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
%%NEW\r
\subsection{Influence of parameters for TSIRM}\r
-In this section we present some experimental results in order to study the influence of some parameters on the TSIRM algorithm. We conducted experiments on $16$ cores to solve 3D problems of size $200,000$ components per core. We solved nonlinear problems token from examples of PETSc. We fixed some parameters of the TSIRM algorithm as follows: the nonlinear systems are solved with a precision of $10^{-8}$, block Jacobi preconditioner is used, the tolerance threshold $\epsilon_{tsirm}$ is $10^{-8}$ , the maximum number of iterations $max\_iter_{tsirm}$ is set to $10,000$ iterations, the FGMRES method is used as the inner solver with a tolerance threshold $\epsilon_{kryl}=10^{-10}$ and the least-squares problem is solved with a precision $\epsilon_{ls}=10^{-40}$ in the minimization process.\r
+In this section we present some experimental results in order to study the influence of some parameters on the TSIRM algorithm which are: the method to solve the linear least-squares problem in the minimization process, the number of inner iterations $max\_iter_{kryl}$, and the size $s$ of matrix $AS$ (i.e. matrix $S$) of the least-squares problem. We conducted experiments on $16$ cores to solve linear and nonlinear problems of size $200,000$ components per core. We solved problems token from examples {\it ksp}~\cite{ksp} and {\it snes}~\cite{snes} of PETSc. We fixed some parameters of the TSIRM algorithm as follows: the nonlinear systems are solved with a precision of $10^{-8}$, the tolerance threshold and the maximum number of iterations of TSIRM algorithm are $\epsilon_{tsirm}=10^{-10}$ and $max\_iter_{tsirm}=10,000$, the FGMRES method is used as the inner solver with a tolerance threshold $\epsilon_{kryl}=10^{-10}$, the additive Schwarz method (ASM) is used as a preconditioner, and the least-squares problem is solved with a precision $\epsilon_{ls}=10^{-40}$ in the minimization process.\r
\r
-%time mpirun ../ex48 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -snes_rtol 1.e-8 -snes_monitor -ksp_type tsirm -ksp_pc_type bjacobi -pc_type ksp -ksp_tsirm_tol 1e-8 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 30 -ksp_tsirm_inner_restarts 30 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 15 -ksp_tsirm_size_ls 10 \r
+%time mpirun ../ex34 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -ksp_type tsirm -ksp_pc_type asm -pc_type ksp -ksp_tsirm_tol 1e-10 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 30 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 20 -ksp_tsirm_size_ls 10\r
\begin{figure}[htbp]\r
\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_cgls_iter_total}\r
-\caption{Number of total iterations using two different methods for the minimization: LSQR and CGLS.}\r
-\label{fig:cgls-iter} \r
+ \includegraphics[width=0.5\textwidth]{ksp_tsirm_cgls}\r
+\caption{Number of total iterations using two different methods for the minimization: CGLS and LSQR.}\r
+\label{fig:cgls} \r
\end{figure}\r
\r
\begin{figure}[htbp]\r
\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_cgls_time}\r
-\caption{Execution time in seconds using two different methods for the minimization: LSQR and CGLS.}\r
-\label{fig:cgls-time} \r
+ \includegraphics[width=0.5\textwidth]{ksp_ex12}\r
+\caption{Total number of iterations in example {\it ksp ex12} of PETSc by varyin the number of inner iterations and the size of the least-squares problem.}\r
+\label{fig:ksp_ex12} \r
\end{figure}\r
\r
-%time mpirun ../ex35 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -snes_rtol 1.e-8 -snes_monitor -ksp_type tsirm -ksp_pc_type bjacobi -pc_type ksp -ksp_tsirm_tol 1e-8 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 30 -ksp_tsirm_inner_restarts 38 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 15 -ksp_tsirm_size_ls 10\r
