X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/8a3bd904941367c04cdeba8a6b18cc411f0c1003..0e6cba866757a4fe9035fabe308ee2cff9181361:/IJHPCN/paper.tex

diff --git a/IJHPCN/paper.tex b/IJHPCN/paper.tex
index 02ab9cf..410b7ad 100644
--- a/IJHPCN/paper.tex
+++ b/IJHPCN/paper.tex
@@ -366,41 +366,48 @@ 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
-the parallelization of CGLS which is  similar to LSQR.
-
-\begin{algorithm}[t]
-\caption{CGLS}
-\begin{algorithmic}[1]
-  \Input $A$ (matrix), $b$ (right-hand side)
-  \Output $x$ (solution vector)\vspace{0.2cm}
-  \State 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}
-\end{algorithm}
+%% 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}
+%% \begin{algorithmic}[1]
+%%   \Input $A$ (matrix), $b$ (right-hand side)
+%%   \Output $x$ (solution vector)\vspace{0.2cm}
+%%   \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}
+%% \end{algorithm}
 
+%%NEW
 
-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.
+The PETSc code we have develop is avaiable here: {\bf a mettre} and it will soon
+be integrated with  the PETSc sources. TSIRM has been  implemented as any solver
+for linear systems in PETSc. As it  requires to use another solver, we have used
+a very interesting  feature of PETSc that  enables to use a  preconditioner as a
+linear system  with the  function {\it  PCKSPGetKSP}.  As  the LSQR  function is
+already implemented in PETSc, we have used  it. CGLS was not implemented yet, so
+we have  implemented it and  we plan  to define it  as a minimization  solver in
+PETSc similarly to LSQR. Both CGLS and LSQR are not complex from the computation
+point of  view. They involves  matrix-vector multiplications and  some classical
+operations:  dot product,  norm,  multiplication, and  addition  on vectors.  As
+presented in Section~\ref{sec:05} the minimization step is scalable.
 
 
 %%%*********************************************************
@@ -409,8 +416,6 @@ little bit longer but it performs more or less the same operations.
 \section{Convergence results}
 \label{sec:04}
 
-%%NEW
-
 
 We suppose in this section that GMRES($m$) is used as solver in the TSIRM algorithm applied on a complex matrix $A$.
 Let us denote $A^\ast$ the conjugate transpose of $A$, and let $\mathfrak{R}(A)=\dfrac{1}{2} \left( A + A^\ast\right)$, $\mathfrak{I}(A)=\dfrac{1}{2i} \left( A - A^\ast\right)$. 
@@ -599,16 +604,18 @@ with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular c
 regarding $A$, 
 that the proposed TSIRM converges while the GMRES($m$) does not.
 
-%%ENDNEW
-
 
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}
 \label{sec:05}
 
-%%NEW
-In this section four kinds of experiments have been performed. First, some experiments on real matrices issued from the sparse matrix florida have been achieved out. Second, some experiments in parallel with some linear problems are reported and analyzed. Third, some experiments in parallÃ¨le with som nonlinear problems are illustrated. Finally some parameters of TSIRM are studied in order to understand their influences.
+In  this section  four kinds  of experiments  have been  performed. First,  some
+experiments on  real matrices issued  from the  sparse matrix florida  have been
+achieved out. Second, some experiments in parallel with some linear problems are
+reported and analyzed.  Third, some experiments in parallÃ¨le  with som nonlinear
+problems are illustrated. Finally some parameters  of TSIRM are studied in order
+to understand their influences.
 
 
 \subsection{Real matrices}
@@ -1164,7 +1171,7 @@ Curie and Juqueen respectively based in France and Germany.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \bibliography{biblio}
-\bibliographystyle{unsrt}
-\bibliographystyle{alpha}
+\bibliographystyle{plain}
+%\bibliographystyle{alpha}
 
 \end{document}
