X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/299e2e52d19b38d91f622d1eb8b7af2bb44c7685..6fda575b88fae086f925cd4dcf960fc2af0d3b4f:/paper.tex

diff --git a/paper.tex b/paper.tex
index 4757973..04ad634 100644
--- a/paper.tex
+++ b/paper.tex
@@ -626,8 +626,8 @@ inner solver. The current approximation of the Krylov method is then stored insi
 $S$ composed by the successive solutions that are computed during inner iterations.
 
 At each $s$ iterations, the minimization step is applied in order to
-compute a new  solution $x$. For that, the previous  residuals are computed 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}
@@ -654,7 +654,7 @@ appropriate than a single direct method in a parallel context.
     \State $S_{k \mod s}=x^k$ \label{algo:store}
     \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
-            \State Solve least-square problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
+            \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
       \State $x^k=S\alpha$  \Comment{compute new solution}
     \EndIf
   \EndFor
@@ -789,7 +789,7 @@ 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
 gmres or fgmres is used to  solve the linear system.  According to the matrices,
 different  preconditioner is used.   With TSIRM,  the same  solver and  the same
-preconditionner is used.  This Table shows that TSIRM can drastically reduce the
+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
@@ -823,12 +823,12 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
-In order to perform larger  experiments, we have tested some example application
+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:
 \begin{itemize}
 \item ex15  is an example  which solves in  parallel an operator using  a finite
-  difference  scheme.   The  diagonal  is  equals to  4  and  4  extra-diagonals
+  difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
   representing the neighbors in each directions  is equal to -1. This example is
   used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
   propagation...
@@ -940,7 +940,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \end{table*}
 
 
-In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
 
 
 \begin{table*}[htbp]
@@ -965,6 +965,13 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 \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}}
+\label{fig:02}
+\end{figure}
+
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