\begin{figure}[htbp]\r
\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_inner_restarts_iter_total}\r
-\caption{Number of total iterations with variation of restarts in the inner solver FGMRES.}\r
-\label{fig:inner_restarts_iter_total} \r
+ \includegraphics[width=0.5\textwidth]{ksp_ex34}\r
+\caption{Total number of iterations in example {\it ksp ex34} of PETSc by varyin the number of inner iterations and the size of the least-squares problem.}\r
+\label{fig:ksp_ex34} \r
\end{figure}\r
\r
\begin{figure}[htbp]\r
\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_inner_restarts_time}\r
-\caption{Execution time in seconds with variation of restarts in the inner solver FGMRES.}\r
-\label{fig:inner_restarts_time} \r
+ \includegraphics[width=0.5\textwidth]{snes_ex14}\r
+\caption{Total number of iterations in example {\it snes ex14} of PETSc by varyin the number of inner iterations and the size of the least-squares problem.}\r
+\label{fig:snes_ex14} \r
\end{figure}\r
\r
-%time mpirun ../ex14 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -snes_rtol 1.e-8 -snes_monitor -ksp_type tsirm -ksp_pc_type bjacobi -pc_type ksp -ksp_tsirm_tol 1e-8 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 1000 -ksp_tsirm_inner_restarts 30 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 15 -ksp_tsirm_size_ls 10\r
\begin{figure}[htbp]\r
\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_max_inner_iter}\r
-\caption{Number of total iterations with variation of number of inner iterations.}\r
-\label{fig:max_inner_iter} \r
+ \includegraphics[width=0.5\textwidth]{snes_ex20}\r
+\caption{Total number of iterations in example {\it snes ex20} of PETSc by varyin the number of inner iterations and the size of the least-squares problem.}\r
+\label{fig:snes_ex20} \r
\end{figure}\r
\r
-\begin{figure}[htbp]\r
-\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_max_inner_time}\r
-\caption{Execution time in seconds with variation of number of inner iterations.}\r
-\label{fig:max_inner_time} \r
-\end{figure}\r
\r
-%time mpirun ../ex14 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -snes_rtol 1.e-8 -snes_monitor -ksp_type tsirm -ksp_pc_type bjacobi -pc_type ksp -ksp_tsirm_tol 1e-8 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 30 -ksp_tsirm_inner_restarts 30 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 5 -ksp_tsirm_size_ls 10\r
-\begin{figure}[htbp]\r
-\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_maxiter_ls_iter}\r
-\caption{Number of total iterations with variation of number of iterations in the minimization process.}\r
-\label{fig:maxiter_ls_iter} \r
-\end{figure}\r
-\r
-\begin{figure}[htbp]\r
-\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_maxiter_ls_time}\r
-\caption{Execution time in seconds with variation of number of iterations in the minimization process.}\r
-\label{fig:maxiter_ls_time} \r
-\end{figure}\r
\r
-%time mpirun ../ex14 -da_grid_x 147 -da_grid_y 147 -da_grid_z 147 -snes_rtol 1.e-8 -snes_monitor -ksp_type tsirm -ksp_pc_type bjacobi -pc_type ksp -ksp_tsirm_tol 1e-8 -ksp_tsirm_maxiter 10000 -ksp_ksp_type fgmres -ksp_tsirm_max_inner_iter 30 -ksp_tsirm_inner_restarts 30 -ksp_tsirm_inner_tol 1e-10 -ksp_tsirm_cgls 0 -ksp_tsirm_tol_ls 1.e-40 -ksp_tsirm_maxiter_ls 15 -ksp_tsirm_size_ls 2\r
-\begin{figure}[htbp]\r
-\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_size_ls_iter}\r
-\caption{Number of total iterations with variation of the size of the least-squares problem in the minimization process.}\r
-\label{fig:size_ls_iter} \r
-\end{figure}\r
\r
-\begin{figure}[htbp]\r
-\centering\r
- \includegraphics[angle=-90,width=0.5\textwidth]{ksp_tsirm_size_ls_time}\r
-\caption{Execution time in seconds with variation of the size of the least-squares problem in the minimization process.}\r
-\label{fig:size_ls_time} \r
-\end{figure}\r
\r
%%ENDNEW\r
\r
