\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{algorithmic}[1]
\Input $A$ (matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
- \State $r=b-Ax$
- \State $p=A'r$
- \State $s=p$
- \State $g=||s||^2_2$
- \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
- \State $q=Ap$
- \State $\alpha=g/||q||^2_2$
- \State $x=x+alpha*p$
- \State $r=r-alpha*q$
- \State $s=A'*r$
- \State $g_{old}=g$
- \State $g=||s||^2_2$
- \State $\beta=g/g_{old}$
+ \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}
\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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