\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{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.
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
\end{proposition}
\begin{proof}
$k$-th iterate of TSIRM.
We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+Each step of the TSIRM algorithm \\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+$\begin{array}{ll}
+& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant ||b-Ax_{k-1}||
+\end{array}$
\end{proof}
+
%%%*********************************************************
%%%*********************************************************
\section{Experiments using PETSc}
\hline
\end{tabular}
-\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
\label{tab:02}
\end{center}
\end{table}
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
+problems) per core 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
small.
In Figure~\ref{fig:01}, the number of iterations per second corresponding to
-Table~\ref{tab:01} is displayed. It can be noticed that the number of
-iterations per second of FMGRES is constant whereas it decrease with TSIRM with
-both preconditioner. This can be explained by the fact that when the number of
-core increases the time for the minimization step also increases but, generally,
+Table~\ref{tab:03} is displayed. It can be noticed that the number of
+iterations per second of FMGRES is constant whereas it decreases with TSIRM with
+both preconditioners. This can be explained by the fact that when the number of
+cores increases the time for the least-squares minimization step also increases but, generally,
when the number of cores increases, the number of iterations to reach the
threshold also increases, and, in that case, TSIRM is more efficient to reduce
the number of iterations. So, the overall benefit of using TSIRM is interesting.
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
+ nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
\cline{3-8}
& & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
\hline
\end{tabular}
-\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
\label{tab:04}
\end{center}
\end{table*}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\
+ nb. cores & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\
\cline{2-7} \cline{9-11}
- & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline
+ & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & FGMRES & TS CGLS & TS LSQR\\\hline \hline
512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\
1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\
2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\
\hline
\end{tabular}
-\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshold 5e-5), time is expressed in seconds.}
\label{tab:05}
\end{center}
\end{table*}
future plan : \\
- study other kinds of matrices, problems, inner solvers\\
-- test the influence of all the parameters\\
+- test the influence of all parameters\\
- adaptative number of outer iterations to minimize\\
- other methods to minimize the residuals?\\
- implement our solver inside PETSc
%%%*********************************************************
\section*{Acknowledgment}
This paper is partially funded by the Labex ACTION program (contract
-ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource
+ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources
Curie and Juqueen respectively based in France and Germany.
% that's all folks
\end{document}
+
