X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7a7e3e4142880bede1e4c831277adaf5e3773160..84e15020344b77e5497c4a516cc20b472b2914cd:/paper.tex

diff --git a/paper.tex b/paper.tex
index d660010..b08750d 100644
--- a/paper.tex
+++ b/paper.tex
@@ -364,6 +364,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
 
 \begin{document}
 %
@@ -380,7 +381,7 @@
 % 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}}
+\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\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
@@ -741,11 +742,22 @@ Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the res
 \begin{equation}
 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
 \end{equation}
-where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves 
+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 $A$.
 \end{proposition}
 
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+\end{proposition}
+
+\begin{proof}
+Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
+$k$-th iterate of TSIRM.
+We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
 
+Each step of the TSIRM algorithm 
+\end{proof}
 
 %%%*********************************************************
 %%%*********************************************************
@@ -846,7 +858,7 @@ chosen because they  are scalable with many cores which is not the case of other
 In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
 
 
-{\bf Description of preconditioners}
+{\bf Description of preconditioners}\\
 
 \begin{table*}[htbp]
 \begin{center}
@@ -867,15 +879,15 @@ 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
+example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
+are  studied ranging  from  2,048  up-to 16,383.   Two  preconditioners have  been
+tested: {\it mg} and {\it sor}.   For those experiments,  the number  of components  (or unknowns  of the
 problems)  per processor  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
@@ -883,11 +895,11 @@ 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
@@ -1048,5 +1060,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
-
-