interesting to solve the least-squares minimization, the CGLS and the LSQR\r
methods.\r
\r
-In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows\r
-more or less the same principle but it takes more place, so we briefly explain\r
-the parallelization of CGLS which is similar to LSQR.\r
-\r
-\begin{algorithm}[t]\r
-\caption{CGLS}\r
-\begin{algorithmic}[1]\r
- \Input $A$ (matrix), $b$ (right-hand side)\r
- \Output $x$ (solution vector)\vspace{0.2cm}\r
- \State Let $x_0$ be an initial approximation\r
- \State $r_0=b-Ax_0$\r
- \State $p_1=A^Tr_0$\r
- \State $s_0=p_1$\r
- \State $\gamma=||s_0||^2_2$\r
- \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}\r
- \State $q_k=Ap_k$\r
- \State $\alpha_k=\gamma/||q_k||^2_2$\r
- \State $x_k=x_{k-1}+\alpha_kp_k$\r
- \State $r_k=r_{k-1}-\alpha_kq_k$\r
- \State $s_k=A^Tr_k$\r
- \State $\gamma_{old}=\gamma$\r
- \State $\gamma=||s_k||^2_2$\r
- \State $\beta_k=\gamma/\gamma_{old}$\r
- \State $p_{k+1}=s_k+\beta_kp_k$\r
- \EndFor\r
-\end{algorithmic}\r
-\label{algo:02}\r
-\end{algorithm}\r
+%% In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows\r
+%% more or less the same principle but it takes more place, so we briefly explain\r
+%% the parallelization of CGLS which is similar to LSQR.\r
+\r
+%% \begin{algorithm}[t]\r
+%% \caption{CGLS}\r
+%% \begin{algorithmic}[1]\r
+%% \Input $A$ (matrix), $b$ (right-hand side)\r
+%% \Output $x$ (solution vector)\vspace{0.2cm}\r
+%% \State Let $x_0$ be an initial approximation\r
+%% \State $r_0=b-Ax_0$\r
+%% \State $p_1=A^Tr_0$\r
+%% \State $s_0=p_1$\r
+%% \State $\gamma=||s_0||^2_2$\r
+%% \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}\r
+%% \State $q_k=Ap_k$\r
+%% \State $\alpha_k=\gamma/||q_k||^2_2$\r
+%% \State $x_k=x_{k-1}+\alpha_kp_k$\r
+%% \State $r_k=r_{k-1}-\alpha_kq_k$\r
+%% \State $s_k=A^Tr_k$\r
+%% \State $\gamma_{old}=\gamma$\r
+%% \State $\gamma=||s_k||^2_2$\r
+%% \State $\beta_k=\gamma/\gamma_{old}$\r
+%% \State $p_{k+1}=s_k+\beta_kp_k$\r
+%% \EndFor\r
+%% \end{algorithmic}\r
+%% \label{algo:02}\r
+%% \end{algorithm}\r
\r
+%%NEW\r
\r
-In each iteration of CGLS, there are two matrix-vector multiplications and some\r
-classical operations: dot product, norm, multiplication, and addition on\r
-vectors. All these operations are easy to implement in PETSc or similar\r
-environment. It should be noticed that LSQR follows the same principle, it is a\r
-little bit longer but it performs more or less the same operations.\r
+The PETSc code we have develop is avaiable here: {\bf a mettre} and it will soon\r
+be integrated with the PETSc sources. TSIRM has been implemented as any solver\r
+for linear systems in PETSc. As it requires to use another solver, we have used\r
+a very interesting feature of PETSc that enables to use a preconditioner as a\r
+linear system with the function {\it PCKSPGetKSP}. As the LSQR function is\r
+already implemented in PETSc, we have used it. CGLS was not implemented yet, so\r
+we have implemented it and we plan to define it as a minimization solver in\r
+PETSc similarly to LSQR. Both CGLS and LSQR are not complex from the computation\r
+point of view. They involves matrix-vector multiplications and some classical\r
+operations: dot product, norm, multiplication, and addition on vectors. As\r
+presented in Section~\ref{sec:05} the minimization step is scalable.\r
\r
\r
%%%*********************************************************\r
\section{Convergence results}\r
\label{sec:04}\r
\r
-%%NEW\r
-\r
\r
We suppose in this section that GMRES($m$) is used as solver in the TSIRM algorithm applied on a complex matrix $A$.\r
