10-10-2014 03

[GMRES2stage.git] / paper.tex

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@@ -241,7 +241,7 @@
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+\usepackage{eqparbox}

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@@ -348,11 +348,14 @@
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-
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}

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+\usepackage{multirow}
+\usepackage{graphicx}

 
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-
+\newtheorem{proposition}{Proposition}

 
 \begin{document}
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 % paper title
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 \begin{document}
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-\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{???}
+\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
+
+
+
+

 
 
 % 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\\
+\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}
 }
 

+
+

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@@ -427,11 +425,20 @@ 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 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 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}
 \end{abstract}
 
 \begin{IEEEkeywords}

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

 \end{IEEEkeywords}
 
 
 \end{IEEEkeywords}
 
 


@@ -538,11 +545,47 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
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-Iterative methods are become more attractive than direct ones to solve 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.

 
 

-The most successful iterative methods currently available are those based on the Krylov subspace which consists in forming a basis of a sequence of successive matrix powers times the initial residual. These methods are based on orthogonality of vectors of the Krylov subspace basis to solve generalized linear systems. The most well-known iterative Krylov subspace methods are Conjugate Gradient method and GMRES method (generalized minimal residual).
-
-However, the 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. 
+Iterative methods have recently become more attractive than  direct ones to  solve very large
+sparse  linear systems.  They are more  efficient  in a  parallel
+context,  supporting  thousands  of  cores,  and they require  less  memory  and  arithmetic
+operations than direct  methods. This is why new iterative  methods are frequently 
+proposed or adapted by researchers, and the increasing need to solve very large sparse
+linear  systems  has triggered the  development  of such efficient iterative  techniques
+suitable for parallel processing.
+
+Most of the successful iterative methods currently available are based on so-called ``Krylov
+subspaces''. They  consist in forming a  basis of successive matrix
+powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov  subspace  basis in order to solve  linear
+systems.  The  most known iterative  Krylov subspace methods  are conjugate
+gradient and GMRES ones (Generalized Minimal RESidual).
+
+
+However,  iterative  methods suffer  from scalability  problems  on parallel
+computing  platforms  with many  processors, due  to  their need  of  reduction
+operations, and to  collective    communications   to  achive   matrix-vector
+multiplications. The  communications on large  clusters with thousands  of cores
+and  large  sizes of  messages  can  significantly  affect the  performances  of these
+iterative methods. As a consequence, Krylov subspace iteration methods are often used
+with preconditioners in practice, to increase their convergence and accelerate their
+performances.  However, most  of the  good preconditioners  are not  scalable on
+large clusters.
+
+In this research work, a two-stage algorithm based on  two nested iterations
+called inner-outer  iterations is proposed.  This algorithm  consists in solving  the sparse
+linear system iteratively  with a small number of  inner iterations, and restarting
+the outer  step with a  new solution minimizing  some error functions  over some
+previous residuals. This algorithm is iterative and easy to parallelize on large
+clusters. Furthermore,  the   minimization  technique   improves  its   convergence  and
+performances.
+
+The present  article is  organized as follows.   Related works are  presented in
+Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
+a  least-square  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.

 
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@@ -552,6 +595,7 @@ However, the iterative methods suffer from scalability problems on parallel comp
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 \section{Related works}
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 \section{Related works}

+\label{sec:02} 

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@@ -560,51 +604,367 @@ However, the iterative methods suffer from scalability problems on parallel comp
 
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-\section{A Krylov two-stage algorithm}
-We propose a two-stage algorithm 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 function over a Krylov sub-space~\cite{saad96}. At every iteration, the sparse linear system $Ax=b$ is solved iteratively with an iterative method as GMRES method~\cite{saad86} and the Krylov sub-space 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
+\section{Two-stage iteration with least-square residuals minimization algorithm}
+\label{sec:03}
+A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
+form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
+nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector,   and
+$b\in\mathbb{R}^n$ is  the right-hand side.  As explained previously, 
+the algorithm is implemented  as an
+inner-outer iteration  solver based  on iterative Krylov  methods. The  main 
+key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
+It can be summarized as follows: the
+inner solver is a Krylov based one. In order to accelerate its convergence, the 
+outer solver periodically applies a least-square 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 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}
 \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}) as CGLS~\cite{hestenes52} or LSQR~\cite{paige82} methods which is 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-square 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]
 
