X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/12080b70975c1166b9faba747f271766a8c8dd75..cce18db44812b9a67dd4d30ca37ea74fd46f16b1:/paper.tex

diff --git a/paper.tex b/paper.tex
index e540970..cf61a9e 100644
--- a/paper.tex
+++ b/paper.tex
@@ -369,7 +369,7 @@
 %
 % paper title
 % can use linebreaks \\ within to get better formatting as desired
-\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
+\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}
 
 
 
@@ -380,8 +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}
@@ -425,20 +425,21 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{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.
+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}
 
 
@@ -546,42 +547,46 @@ 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 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
+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 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.
+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-square  residual   minimization,  while  Section~\ref{sec:04}  provides
+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
@@ -596,7 +601,13 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 
-%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+%GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers. 
+
+%The next two chapters explore a few methods which are considered currently to be among the most important iterative techniques available for solving large linear systems. These techniques are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these techniques approximate A â1 b by p(A)b, where p is a âgoodâ polynomial. This chapter covers methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers methods based on Lanczos biorthogonalization.
+
+%Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.
+
+%Preconditioned Krylov-subspace iterations are a key ingredient in many modern linear solvers, including in solvers that employ support preconditioners. 
 %%%*********************************************************
 %%%*********************************************************
 
@@ -604,7 +615,7 @@ is summarized while intended perspectives are provided.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Two-stage iteration with least-square residuals minimization algorithm}
+\section{Two-stage iteration with least-squares residuals minimization algorithm}
 \label{sec:03}
 A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
@@ -615,28 +626,29 @@ 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 
+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-square method  such as
+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.
 
@@ -647,52 +659,55 @@ appropriate than a single direct method in a parallel context.
 \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$<\epsilon_{tsirm}$)} \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 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-square 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 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.
+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 $\epsilon_{tsirm}$: 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.
+\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
+\item $\epsilon_{ls}$: the threshold used to stop the least-squares method.
 \end{itemize}
 
 
-The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
-parts. More  precisely, except  the least-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 takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
@@ -702,19 +717,21 @@ less the same principle but it takes more place, so we briefly explain the paral
 \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}
@@ -732,17 +749,79 @@ these operations are easy to implement in PETSc or similar environment.
 
 \section{Convergence results}
 \label{sec:04}
-Let us recall the following result, see~\cite{Saad86}.
+
+
+We can now claim that,
 \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:
+\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_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{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 $A$.
+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_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{k/2} \|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 such assumptions regarding $A$.
+These 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 $r_1$ that follows the inductive hypothesis due, to the results recalled above.
+
+Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ in the definite positive one.
+We will show that the statement holds too for $r_k$. Two situations can occur:
+\begin{itemize}
+\item If $k \not\equiv 0 ~(\textrm{mod}\ m)$, then the TSIRM algorithm consists in executing GMRES once. In that case and by using the inductive hypothesis, we obtain either $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ if $A$ is positive, or $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite case.
+\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies:
+\begin{itemize}
+\item $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, 
+\item $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite one,
+\end{itemize}
+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||, \textrm{ if $A$ is positive,}\\
+& \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|, \textrm{ if $A$ is}\\
+& \textrm{positive definite,} 
+\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.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -750,11 +829,13 @@ the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \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.
+In order to see the behavior of the proposal when considering only one processor, a first
+comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented. 
+Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in 
+Table~\ref{tab:01}.  These latter, which are real-world applications matrices, 
+have been extracted 
+ from   the  Davis  collection,   University  of
+Florida~\cite{Dav97}.
 
 \begin{table}[htbp]
 \begin{center}
@@ -774,8 +855,9 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
-
-The following  parameters have been chosen  for our experiments.   As by default
+Chosen parameters are detailed below.
+%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:
@@ -787,13 +869,14 @@ 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.
+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}[htbp]
@@ -814,7 +897,7 @@ 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}
@@ -823,29 +906,48 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
-In order to perform larger  experiments, we have tested some example applications
+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
+  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...
+  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.
+
+
+In  many situations, using  preconditioners is  essential in  order to  find the
+solution of a linear system.  There are many preconditioners available in PETSc.
+For parallel applications all  the preconditioners based on matrix factorization
+are  not  available. In  our  experiments, we  have  tested  different kinds  of
+preconditioners, however  as it is  not the subject  of this paper, we  will not
+present results with many preconditioners. In  practise, we have chosen to use a
+multigrid (mg)  and successive  over-relaxation (sor). For  more details  on the
+preconditioner in PETSc please consult~\cite{petsc-web-page}.
 
