X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/176107a654cef5a1e3e376218e5da656e515453b..9300a8195f1fd0aba96819dbd29244ca8bc90a33:/paper.tex

diff --git a/paper.tex b/paper.tex
index 18340f6..812326f 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:
@@ -348,7 +348,8 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
-
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{amsmath}
@@ -362,17 +363,14 @@
 \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{TSARM: A Two-Stage Algorithm with least-square Residual Minimization 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}
+
 
 
 
@@ -382,8 +380,8 @@
 % 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}}
-\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, 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-Comt\'e, France\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
 Email: lilia.ziane@inria.fr}
@@ -427,20 +425,21 @@ Email: lilia.ziane@inria.fr}
 
 
 \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}
-Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %Ã  voir... 
+Iterative Krylov methods; sparse linear systems; two stage iteration; least-squares residual minimization; PETSc
 \end{IEEEkeywords}
 
 
@@ -548,45 +547,51 @@ 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)
 
-Iterative methods  became more attractive than  direct ones to  solve very large
-sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
-context,  with  thousands  of  cores,  and  require  less  memory  and  arithmetic
-operations than direct  methods. A number of iterative  methods are proposed and
-adapted by many researchers and the increased need for solving very large sparse
-linear  systems  triggered the  development  of  efficient iterative  techniques
-suitable for the parallel processing.
-
-Most of the successful iterative methods currently available are based on Krylov
-subspaces which  consist in forming a  basis of a sequence  of successive matrix
-powers times an initial vector for example the residual. These methods are based
-on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
-systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
-Gradient method and GMRES method (generalized minimal residual).
-
-However,  iterative  methods suffer  from scalability  problems  on parallel
-computing  platforms  with many  processors  due  to  their need  for  reduction
-operations    and   collective    communications   to    perform   matrix-vector
+Iterative methods have recently become more attractive than direct ones to solve
+very large sparse  linear systems\cite{Saad2003}.  They are more  efficient in a
+parallel context,  supporting thousands of  cores, and they require  less memory
+and  arithmetic operations than  direct methods~\cite{bahicontascoutu}.  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   achieve  matrix-vector
 multiplications. The  communications on large  clusters with thousands  of cores
-and  large  sizes of  messages  can  significantly  affect the  performances  of
-iterative methods. In practice, Krylov subspace iteration methods are often used
-with preconditioners in order to increase their convergence and accelerate their
-performances.  However, most  of the  good preconditioners  are not  scalable on
-large clusters.
-
-In this  paper we propose a  two-stage algorithm based on  two nested iterations
-called inner-outer  iterations.  This algorithm  consists in solving  the sparse
-linear system iteratively  with a small number of  inner iterations and restarts
-the outer  step with a  new solution minimizing  some error functions  over some
-previous residuals. This algorithm is iterative and easy to parallelize on large
-clusters   and  the   minimization  technique   improves  its   convergence  and
-performances.
-
-The present paper is organized  as follows. In Section~\ref{sec:02} some related
-works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
-a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
-convergence  results  on this  method.   Section~\ref{sec:05}  shows  some  experimental
-results  obtained on large  clusters of  our algorithm  using routines  of PETSc
-toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
+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. For  further information  on two-stage
+iteration      methods,     interested      readers      are     invited      to
+consult~\cite{Nichols:1973:CTS}. Two-stage algorithms are easy to parallelize on
+large clusters.  Furthermore,  the least-squares 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.
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -604,113 +609,123 @@ 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
-nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
-$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
-inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
-points of our solver are given in Algorithm~\ref{algo:01}.
-
-In order to accelerate the convergence, the outer iteration periodically applies
-a least-square minimization  on the residuals computed by  the inner solver. The
-inner solver is based on a Krylov method which does not require to be changed.
-
-At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
-iterations, using an iterative method restarting with the previous solution. For
-example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
-inner solver. The current solution of the Krylov method is saved inside a matrix
-$S$ composed of successive solutions computed by the inner iteration.
-
-Periodically, every $s$ iterations, the minimization step is applied in order to
-compute a new  solution $x$. For that, the previous  residuals are computed with
-$(b-AS)$. The minimization of the residuals is obtained by 
+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 last obtained approximation.  GMRES method~\cite{Saad86}, or any of its
+variants, can potentially be used  as inner solver. The current approximation of
+the Krylov  method is then  stored inside  a $n \times  s$ matrix $S$,  which is
+composed by  the $s$  last solutions  that have been  computed during  the inner
+iterations phase.   In the remainder,  the $i$-th column  vector of $S$  will be
+denoted by $S_i$.
+
+At each $s$ iterations, another kind of 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}
-with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+with $R=AS$. The new solution $x$ is then 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.
 
