update param algo

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 8232b7e757e45ce5353adf25a7c24cac2c465d86..18340f66fde077e613da5a6fa0effbf69a719bb9 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -348,35 +348,49 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 

+
+\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]}
+
+
+

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

+
+
+

 % author names and affiliations
 % use a multiple column layout for up to two different
 % affiliations
 
 % author names and affiliations
 % use a multiple column layout for up to two different
 % affiliations
 

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


@@ -413,11 +427,20 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR 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  saved 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}
 \end{abstract}
 
 \begin{IEEEkeywords}

-Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à voir... 
+Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir... 

 \end{IEEEkeywords}
 
 
 \end{IEEEkeywords}
 
 


@@ -524,6 +547,46 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 % no \IEEEPARstart
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
 % 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.

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


@@ -532,6 +595,7 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{Related works}
 %%%*********************************************************
 %%%*********************************************************
 \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.
 %%%*********************************************************
 %%%*********************************************************
 %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.
 %%%*********************************************************
 %%%*********************************************************


@@ -540,15 +604,320 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 
 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************

-\section{A Krylov two-stage algorithm}
+\section{Two-stage algorithm with least-square residuals minimization}
+\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 based on a Krylov method which does not require 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 (error$<\epsilon_{tsarm}$)} \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$>\epsilon_{tsarm}$}
+      \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
+      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \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 TSARM  algorithm  (i.e.
+$\epsilon_{tsarm}$).  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 iteration and the threshold to stop the
+method.
+
+To summarize, the important parameters of TSARM are:
+\begin{itemize}
+\item $\epsilon_{tsarm}$ the threshold to stop the TSARM 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-square method
+\item $\epsilon_{ls}$ the threshold to stop the least-square method
+\end{itemize}
+
+
+The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
+parts. More  precisely, except  the least-square 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-square minimization, CGLS and LSQR.
+
+In the following  we remind the CGLS algorithm. The LSQR  method follows more or
+less the same principle but it take 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 $r=b-Ax$
+  \State $p=A'r$
+  \State $s=p$
+  \State $g=||s||^2_2$
+  \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
+    \State $q=Ap$
+    \State $\alpha=g/||q||^2_2$
+    \State $x=x+alpha*p$
+    \State $r=r-alpha*q$
+    \State $s=A'*r$
+    \State $g_{old}=g$
+    \State $g=||s||^2_2$
+    \State $\beta=g/g_{old}$
+  \EndFor
+\end{algorithmic}
+\label{algo:02}
+\end{algorithm}
+
+
+In each iteration  of CGLS, there is two  matrix-vector multiplications and some
+classical operations:  dots, norm, multiplication  and addition on  vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
+
+

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

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

 
 
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
 
 
 %%%*********************************************************
 %%%*********************************************************
 \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, $max\_iter_{kryl}=30$).  $s$ 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
+$\epsilon_{tsarm}=1e-10$.  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 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} & \multicolumn{2}{c|}{TSARM 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   the   following  we   describe   the   applications   of  PETSc   we   have
+experimented. 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  equals to  4  and  4  extra-diagonals
+  representing the neighbors in each directions  is equal to -1. This example is
+  used  in many  physical phenomena  , for  exemple, heat  and fluid  flow, wave
+  propagation...
+\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, it is called $\alpha$.
+\end{itemize}
+For more technical details on  these applications, interested reader are invited
+to  read the  codes available  in the  PETSc sources.   Those problem  have been
+chosen because they  are scalable with many cores. We  have tested other problem
+but they are not scalable with many cores.
+
+
+
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & precond   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM 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{figure}
+\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}}
+\label{fig:01}
+\end{figure}
+
+
+
+
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM 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*}
+
+
+
+
+
+\begin{table*}
+\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|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM 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*}
+

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


@@ -557,11 +926,19 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

+\label{sec:06}

 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 
 
 %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
 
 
 % conference papers do not normally have an appendix
 


@@ -571,10 +948,10 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section*{Acknowledgment}
 %%%*********************************************************
 %%%*********************************************************
 \section*{Acknowledgment}

-%The authors would like to thank...
-%more thanks here
-%%%*********************************************************
-%%%*********************************************************
+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
 
 
 % trigger a \newpage just before the given reference


@@ -592,20 +969,23 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %à
 % 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/
 % 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}
+\bibliographystyle{IEEEtran}

 % argument is your BibTeX string definitions and bibliography database(s)
 % argument is your BibTeX string definitions and bibliography database(s)

-%\bibliography{IEEEabrv,../bib/paper}
+\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)
 %
 % <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}
+%% \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{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
-%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+%% \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.

 
 

-\end{thebibliography}
+%% \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}