 \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 $x^0$
 \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 Add vector $x^k$ to Krylov subspace basis $S$
-    \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 Reinitialize Krylov subspace basis $S$
+  \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 Solve least-square 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}
     \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 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-square method;
+\item $\epsilon_{ls}$: the threshold used to stop the least-square method.
+\end{itemize}
+
+
+The  parallelisation  of  TSIRM  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 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 $r=b-Ax$
+  \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:  dot product, norm, multiplication  and addition on  vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
+
+

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+\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}
+

 
 
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-\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 are given.
+
+\begin{table}[htbp]
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|} 
+\hline
+Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
+crashbasis         & Optimization      & 160,000  &  1,750,416  \\
+parabolic\_fem     & Comput. fluid dynamics  & 525,825 & 2,100,225 \\
+epb3               & Thermal problem   & 84,617  & 463,625  \\
+atmosmodj          & Comput. fluid dynamics  & 1,270,432 & 8,814,880 \\
+bfwa398            & Electromagnetics pb & 398 & 3,678 \\
+torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\end{center}
+\end{table}
+
+The following  parameters have been chosen  for our experiments.   As by default
+the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
+the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$).  $s$ is set to 8. CGLS is
+chosen  to minimize  the least-squares  problem with  the  following parameters:
+$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to
+$\epsilon_{tsirm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+
+
+In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear
+systems obtained with the previous matrices  with a GMRES variant and with out 2
+stage algorithm are  given. In the second column, it can  be noticed that either
+gmres or fgmres is used to  solve the linear system.  According to the matrices,
+different  preconditioner is used.   With TSIRM,  the same  solver and  the same
+preconditionner are used.  This Table shows that TSIRM can drastically reduce the
+number of iterations to reach the  convergence when the number of iterations for
+the normal GMRES is more or less  greater than 500. In fact this also depends on
+tow  parameters: the  number  of iterations  to  stop GMRES  and  the number  of
+iterations to perform the minimization.
+
+
+\begin{table}[htbp]
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|} 
+\hline
+
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\ 
+\cline{3-6}
+       &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
+
+crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
+parabolic\_fem     & gmres / ilu           & 1009.94   & 7573 & 401.52 & 2970 \\
+epb3               & fgmres / sor             &  8.67     & 600  &  8.21 & 540  \\
+atmosmodj          &  fgmres / sor & 104.23  & 451 & 88.97 & 366  \\
+bfwa398            & gmres / none  & 1.42 & 9612 & 0.28 & 1650 \\
+torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of (F)GMRES and 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  is equal to -1. This example is
+  used  in many  physical phenomena, for  example, 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.
+
+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 preconditioner (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. Differents  number of cores
+are  studied rangin  from  2,048  upto 16,383.   Two  preconditioners have  been
+tested.   For those experiments,  the number  of components  (or unknown  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 this Table, 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  alsways  faster than
+FGMRES. For this example, the  multigrid preconditionner 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}}
+\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 decrease with TSIRM with
+both preconditioner. This  can be explained by the fact that  when the number of
+core increases the time for the 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*}
+

 %%%*********************************************************
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@@ -613,11 +973,19 @@ such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\l
 %%%*********************************************************
 %%%*********************************************************
 \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
+

 
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 % conference papers do not normally have an appendix
 


@@ -627,10 +995,10 @@ such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\l
 %%%*********************************************************
 %%%*********************************************************
 \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.
+

 
 
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@@ -648,23 +1016,23 @@ such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\l
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-%\bibliographystyle{IEEEtran}
+\bibliographystyle{IEEEtran}

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-%\bibliography{IEEEabrv,../bib/paper}
+\bibliography{biblio}

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-\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}