
convergence of Krylov iterative methods,  typically those of GMRES variants. The
principle of  our approach  is to  build an external  iteration over  the Krylov
method  and to  save  the current  residual  frequently (for  example, for  each
restart of GMRES). Then after a given number of outer iterations, a minimization
step is applied on the matrix composed of the save residuals in order to compute
a  better solution and  make a  new iteration  if necessary.  We prove  that our
method  has the  same  convergence property  than  the inner  method used.  Some
experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
can be around 7 times faster.
\end{abstract}

\begin{IEEEkeywords}
Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir... 
\end{IEEEkeywords}


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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  became more attractive than  direct ones to  solve very large
sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
context,  with  thousands  of  cores,  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 some
previous residuals. 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 using
a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
convergence   results   on  this   method.    Section~\ref{sec:05}  shows   some
experimental results obtained on large  clusters of our algorithm using routines
of  PETSc  toolkit.  Finally   Section~\ref{sec:06}  concludes  and  gives  some
perspectives.
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Related works}
\label{sec:02} 
%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{A Krylov two-stage algorithm}
\label{sec:03}
A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
points of our solver are given in Algorithm~\ref{algo:01}.

In order to accelerate the convergence, the outer iteration periodically applies
a least-square minimization  on the residuals computed by  the inner solver. The
inner solver is a Krylov based solver which does not required to be changed.

At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
iterations, using an iterative method restarting with the previous solution. For
example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
inner solver. The current solution of the Krylov method is saved inside a matrix
$S$ composed of successive solutions computed by the inner iteration.

Periodically, every $s$ iterations, the minimization step is applied in order to
compute a new  solution $x$. For that, the previous  residuals are computed 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 in  $\mathbb{R}^{n\times s}$,
$s\ll n$.   In order  to minimize~(\ref{eq:01}), a  least-square method  such as
CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Those  methods are more
appropriate than a direct method in a parallel context.

\begin{algorithm}[t]
\caption{TSARM}
\begin{algorithmic}[1]
  \Input $A$ (sparse matrix), $b$ (right-hand side)
  \Output $x$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
    \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}
    \State $S_{k~mod~s}=x^k$ \label{algo:store}
    \If {$k$ mod $s=0$ {\bf and} not convergence}
      \State $R=AS$ \Comment{compute dense matrix}
      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
      \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 $m$ iterations.  In practice, we  suggest to choose $m$
equals to  the restart number  of the GMRES like  method. 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.

%%%*********************************************************
%%%*********************************************************

\section{Convergence results}
\label{sec:04}

%%%*********************************************************
%%%*********************************************************
\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 is given.

\begin{table*}
\begin{center}
\begin{tabular}{|c|c|r|r|r|} 
\hline
Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
crashbasis         & Optimization      & 160,000  &  1,750,416  \\
parabolic\_fem     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
epb3               & Thermal problem   & 84,617  & 463,625  \\
atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
bfwa398            & Electromagnetics problem & 398 & 3,678 \\
torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
\hline

\end{tabular}
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
\end{table*}

The following  parameters have been chosen  for our experiments.   As by default
the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
the     GMRES    every     30    iterations     (line     \ref{algo:solve}    in
Algorithm~\ref{algo:01}).   $s$ is  set to  8. CGLS  is chosen  to  minimize the
least-squares  problem.  Two  conditions  are  used to  stop  CGLS,  either  the
precision is under $1e-40$ or the  number of iterations is greater to $20$.  The
external   precision    is   set    to   $1e-10$   (line    \ref{algo:conv}   in
Algorithm~\ref{algo:01}).  Those  experiments have been performed  on a Intel(R)
Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.


In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear
systems obtained with the previous matrices  with a GMRES variant and with out 2
stage algorithm are  given. In the second column, it can  be noticed that either
gmres or fgmres is used to  solve the linear system.  According to the matrices,
different preconditioner is  used.  With the 2 stage  algorithm, the same solver
and  the same  preconditionner  is used.   This  Table shows  that  the 2  stage
algorithm  can  drastically  reduce  the  number  of  iterations  to  reach  the
convergence when the  number of iterations for the normal GMRES  is more or less
greater than  500. In fact  this also depends  on tow parameters: the  number of
iterations  to  stop  GMRES  and   the  number  of  iterations  to  perform  the
minimization.


\begin{table}
\begin{center}
\begin{tabular}{|c|c|r|r|r|r|} 
\hline

 \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage 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}




Larger experiments ....

\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores & precond   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage 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 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
\label{tab:03}
\end{center}
\end{table*}


\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores & threshold   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage 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                  & 792.11 & 109,590 & 76.83  &  10,470  & 65.20  & 9,030  & 12.14 \\
  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*}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\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\\
- 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.



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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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