-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|} 
@@ -865,27 +967,26 @@ In the following larger experiments are described on two large scale architectur
 \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.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
 \label{tab:03}
 \end{center}
 \end{table*}
 
 Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
-example ex15  of PETSc on the  Juqueen architecture. 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
+example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
+are  studied ranging  from  2,048  up-to 16,383 with the two preconditioners {\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 this Table, we  can notice that TSIRM is always faster  than FGMRES. The last
+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  alsways  faster than
-FGMRES. For this example, the  multigrid preconditionner is faster than SOR. The
+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
@@ -897,16 +998,16 @@ 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: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,
+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.
@@ -921,7 +1022,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \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 \\ 
+  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 \\
@@ -934,23 +1035,55 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \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
-
+In  Table~\ref{tab:04},  some  experiments   with  example  ex54  on  the  Curie
+architecture are reported.  For this  application, we fixed $\alpha=0.6$.  As it
+can be seen in that Table, the size of the problem has a strong influence on the
+number of iterations to reach the  convergence. That is why we have preferred to
+change the threshold.  If we set  it to $1e-3$ as with the previous application,
+only one iteration is necessray  to reach the convergence. So Table~\ref{tab:04}
+shows the results  of differents executions with differents  number of cores and
+differents thresholds. As  with the previous example, we  can observe that TSIRM
+is faster than FGMRES. The ratio greatly depends on the number of iterations for
+FMGRES to reach the threshold. The greater the number of iterations to reach the
+convergence is, the  better the ratio between our algorithm  and FMGRES is. This
+experiment is  also a  weak scaling with  approximately $25,000$  components per
+core. It can also  be observed that the difference between CGLS  and LSQR is not
+significant. Both can be good but it seems not possible to know in advance which
+one will be the best.
+
+Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
+Curie  architecture. So  in  this case,  the  number of  unknownws  is fixed  to
+$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
+of two.  The  threshold is fixed to $5e-5$ and only  the $mg$ preconditioner has
+been tested. Here  again we can see that TSIRM is  faster that FGMRES. Efficiecy
+of each algorithms is reported. It  can be noticed that FGMRES is more efficient
+than TSIRM except with $8,192$ cores and that its efficiency is greater that one
+whereas the  efficiency of TSIRM is  lower than one. Nevertheless,  the ratio of
+TSIRM  with any  version  of the  least-squares  method is  always faster.  With
+$8,192$ cores when the number of iterations is far more important for FGMRES, we
+can see that it is only slightly more important for TSIRM.
+
+In  Figure~\ref{fig:02}  we report  the  number  of  iterations per  second  for
+experiments  reported in  Table~\ref{tab:05}.  This Figure  highlights that  the
+number of iterations per  seconds is more of less the same  for FGMRES and TSIRM
+with a little advantage for FGMRES. It  can be explained by the fact that, as we
+have previously explained, that the iterations of the least-sqaure steps are not
+taken into account with TSIRM.
 
 \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} \\ 
+  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\\
@@ -960,11 +1093,38 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 \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}
+
+
+Concerning the  experiments some  other remarks are  interesting.
+\begin{itemize}
+\item We  can tested other examples of  PETSc (ex29, ex45, ex49).  For all these
+  examples,  we also obtained  similar gain  between GMRES  and TSIRM  but those
+  examples are  not scalable with many  cores. In general, we  had some problems
+  with more than $4,096$ cores.
+\item We have tested many iterative  solvers available in PETSc.  In fast, it is
+  possible to use most of them with TSIRM. From our point of view, the condition
+  to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
+  feature. More  precisely, the solver must  support to be  stoped and restarted
+  without decrease its  converge. That is why  with GMRES we stop it  when it is
+  naturraly  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate
+  Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
+  so they  are not  efficient.  They  will converge with  TSIRM but  not quickly
+  because if  we compare  a normal CG  with a CG  for which  we stop it  each 16
+  iterations  for example,  the  normal CG  will  be for  more efficient.   Some
+  restarted CG  or CG variant versions exist  and may be interested  to study in
+  future works.
+\end{itemize}
 %%%*********************************************************
 %%%*********************************************************
 
@@ -978,13 +1138,23 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 %%%*********************************************************
 %%%*********************************************************
 
+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. This  would be very interesting because it would  allow us to test
+all the non-linear  examples and compare our algorithm  with the other algorithm
+implemented in PETSc.
 
 
 % conference papers do not normally have an appendix
@@ -996,7 +1166,7 @@ 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.
 
 
@@ -1039,5 +1209,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
-
-
