X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/af93eecf8bd45d51b68e4c9ac79d9e2bf7100db4..c5fd74103dc91067defdff61917384d6bbf68d1e:/paper.tex

diff --git a/paper.tex b/paper.tex
index 2d34b8f..f141e2b 100644
--- a/paper.tex
+++ b/paper.tex
@@ -548,7 +548,7 @@ Iterative Krylov methods; sparse linear systems; two stage iteration; least-squa
 % (should never be an issue)
 
 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
+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
@@ -560,7 +560,7 @@ 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
+in  order to solve  linear systems.   The best  known iterative  Krylov subspace
 methods are conjugate gradient and GMRES ones (Generalized Minimal RESidual).
 
 
@@ -601,36 +601,62 @@ 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,  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
+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  for  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  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
-noticed 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. However in practice the 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}. 
+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  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  itself   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 Flexible GMRES (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.
+methods,  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.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -641,29 +667,30 @@ a given number of outer iterations.
 %%%*********************************************************
 \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
+A two-stage  algorithm is proposed to  solve large sparse linear  systems of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
-nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector,   and
-$b\in\mathbb{R}^n$ is  the right-hand side.  As explained previously, 
-the algorithm is implemented  as an
-inner-outer iteration  solver based  on iterative Krylov  methods. The  main 
-key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
-It can be summarized as follows: the
-inner solver is a Krylov based one. In order to accelerate its convergence, the 
-outer solver periodically applies a least-squares minimization  on the residuals computed by  the inner one. %Tsolver which does not required to be changed.
+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.
 
 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
+with the last obtained approximation.  The 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  
+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}
@@ -671,10 +698,11 @@ the inner iterations with $(b-AS)$. The minimization of the residuals is obtaine
 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 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. CGLS has recently been used to improve the performance of multisplitting algorithms \cite{cz15:ij}.
 
 
 
@@ -705,7 +733,7 @@ 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 $||Ax_k-b||_2$.
+$error$, 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
@@ -716,7 +744,7 @@ $error$ which is defined by $||Ax_k-b||_2$.
 
 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;
@@ -725,13 +753,14 @@ 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
+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
-columns in  practice. As explained  previously, at least  two methods seem  to be
-interesting to solve the least-squares minimization, CGLS and LSQR.
+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,  the CGLS  and the  LSQR
+methods.
 
 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
@@ -763,8 +792,8 @@ the parallelization of CGLS which is  similar to LSQR.
 \end{algorithm}
 
 
-In each iteration  of CGLS, there is two  matrix-vector multiplications and some
-classical  operations:  dot  product,   norm,  multiplication  and  addition  on
+In each iteration  of CGLS, there are 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.  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.
@@ -780,7 +809,7 @@ little bit longer but it performs more or less the same operations.
 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. 
+If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as a solver, then the TSIRM algorithm is convergent. 
 
 Furthermore, let $r_k$ be the
 $k$-th residue of TSIRM, then
@@ -816,7 +845,7 @@ 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.
+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:
@@ -846,8 +875,11 @@ $\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.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -881,24 +913,24 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
-Chosen parameters  are detailed below.   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 restart parameter.  The parameter $s$ 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 TSIRM
-are given. In the  second column, it can be noticed that  either GMRES or FGMRES
-(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system.  According
-to  the matrices,  different preconditioners  are  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
+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 has been used for linear systems solving.  
+Different preconditioners  have been used according to the matrices.  With  TSIRM, the  same
+solver and the  same preconditioner 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 to stop GMRES
+fact this also depends on two parameters: the number of iterations before stopping GMRES
 and the number of iterations to perform the minimization.
 
 
@@ -920,7 +952,7 @@ 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}
@@ -930,43 +962,44 @@ 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 \emph{ksp}  part  which is
+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
+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.
+In  the  following,   larger  experiments  are  described  on   two  large  scale
+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 Center in Germany.  They belong with other similar architectures
+to 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
+architectures are equipped 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
+preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
 
 
 
@@ -989,50 +1022,54 @@ 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 ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  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  at  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
-  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
-\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
+  \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters as 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.
+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.
 
 
 
@@ -1065,13 +1102,13 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 
 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
+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
@@ -1079,23 +1116,24 @@ 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
+Table~\ref{tab:05} shows  a strong scaling  experiment with example ex54  on the
+Curie  architecture. So,  in  this case,  the  number of  unknowns  is fixed  at
 $204,919,225$ 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
+of two.  The  threshold is fixed at $5e-5$ and only  the $mg$ preconditioner has
+been  tested. Here  again  we can  see that  TSIRM  is faster  than FGMRES.  The
+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 than 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 the
+experiments  reported in  Table~\ref{tab:05}.  This  figure highlights  that the
+number of iterations  per second is more  or 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, the  iterations of the  least-squares steps  are not
 taken into account with TSIRM.
 
 \begin{table*}[htbp]
@@ -1115,37 +1153,37 @@ 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 ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  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*}
 
 \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)}
+  \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex54_curie}
+\caption{Number of iterations per second with ex54 and the same parameters as 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
+\item We have tested other examples  of PETSc (ex29, ex45, ex49).  For all these
+  examples,  we have also  obtained similar  gains 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 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
-  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.
+  feature. More precisely,  the solver must support to  be stopped and restarted
+  without decreasing its convergence. That is  why with GMRES we stop it when it
+  is  naturally restarted (\emph{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  which is  stopped and
+  restarted every  16 iterations (for example),  the normal CG will  be far more
+  efficient.   Some  restarted  CG or  CG  variant  versions  exist and  may  be
+  interesting to study in future works.
 \end{itemize}
 %%%*********************************************************
 %%%*********************************************************
@@ -1160,8 +1198,8 @@ Concerning the  experiments some  other remarks are  interesting.
 %%%*********************************************************
 %%%*********************************************************
 
-A novel two-stage iterative  algorithm has been proposed in this article,
-in order to accelerate the convergence Krylov iterative  methods.
+A new two-stage iterative  algorithm TSIRM has been proposed in this article,
+in order to accelerate the convergence of 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 
@@ -1170,11 +1208,13 @@ 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
+number of outer  iterations to minimize should become  adaptive 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  allows us to test
 all the non-linear  examples and compare our algorithm  with the other algorithm
 implemented in PETSc.
 
@@ -1188,7 +1228,7 @@ implemented in PETSc.
 %%%*********************************************************
 \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 resources
+ANR-11-LABX-01-01).   We acknowledge the Mesocentre of Franche-Comte and PRACE  for awarding  us access  to resources
 Curie and Juqueen respectively based in France and Germany.
 
 