 
-In  practice, $R$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
-$s\ll n$.   In order  to minimize~(\ref{eq:01}), a  least-square method  such as
-CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Those  methods are more
-appropriate than a direct method in a parallel context.
 
 \begin{algorithm}[t]
-\caption{TSARM}
+\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 (error$<\epsilon_{tsarm}$)} \label{algo:conv}
-    \State  $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$   \label{algo:solve}
-    \State retrieve error
-    \State $S_{k~mod~s}=x^k$ \label{algo:store}
-    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$}
+  \State Set the initial guess $x_0$
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
+    \State  $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$   \label{algo:solve}
+    \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column (k mod s) of S}
+    \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
       \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}
 \label{algo:01}
 \end{algorithm}
 
-Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
-iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
-called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
-equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
-threshold must be specified for the  solver. In practice, this threshold must be
-much  smaller  than the  convergence  threshold  of  the TSARM  algorithm  (i.e.
-$\epsilon_{tsarm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
-solution  $x_k$  into the  column  $k~  mod~Â s$ of  the  matrix  $S$. After  the
-minimization, the matrix $S$ is reused with the new values of the residuals.  To
-solve the minimization problem, an  iterative method is used. Two parameters are
-required for that: the maximum number of iteration and the threshold to stop the
-method.
-
-To summarize, the important parameters of TSARM are:
+Algorithm~\ref{algo:01} summarizes  the principle  of the proposed  method.  The
+outer iteration is inside the \emph{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 equal to the restart  number in 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.}, $\epsilon_{tsirm}$).  We also  consider that
+after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
+which is defined by $||Ax^k-b||_2$.
+
+  Line~\ref{algo:store},
+$S_{k \mod  s}=x^k$ consists in  copying the solution  $x_k$ into the  column $k
+\mod s$ of $S$.   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 $\epsilon_{tsarm}$ the threshold to stop the TSARM method
-\item $max\_iter_{kryl}$ the maximum number of iterations for the krylov method
-\item $s$ the number of outer iterations before applying the minimization step
-\item $max\_iter_{ls}$ the maximum number of iterations for the iterative least-square method
-\item $\epsilon_{ls}$ the threshold to stop the least-square method
+\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  TSARM  relies   on  the  parallelization  of  all  its
-parts. More  precisely, except  the least-square step,  all the other  parts are
+The  parallelization  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-square minimization, CGLS and LSQR.
+columns 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 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}
-  \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}
@@ -718,7 +733,7 @@ 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
-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.
 
 
@@ -730,30 +745,94 @@ these operations are easy to implement in PETSc or similar environment.
 \label{sec:04}
 
 
+We can now claim that,
+\begin{proposition}
+\label{prop:saad}
+If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, 
+let $r_k$ be the
+$k$-th residue of TSIRM, then
+we have the following boundaries:
+\begin{itemize}
+\item when $A$ is positive:
+\begin{equation}
+||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
+\end{equation}
+where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$;
+\item when $A$ is positive definite:
+\begin{equation}
+\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|.
+\end{equation}
+\end{itemize}
+%In the general case, where A is not positive definite, we have
+%$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .$
+\end{proposition}
+
+\begin{proof}
+Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows:
+\begin{equation*}
+\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .
+\end{equation*}
+Additionally, when $A$ 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 $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves 
+the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
+Such well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.
+
+We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
+$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite.
+
+The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.
+
+Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
+We will show that the statement holds too for $r_k$. Two situations can occur:
+\begin{itemize}
+\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis.
+\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
+Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+
+$\begin{array}{ll}
+& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\
+& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
+& \leqslant ||b-Ax_{k}||_2\\
+& = ||r_k||_2\\
+& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||,
+\end{array}$
+\end{itemize}
+which concludes the induction and the proof.
+\end{proof}
 
