X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/16f0ca498711e0af6a2de8200c2fbee0e5f8810e..8aeef74d04b37c2601749676e053a538ba3785cd:/paper.tex

diff --git a/paper.tex b/paper.tex
index 35d9ed7..f1becbd 100644
--- a/paper.tex
+++ b/paper.tex
@@ -241,7 +241,7 @@
 % quality.
 
 
-%\usepackage{eqparbox}
+\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:
@@ -348,12 +348,14 @@
 \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]}
@@ -361,37 +363,32 @@
 \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{A Krylov two-stage algorithm to solve large sparse linear systems}
-%oÃ¹
-%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
-%oÃ¹
-%\title{???}
+\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}
-\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}
 }
 
+
+
 % 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
@@ -428,11 +425,21 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
-%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR 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; error minimization; PETSc; %Ã  voir... 
+Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %Ã  voir... 
 \end{IEEEkeywords}
 
 
@@ -539,42 +546,48 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %Ã 
 % 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).
+
+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  for  reduction
-operations    and   collective    communications   to    perform   matrix-vector
+computing  platforms  with many  processors, due  to  their need  of  reduction
+operations, and to  collective    communications   to  achive   matrix-vector
 multiplications. The  communications on large  clusters with thousands  of cores
-and  large  sizes of  messages  can  significantly  affect the  performances  of
-iterative methods. In practice, Krylov subspace iteration methods are often used
-with preconditioners in order to increase their convergence and accelerate their
+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  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.
+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.
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -592,95 +605,185 @@ of PETSc toolkit.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{A Krylov two-stage algorithm}
+\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.  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
+nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector,   and
+$b\in\mathbb{R}^n$ is  the right-hand side.  As explained previously, 
+the algorithm is implemented  as an
+inner-outer iteration  solver based  on iterative Krylov  methods. The  main 
+key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
+It can be summarized as follows: the
+inner solver is a Krylov based one. In order to accelerate its convergence, the 
+outer solver periodically applies a least-squares minimization  on the residuals computed by  the inner one. %Tsolver which does not required to be changed.
+
+At each outer iteration, the sparse linear system $Ax=b$ is partially 
+solved using only $m$
+iterations of an iterative method, this latter being initialized with the 
+best known approximation previously obtained. 
+GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
+inner solver. The current approximation of the Krylov method is then stored inside a matrix
+$S$ composed by the successive solutions that are computed during inner iterations.
+
+At each $s$ iterations, the minimization step is applied in order to
+compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
+the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by  
 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}
-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.
+with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+
+
+In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
+with $s\ll n$.   In order  to minimize~\eqref{eq:01}, a  least-squares method  such as
+CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Remark that these  methods are more
+appropriate than a single direct method in a parallel context.
+
+
 
 \begin{algorithm}[t]
-\caption{A Krylov two-stage algorithm}
+\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} \label{algo:conv}
-    \State Solve iteratively $Ax^k=b$  \label{algo:solve}
-    \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 $x^k=S\alpha$
+  \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}
 
-Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ into the
-column $k~ mod~Â s$ of the matrix  $S$. After the minimization, the matrix $S$ is
-reused with the new values of the residuals.
+Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
+iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
+called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
+equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
+threshold must be specified for the  solver. In practice, this threshold must be
+much  smaller  than the  convergence  threshold  of  the TSIRM  algorithm  (\emph{i.e.}
+$\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+solution  $x_k$  into the  column  $k~ mod~ s$ of  the  matrix  $S$. After  the
+minimization, the matrix $S$ is reused with the new values of the residuals.  To
+solve the minimization problem, an  iterative method is used. Two parameters are
+required for that: the maximum number of iterations and the threshold to stop the
+method.
+
+Let us summarize the most important parameters of TSIRM:
+\begin{itemize}
+\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
+\item $s$: the number of outer iterations before applying the minimization step;
+\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
+\item $\epsilon_{ls}$: the threshold used to stop the least-squares method.
+\end{itemize}
+
+
+The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
+parts. More  precisely, except  the least-squares step,  all the other  parts are
+obvious to  achieve out in parallel. In  order to develop a  parallel version of
+our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
+line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
+efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
+colums in  practice. As explained  previously, at least  two methods seem  to be
+interesting to solve the least-squares minimization, CGLS and LSQR.
+
+In the following  we remind the CGLS algorithm. The LSQR  method follows more or
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+
+\begin{algorithm}[t]
+\caption{CGLS}
+\begin{algorithmic}[1]
+  \Input $A$ (matrix), $b$ (right-hand side)
+  \Output $x$ (solution vector)\vspace{0.2cm}
+  \State Let $x_0$ be an initial approximation
+  \State $r_0=b-Ax_0$
+  \State $p_1=A^Tr_0$
+  \State $s_0=p_1$
+  \State $\gamma=||s_0||^2_2$
+  \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
+    \State $q_k=Ap_k$
+    \State $\alpha_k=\gamma/||q_k||^2_2$
+    \State $x_k=x_{k-1}+\alpha_kp_k$
+    \State $r_k=r_{k-1}-\alpha_kq_k$
+    \State $s_k=A^Tr_k$
+    \State $\gamma_{old}=\gamma$
+    \State $\gamma=||s_k||^2_2$
+    \State $\beta_k=\gamma/\gamma_{old}$
+    \State $p_{k+1}=s_k+\beta_kp_k$
+  \EndFor
+\end{algorithmic}
+\label{algo:02}
+\end{algorithm}
+
+
+In each iteration  of CGLS, there is two  matrix-vector multiplications and some
+classical operations:  dot product, norm, multiplication  and addition on  vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
+
 
 %%%*********************************************************
 %%%*********************************************************
 
+\section{Convergence results}
+\label{sec:04}
+Let us recall the following result, see~\cite{Saad86}.
+\begin{proposition}
+Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
+\begin{equation}
+||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+\end{equation}
+where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves 
+the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
+\end{proposition}
+
+<<<<<<< HEAD
 
+=======
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+\end{proposition}
+
+\begin{proof}
+Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
+$k$-th iterate of TSIRM.
+We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+
+Each step of the TSIRM algorithm 
+\end{proof}
+>>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Experiments using petsc}
-\label{sec: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
+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.
+and the number of nonzero elements are given.
 
-\begin{table}
+\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     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
+parabolic\_fem     & Comput. 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 \\
+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
 
@@ -692,12 +795,10 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 
 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)
+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.
 
 
@@ -705,21 +806,20 @@ 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.
+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}
+\begin{table}[htbp]
 \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} \\ 
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\ 
 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
@@ -728,17 +828,167 @@ 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 \\
+torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
-\caption{Comparison of GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
 \label{tab:02}
 \end{center}
 \end{table}
 
 
 
