X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/2872059db6be7a868af7890308cff5d0ca4db357..348f3de01d413c95e0ee5a51971ad04d01d9fec5:/paper.tex

diff --git a/paper.tex b/paper.tex
index 7957f80..3d4826c 100644
--- a/paper.tex
+++ b/paper.tex
@@ -601,13 +601,36 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 
-Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
-
-GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
-
-In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned.     
-
-% two-stage, communication avoiding 
+Krylov subspace iteration methods have increasingly become key
+techniques  for  solving  linear and nonlinear systems,  or  eigenvalue  problems,
+especially      since       the      increasing      development       of      
+preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of
+these methods is their generality, simplicity, and efficiency to solve systems of
+equations arising from very large and complex problems.
+
+GMRES is one of the most  widely used Krylov iterative method for solving sparse
+and large  linear systems. It  has been developed by  Saad \emph{et al.}~\cite{Saad86}  as a
+generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and
+indefinite symmetric problems too. In its original version called full GMRES, this algorithm
+minimizes the residual over the  current Krylov subspace until convergence in at
+most $n$ iterations,  where $n$ is the  size of the sparse matrix.  
+Full GMRES is however too much expensive in the case of large matrices, since the
+required orthogonalization  process per  iteration grows quadratically  with the
+number of iterations. For that reason, GMRES is restarted in practice after each
+$m\ll n$ iterations, to avoid the  storage of a large orthonormal basis. However,
+the  convergence behavior  of the  restarted GMRES,  called GMRES($m$),  in many
+cases depends quite critically on  the $m$ value~\cite{Huang89}. Therefore in
+most cases, a preconditioning technique is applied to the restarted GMRES method
+in order to improve its convergence.
+
+To enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has for instance proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, leading to the so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process is referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However, in practice, good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.  
+
+Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding techniques for the GMRES method, so-called CA-GMRES, on multicore processors and multi-GPU machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}. 
+
+Compared  to all these  works and  to all  the other  works on  Krylov iterative
+method, the originality of our work is to build a second iteration over a Krylov
+iterative method and to minimize the residuals with a least-squares method after
+a given number of outer iterations.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -616,7 +639,7 @@ In order to enhance the robustness of Krylov iterative solvers, some techniques
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Two-stage iteration with least-squares residuals minimization algorithm}
+\section{TSIRM: 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
@@ -680,20 +703,20 @@ 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 $||Ax_k-b||_2$.
+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.
+  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  ($max\_iter_{ls}$) and the threshold to stop
+  the method ($\epsilon_{ls}$).
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
-\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $\epsilon_{tsirm}$: the threshold that stops 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;
@@ -704,14 +727,15 @@ Let us summarize the most important parameters of TSIRM:
 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
+our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    In
+line~\ref{algo:matrix_mul}, the  matrix-matrix multiplication is  implemented and
+efficient since the  matrix $A$ is sparse and the  matrix $S$ contains few
 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.
+In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows
+more or less the  same principle but it takes more place,  so we briefly explain
+the parallelization of CGLS which is  similar to LSQR.
 
 \begin{algorithm}[t]
 \caption{CGLS}
@@ -740,9 +764,10 @@ less the same principle but it takes more place, so we briefly explain the paral
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
-classical operations:  dot product, norm, multiplication  and addition on  vectors. All
-these operations are easy to implement in PETSc or similar environment.
-
+classical  operations:  dot  product,   norm,  multiplication,  and  addition  on
+vectors.  All  these  operations are  easy  to  implement  in PETSc  or  similar
+environment.  It should be noticed that LSQR follows the same principle, it is a
+little bit longer but it performs more or less the same operations.
 
 
 %%%*********************************************************
@@ -821,8 +846,11 @@ $\begin{array}{ll}
 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.
+Remark that a similar proposition can be formulated at each time
+the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
+with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases 
+regarding $A$, 
+that the proposed TSIRM converges while the GMRES($m$) does not.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -830,12 +858,12 @@ which concludes the induction and the proof.
 \label{sec:05}
 
 
-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
+In order to see the behavior of our approach 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]
@@ -856,28 +884,26 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
-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:
+Chosen parameters  are detailed below.   
+We have  stopped  the  GMRES every  30
+iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default 
+setting of GMRES.  $s$, for its part, has been set to 8. CGLS 
+ minimizes  the   least-squares  problem   with  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.
+$\epsilon_{tsirm}=1e-10$.  These  experiments have been performed  on an Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version  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
-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.
+Experiments comparing 
+a GMRES variant with TSIRM in the resolution of linear systems are given in  Table~\ref{tab:02}. 
+The  second column describes whether GMRES or FGMRES
+(Flexible GMRES~\cite{Saad:1993}) has been used for linear systems solving.  
+Different preconditioners  have been used according to the matrices.  With  TSIRM, the  same
+solver and the  same preconditionner are used.  This table  shows that TSIRM can
+drastically reduce  the number of iterations needed 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 two parameters: the number of iterations before stopping GMRES
+and the number of iterations to perform the minimization.
 
