X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/091a1e083b6822c080f3c7524f05e0ee3a87cad4..af93eecf8bd45d51b68e4c9ac79d9e2bf7100db4:/paper.tex

diff --git a/paper.tex b/paper.tex
index e4d146a..2d34b8f 100644
--- a/paper.tex
+++ b/paper.tex
@@ -703,16 +703,16 @@ 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}
@@ -733,8 +733,9 @@ 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.
 
-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}
@@ -763,9 +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.
 
 
 %%%*********************************************************
@@ -853,12 +855,12 @@ 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]
@@ -879,12 +881,10 @@ 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.   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:
 $\epsilon_{ls}=1e-40$ 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.
@@ -892,13 +892,13 @@ 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 TSIRM
-are given. In the  second column, it can be noticed that  either GRMES or FGMRES
+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 preconditioner  is used.  With TSIRM, the same solver
-and  the  same  preconditionner are  used.   This  Table  shows that  TSIRM  can
+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
 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
+fact this also depends on two parameters: the number of iterations to stop GMRES
 and the number of iterations to perform the minimization.
 
 
@@ -930,8 +930,8 @@ 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. Those  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
   difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
@@ -944,8 +944,7 @@ scalable linear equations solvers:
 \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.
+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