+We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
+than the one of the GMRES method.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Experiments using petsc}
+\section{Experiments using PETSc}
 \label{sec:05}
 
 
 In order to see the influence of our algorithm with only one processor, we first
-show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
-table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
-characteristics. For all  the matrices, the name, the field,  the number of rows
-and the number of nonzero elements is given.
-
-\begin{table*}
+show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01},
+we  show the  matrices we  have  used and  some of  them characteristics.  Those
+matrices  are   chosen  from   the  Davis  collection   of  the   University  of
+Florida~\cite{Dav97}. They are matrices arising in real-world applications.  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     & 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
 
@@ -761,36 +840,36 @@ 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}
-\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 GMRES every 30 iterations, $max\_iter_{kryl}=30$).  $s$ is set to 8. CGLS is
+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
-$\epsilon_{tsarm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
+$\epsilon_{tsirm}=1e-10$.  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 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
+GRMES or  FGMRES (Flexible  GMRES)~\cite{Saad:1993} 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}
+\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|}{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
 
@@ -803,7 +882,7 @@ 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.}
+\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
 \label{tab:02}
 \end{center}
 \end{table}
@@ -812,33 +891,46 @@ 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}
 \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...
+  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, it is called $\alpha$.
+  coefficient in embedded circle 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.
+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.  Both  these architectures  are supercomputer
+composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
+are respectively hosted  by GENCI in France and  JÃ¼lich Supercomputing Centre in
+Germany. They belongs with other similar architectures of the PRACE initiative (
+Partnership  for Advanced  Computing in  Europe)  which aims  at proposing  high
+performance supercomputing architecture to enhance research in Europe. The Curie
+architecture is composed of Intel E5-2680  processors at 2.7 GHz with 2Gb memory
+by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with
+1Gb memory per  core. Both those architecture are equiped  with a dedicated high
+speed network.
 
 
 
-\begin{table*}
+{\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|}{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 \\
@@ -852,29 +944,63 @@ but they are not scalable with many cores.
 \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 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*}
 
-
-\begin{figure}
+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 core  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}}
+\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:03}  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*}
+\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|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
+  nb. cores & threshold   & \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      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
@@ -887,23 +1013,23 @@ but they are not scalable with many cores.
 \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.}
+\caption{Comparison of FGMRES  and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 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*}
+\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|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+  nb. cores   & \multicolumn{2}{c|}{FGMRES} & \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
+                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & FGMRES & 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\\
@@ -913,11 +1039,18 @@ but they are not scalable with many cores.
 \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.}
+\caption{Comparison of FGMRES  and TSIRM with FGMRES 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, threshold 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}
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -931,13 +1064,22 @@ but they are not scalable with many cores.
 %%%*********************************************************
 %%%*********************************************************
 
+A novel two-stage iterative  algorithm has been proposed in this article,
+in order to accelerate the convergence Krylov iterative  methods.
+Our TSIRM proposal acts as a merger between Krylov based solvers and
+a least-squares minimization step.
+The convergence of the method has been proven in some situations, while 
+experiments up to 16,394 cores have been led to verify that TSIRM runs
+5 or  7 times  faster than GMRES.
 
-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
+
+For future work, the authors' intention is to investigate 
+other kinds of matrices, problems, and inner solvers. The 
+influence of all parameters must be tested too, while 
+other methods to minimize the residuals must be regarded.
+The number of outer iterations to minimize should become 
+adaptative to improve the overall performances of the proposal.
+Finally, this solver will be implemented inside PETSc.
 
 
 % conference papers do not normally have an appendix
@@ -949,7 +1091,7 @@ future plan : \\
 %%%*********************************************************
 \section*{Acknowledgment}
 This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
-ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resource
+ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resources
 Curie and Juqueen respectively based in France and Germany.
 
 
@@ -992,5 +1134,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
-
-
