new

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 23bb18b3a90f3fac530a83857a116ef1467738b7..acb46bbfc4d18076e7ec6a4f85b32c3c5df5e3c8 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -644,12 +644,13 @@ appropriate than a direct method in a parallel context.
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$

-  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon$)} \label{algo:conv}

     \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}
     \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}

+    \State retrieve error

     \State $S_{k~mod~s}=x^k$ \label{algo:store}
     \State $S_{k~mod~s}=x^k$ \label{algo:store}

-    \If {$k$ mod $s=0$ {\bf and} not convergence}
+    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon$}

       \State $R=AS$ \Comment{compute dense matrix}
       \State $R=AS$ \Comment{compute dense matrix}

-      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}

       \State $x^k=S\alpha$  \Comment{compute new solution}
     \EndIf
   \EndFor
       \State $x^k=S\alpha$  \Comment{compute new solution}
     \EndIf
   \EndFor


@@ -660,10 +661,19 @@ appropriate than a direct method in a parallel context.
 Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
 iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
 called for a  maximum of $m$ iterations.  In practice, we  suggest to choose $m$
 Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
 iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
 called for a  maximum of $m$ iterations.  In practice, we  suggest to choose $m$

-equals to  the restart number  of the GMRES like  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 the matrix $S$. After the minimization, the matrix $S$ is reused with
-the new values of the residuals.
+equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
+threshold must be specified for the  solver. In practise, this threshold must be
+much   smaller  than   the  convergence   threshold  of   the   TSARM  algorithm
+(i.e.  $\epsilon$).  Line~\ref{algo:store},  $S_{k~  mod~  s}=x^k$  consists  in
+copying the solution $x_k$ into the  column $k~ mod~ s$ of the matrix $S$. After
+the minimization, the matrix $S$ is reused with the new values of the residuals. % à continuer Line
+
+To summarize, the important parameters of are:
+\begin{itemize}
+\item $\epsilon$ the threshold to stop the method
+\item $m$ the number of iterations for the krylov method
+\item $s$ the number of outer iterations before applying the minimization step
+\end{itemize}

 
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