 
 \begin{table}[htbp]
@@ -898,7 +924,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
-\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
 \label{tab:02}
 \end{center}
 \end{table}
@@ -908,44 +934,44 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 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:
+of  PETSc. These  applications are  available in  the \emph{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
+\item ex15  is an example  that 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
+\item ex54 is another example based on a 2D problem discretized with quadrilateral
+  finite elements. In this example, the user can define the scaling of material
   coefficient in embedded circle called $\alpha$.
 \end{itemize}
 For more technical details on these applications, interested readers are invited
-to read  the codes  available in  the PETSc sources.   Those problems  have been
-chosen because they are scalable with many  cores which is not the case of other
-problems that we have tested.
+to read  the codes  available in  the PETSc sources.   These problems  have been
+chosen because they are scalable with many  cores.
 
 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.
+architectures: Curie  and Juqueen.   Both these architectures  are supercomputers
+respectively  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 belong 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,
+for its part, is
+composed by 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
+However, 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}.
+preconditioners, but  as it is  not the subject  of this paper, we  will not
+present results with many preconditioners. In  practice, we have chosen to use a
+multigrid (mg)  and successive  over-relaxation (sor). For  further details  on the
+preconditioner in PETSc, reader is referred to~\cite{petsc-web-page}.
 
 
 
@@ -968,33 +994,36 @@ preconditioner in PETSc please consult~\cite{petsc-web-page}.
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) having 25,000 components per core on Juqueen ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
 \label{tab:03}
 \end{center}
 \end{table*}
 
 Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
 example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
-are  studied ranging  from  2,048  up-to 16,383 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 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
+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. Other parameters  for this application  are described in
+the legend of this Table.
+
+
+
+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.
+recorded  but they  are  time-consuming.  In  general each  $max\_iter_{kryl}*s$
+iterations which corresponds to 30*12, there are $max\_iter_{ls}$ iterations for
+the least-squares method which corresponds to 15.
 
 \begin{figure}[htbp]
 \centering
@@ -1023,7 +1052,7 @@ 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|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ 
+  nb. cores & $\epsilon_{tsirm}$  & \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 \\
@@ -1036,7 +1065,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \hline
 
 \end{tabular}
-\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.}
+\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 ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
 \label{tab:04}
 \end{center}
 \end{table*}
@@ -1047,10 +1076,10 @@ 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
+only one iteration is necessary  to reach the convergence. So Table~\ref{tab:04}
+shows the  results of  different executions with  different number of  cores and
+different 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
@@ -1062,19 +1091,20 @@ 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.
+been tested. Here again we can  see that TSIRM is faster that FGMRES. Efficiency
+of each algorithm  is reported. It can be noticed that  the efficiency of FGMRES
+is better than  the TSIRM one 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
+experiments  reported in  Table~\ref{tab:05}.  This  Figure highlights  that the
+number of iterations  per second 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
+have previously explained, that the iterations of the least-squares steps are not
 taken into account with TSIRM.
 
 \begin{table*}[htbp]
@@ -1094,7 +1124,7 @@ taken into account with TSIRM.
 \hline
 
 \end{tabular}
-\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.}
+\caption{Comparison of FGMRES  and TSIRM for ex54 of PETSc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
 \label{tab:05}
 \end{center}
 \end{table*}
@@ -1109,16 +1139,16 @@ taken into account with TSIRM.
 
 Concerning the  experiments some  other remarks are  interesting.
 \begin{itemize}
-\item We  can tested other examples of  PETSc (ex29, ex45, ex49).  For all these
+\item We  have 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
+\item We have tested many iterative  solvers available in PETSc.  In fact, 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
+  feature. More  precisely, the solver must  support to be  stopped 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
+  naturally  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
@@ -1149,11 +1179,13 @@ experiments up to 16,394 cores have been led to verify that TSIRM runs
 
 
 For  future  work, the  authors'  intention is  to  investigate  other kinds  of
-matrices, problems, and  inner solvers. The influence of  all parameters must be
+matrices, problems, and  inner solvers. In particular, the possibility 
+to obtain a convergence of TSIRM in situations where the GMRES is divergent will be
+investigated. 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
+inside PETSc, which would be of interest as it would  allow us to test
 all the non-linear  examples and compare our algorithm  with the other algorithm
 implemented in PETSc.
 