+
+
+In order to perform larger  experiments, we have tested some example applications
+of PETSc. Those  applications are available in the ksp part  which is suited for
+scalable linear equations solvers:
+\begin{itemize}
+\item ex15  is an example  which solves in  parallel an operator using  a finite
+  difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
+  representing the neighbors in each directions  are equal to -1. This example is
+  used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
+  propagation, etc.
+\item ex54 is another example based on 2D problem discretized with quadrilateral
+  finite elements. For this example, the user can define the scaling of material
+  coefficient in embedded circle called $\alpha$.
+\end{itemize}
+For more technical details on  these applications, interested readers are invited
+to  read the  codes available  in the  PETSc sources.   Those problems  have been
+chosen because they  are scalable with many cores which is not the case of other problems that we have tested.
+
+In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
+
+
+{\bf Description of preconditioners}\\
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & precond   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ 
+\cline{3-8}
+             &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
+  2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
+  2,048      & sor                   & 745.37   & 57,060    & 87.31  & 6,150   & 104.21 & 7,230  & 8.53 \\
+  4,096      & mg                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
+  4,096      & sor                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
+  8,192      & mg                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
+  8,192      & sor                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
+  16,384     & mg                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
+  16,384     & sor                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
+Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
+example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
+are  studied ranging  from  2,048  up-to 16,383.   Two  preconditioners have  been
+tested: {\it mg} and {\it sor}.   For those experiments,  the number  of components  (or unknowns  of the
+problems)  per processor  is fixed  to 25,000,  also called  weak  scaling. This
+number can seem relatively small. In fact, for some applications that need a lot
+of  memory, the  number of  components per  processor requires  sometimes  to be
+small.
+
+
+
+In Table~\ref{tab:03}, we  can notice that TSIRM is always faster  than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSIRM according to
+the minimization  procedure: CGLS or  LSQR. Even if  we have computed  the worst
+case  between CGLS  and LSQR,  it is  clear that  TSIRM is  always  faster than
+FGMRES. For this example, the  multigrid preconditioner is faster than SOR. The
+gain  between   TSIRM  and  FGMRES  is   more  or  less  similar   for  the  two
+preconditioners.  Looking at the number  of iterations to reach the convergence,
+it is  obvious that TSIRM allows the  reduction of the number  of iterations. It
+should be noticed  that for TSIRM, in those experiments,  only the iterations of
+the Krylov solver  are taken into account.  Iterations of CGLS  or LSQR were not
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
+corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
+\label{fig:01}
+\end{figure}
+
+
+In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
+Table~\ref{tab:01}  is  displayed.   It  can  be  noticed  that  the  number  of
+iterations per second of FMGRES is  constant whereas it decreases with TSIRM with
+both preconditioners. This  can be explained by the fact that  when the number of
+cores increases the time for the least-squares minimization step also increases but, generally,
+when  the number  of cores  increases,  the number  of iterations  to reach  the
+threshold also increases,  and, in that case, TSIRM is  more efficient to reduce
+the number of iterations. So, the overall benefit of using TSIRM is interesting.
+
+
+
+
+
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ 
+\cline{3-8}
+             &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
+  2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
+  2,048      & 6e-5                  & 194.01 & 30,270  & 35.50  &  5,430  & 27.74  & 4,350   & 6.99 \\
+  4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
+  4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
+  8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
+  8,192      & 5e-5                  & 785.04 & 109,590 & 76.07  &  10,470  & 69.42 & 9,030  & 11.30 \\
+  16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12),  time is expressed in seconds.}
+\label{tab:04}
+\end{center}
+\end{table*}
+
+
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
+
+
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+\cline{2-7} \cline{9-11}
+                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
+   512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
+   1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
+   2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\
+   4096             & 405.60    & 28,380 & 111.67 & 7,590  & 91.72  & 6,510 & 4.42  & 1.22   &  .79     &   .84 \\
+   8192             & 785.04   & 109,590 & 76.07  & 10,470 & 69.42 & 9,030  & 11.30 &   .32  &   .58    &  .56 \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5),  time is expressed in seconds.}
+\label{tab:05}
+\end{center}
+\end{table*}
+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
+\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
+\label{fig:02}
+\end{figure}
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -747,12 +997,19 @@ torso3             & fgmres/sor  & 37.70 & 565 & 34.97 & 510 \\
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
-\label{sec:05}
+\label{sec:06}
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 
 
+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- test the influence of all the parameters\\
+- adaptative number of outer iterations to minimize\\
+- other methods to minimize the residuals?\\
+- implement our solver inside PETSc
+
 
 % conference papers do not normally have an appendix
 
@@ -762,10 +1019,10 @@ torso3             & fgmres/sor  & 37.70 & 565 & 34.97 & 510 \\
 %%%*********************************************************
 %%%*********************************************************
 \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
@@ -783,23 +1040,23 @@ torso3             & fgmres/sor  & 37.70 & 565 & 34.97 & 510 \\
 % 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)
-%\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)
-\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{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{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{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}
+%% \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}
 
 
 
