\algnewcommand\Output{\item[\algorithmicoutput]}
\newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
\begin{document}
%
% 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\\
\begin{abstract}
-In this article, a two-stage iterative algorithm is proposed to improve the
-convergence of Krylov based iterative methods, typically those of GMRES variants. The
-principle of the proposed approach is to build an external iteration over the Krylov
-method, and to frequently store its current residual (at each
-GMRES restart for instance). After a given number of outer iterations, a minimization
-step is applied on the matrix composed by the saved residuals, in order to
-compute a better solution and to make new iterations if required. It is proven that
-the proposal has the same convergence properties than the inner embedded method itself.
-Experiments using up to 16,394 cores also show that the proposed algorithm
-runs around 5 or 7 times faster than GMRES.
+In this article, a two-stage iterative algorithm is proposed to improve the
+convergence of Krylov based iterative methods, typically those of GMRES
+variants. The principle of the proposed approach is to build an external
+iteration over the Krylov method, and to frequently store its current residual
+(at each GMRES restart for instance). After a given number of outer iterations,
+a least-squares minimization step is applied on the matrix composed by the saved
+residuals, in order to compute a better solution and to make new iterations if
+required. It is proven that the proposal has the same convergence properties
+than the inner embedded method itself. Experiments using up to 16,394 cores
+also show that the proposed algorithm runs around 5 or 7 times faster than
+GMRES.
\end{abstract}
\begin{IEEEkeywords}
\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, then the TSIRM algorithm is convergent.
+\end{proposition}
+\begin{proof}
+\end{proof}
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\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
-\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
\label{fig:01}
\end{figure}
\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} (strong scaling)}
+\label{fig:02}
+\end{figure}
+
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% that's all folks
\end{document}
-
