Je sais pas trop

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 8ffd3879c0596f27d8124b1e633842bd3728b0d4..e512018b8fd2b0bf94de9dbf5cdd4b83a8264a8f 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -241,7 +241,7 @@
 % quality.
 
 
 % 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:
 % 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}
 
 
 \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{algorithm}
 \usepackage{algpseudocode}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{multirow}

+\usepackage{graphicx}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -361,17 +363,15 @@
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 

-
+\newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}

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

 
 
 
 
 
 


@@ -381,7 +381,7 @@
 % use a multiple column layout for up to two different
 % affiliations
 
 % use a multiple column layout for up to two different
 % affiliations
 

-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2} and Christophe Guyeux\IEEEauthorrefmark{1}}
+\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\\
 \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\\


@@ -426,16 +426,17 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-In  this paper  we propose  a  two stage  iterative method  which increases  the
-convergence of Krylov iterative methods,  typically those of GMRES variants. The
-principle of  our approach  is to  build an external  iteration over  the Krylov
-method  and to  save  the current  residual  frequently (for  example, for  each
-restart of GMRES). Then after a given number of outer iterations, a minimization
-step  is applied  on the  matrix composed  of the  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.
+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}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -547,46 +548,47 @@ Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
 
 % 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).
+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
 
 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
 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.
 
 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
+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
 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
+clusters. Furthermore,  the   minimization  technique   improves  its   convergence  and

 performances.
 
 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.   In Section~\ref{sec:05},  parallization
-details  of  TSARM  are  given.  Section~\ref{sec:06}  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.
+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.
+

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


@@ -604,28 +606,30 @@ toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 
 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************

-\section{Two-stage algorithm with least-square residuals minimization}
+\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
 \label{sec:03}
 A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square

-nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
-$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
-inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
-points of our solver are given in Algorithm~\ref{algo:01}.
-
-In order to accelerate the convergence, the outer iteration periodically applies
-a least-square minimization  on the residuals computed by  the inner solver. The
-inner solver is a Krylov based solver which does not required to be changed.
-
-At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
-iterations, using an iterative method restarting with the previous solution. For
-example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
-inner solver. The current solution of the Krylov method is saved inside a matrix
-$S$ composed of successive solutions computed by the inner iteration.
-
-Periodically, every $s$ iterations, the minimization step is applied in order to
-compute a new  solution $x$. For that, the previous  residuals are computed with
-$(b-AS)$. The minimization of the residuals is obtained by 
+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}
 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}


@@ -633,25 +637,27 @@ $(b-AS)$. The minimization of the residuals is obtained by
 with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
 
 
 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.
+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]
 
 \begin{algorithm}[t]

-\caption{TSARM}
+\caption{TSIRM}

 \begin{algorithmic}[1]
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
 \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$<\epsilon_{kryl}$)} \label{algo:conv}
-    \State  $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$   \label{algo:solve}
+  \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 retrieve error

-    \State $S_{k~mod~s}=x^k$ \label{algo:store}
-    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
+    \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 $R=AS$ \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}
+            \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
+      \State $x_k=S\alpha$  \Comment{compute new solution}

     \EndIf
   \EndFor
 \end{algorithmic}
     \EndIf
   \EndFor
 \end{algorithmic}


@@ -663,66 +669,56 @@ 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
 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.
-$\epsilon$).  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
+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
 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
+required for that: the maximum number of iterations and the threshold to stop the

 method.
 
 method.
 

-To summarize, the important parameters of TSARM are:
+Let us summarize the most important parameters of TSIRM:

 \begin{itemize}
 \begin{itemize}

-\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov 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
+\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}
 
 \end{itemize}
 

-%%%*********************************************************
-%%%*********************************************************
-
-\section{Convergence results}
-\label{sec:04}

 
 

-
-
-%%%*********************************************************
-%%%*********************************************************
-\section{Parallelization}
-\label{sec:05}
-
-The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
-parts. More  precisely, except  the least-square step,  all the other  parts are
+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
 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.
+interesting to solve the least-squares minimization, CGLS and LSQR.

 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or
 
 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.
+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}
 
 \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 $\beta=g/g_{old}$
+  \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}
   \EndFor
 \end{algorithmic}
 \label{algo:02}


@@ -730,31 +726,56 @@ less the same principle but it take more place, so we briefly explain the parall
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some

-classical operations:  dots, norm, multiplication  and addition on  vectors. All
+classical operations:  dot product, norm, multiplication  and addition on  vectors. All

 these operations are easy to implement in PETSc or similar environment.
 
 these operations are easy to implement in PETSc or similar environment.
 

+
+

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

-\section{Experiments using petsc}
-\label{sec:06}
+
+\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}
+
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+\end{proposition}
+
+\begin{proof}
+\end{proof}
+
+%%%*********************************************************
+%%%*********************************************************
+\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
 
 
 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
 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  \\
 \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  \\
 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
 
 torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \hline
 


@@ -762,16 +783,14 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}

-\end{table*}
+\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 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.
 
 
 Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
 
 


@@ -779,21 +798,20 @@ 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,
 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
 
 \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} \\ 
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\ 

 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 


@@ -815,28 +833,34 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
 
 
 

-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:
+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}
 \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 $\alpha$.
+\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}
 \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.
+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*}
+\begin{table*}[htbp]

 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 \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 \\ 
+  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 \\
 \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 \\


@@ -850,18 +874,63 @@ but they are not scalable with many cores.
 \hline
 
 \end{tabular}
 \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.}
+\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*}
 
 \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{table*}
+\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 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
 
 \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 \\ 
+  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 \\
 \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 \\


@@ -880,15 +949,15 @@ but they are not scalable with many cores.
 \end{table*}
 
 
 \end{table*}
 
 

+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.

 
 
 
 

-
-\begin{table*}
+\begin{table*}[htbp]

 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 \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} \\ 
+  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     \\
 \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     \\


@@ -905,6 +974,13 @@ but they are not scalable with many cores.
 \end{center}
 \end{table*}
 
 \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}
+

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


@@ -913,7 +989,7 @@ but they are not scalable with many cores.
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

-\label{sec:07}
+\label{sec:06}

 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************


@@ -980,4 +1056,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}
 

-