Let us denote $A^\ast$ the conjugate transpose of $A$, and let $\mathfrak{R}(A)=\dfrac{1}{2} \left( A + A^\ast\right)$, $\mathfrak{I}(A)=\dfrac{1}{2i} \left( A - A^\ast\right)$. \r
regarding $A$, \r
that the proposed TSIRM converges while the GMRES($m$) does not.\r
\r
-%%ENDNEW\r
-\r
\r
%%%*********************************************************\r
%%%*********************************************************\r
\section{Experiments using PETSc}\r
\label{sec:05}\r
\r
-%%NEW\r
-In this section four kinds of experiments have been performed. First, some experiments on real matrices issued from the sparse matrix florida have been achieved out. Second, some experiments in parallel with some linear problems are reported and analyzed. Third, some experiments in parallèle with som nonlinear problems are illustrated. Finally some parameters of TSIRM are studied in order to understand their influences.\r
+In this section four kinds of experiments have been performed. First, some\r
+experiments on real matrices issued from the sparse matrix florida have been\r
+achieved out. Second, some experiments in parallel with some linear problems are\r
+reported and analyzed. Third, some experiments in parallèle with som nonlinear\r
+problems are illustrated. Finally some parameters of TSIRM are studied in order\r
+to understand their influences.\r
\r
\r
\subsection{Real matrices}\r
\r
\r
%%NEW\r
+It is well-known that preconditioners have a very strong influence on the\r
+convergence of linear systems. Previously, we have used some classical\r
+preconditioners provided by PETSc. HYPRE~\cite{Falgout06} is a very efficient\r
+preconditioner based on structured multigrid and element-based algebraic\r
+multigrid algorithms. In Table~\ref{tab:06} we report an experiment that show it\r
+reduces drastivally the number of iterations but sometimes it is very\r
+time-consuming compared to other simpler precondititioners. In this table, we\r
+can see that for $512$ and $2,048$ cores, HYPRE reduces drastically the number\r
+of iterations for FGMRES to reach the convergence. However, it is very\r
+time-consuming compared to TSIRM and FGMRES with the ASM preconditioner. For\r
+$4,096$ and $8,192$ cores, FGMRES with HYPRE did not converge in less than 1000s\r
+where FGMRES and TSIRM with the ASM converge very quickly. Finally, it can be\r
+noticed that TSIRM is also faster than FGMRES and it requires less iterations.\r
\r
-{\bf example ex45/ksp à décrire et commenter en montrant que hypre est pourri avec cet exemple}\r
\r
\begin{table*}[htbp]\r
\begin{center}\r
512 & 5.54 & 685 & 2.5 & 570 & 2.21 & 128.9 & 9 \\\r
2048 & 14.95 & 1,560 & 4.32 & 746 & 3.48 & 335.7 & 9 \\\r
4096 & 25.13 & 2,369 & 5.61 & 859 & 4.48 & >1000 & -- \\\r
- 8192 & 44.35 & 3,197 & 7.6 & 1083 & 5.84 & >1000 & -- \\\r
+ 8192 & 44.35 & 3,197 & 7.6 & 1,083 & 5.84 & >1000 & -- \\\r
\r
\hline\r
\r
\end{center}\r
\end{table*}\r
\r
-In Table~\cite{tab:08}, the results of the experiments with the example ex20 are\r
+In Table~\ref{tab:08}, the results of the experiments with the example ex20 are\r
reported. The block Jacobi preconditioner has also been used and CGLS to solve\r
the minimization step for TSIRM. For this example, we can observ that the number\r
of iterations for FMGRES increase drastically when the number of cores\r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
\r
\bibliography{biblio}\r
-\bibliographystyle{unsrt}\r
-\bibliographystyle{alpha}\r
+\bibliographystyle{plain}\r
+%\bibliographystyle{alpha}\r
\r
\end{document}\r
