year = "2006",
keywords = "parallel software tools,",
note = "ch 8,",
-}
\ No newline at end of file
+}
+
+@misc{ksp,
+ title = {{K}rylov {M}ethods - {KSP}: {E}xamples},
+ howpublished = {\url{http://www.mcs.anl.gov/petsc/petsc-dev/src/ksp/ksp/examples/tutorials/}},
+ note = {Accessed: 2015-09-23}
+}
+
+@misc{snes,
+ title = {{N}onlinear solvers - {SNES}: {E}xamples},
+ howpublished = {\url{http://www.mcs.anl.gov/petsc/petsc-current/src/snes/examples/tutorials/index.html}},
+ note = {Accessed: 2015-09-23}
+}
\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} and {\it snes} of PETSc~\cite{ksp,snes}. 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 ../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
\label{fig:cgls} \r
\end{figure}\r
\r
+\begin{figure}[htbp]\r
+\centering\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
+\begin{figure}[htbp]\r
+\centering\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[width=0.5\textwidth]{snes_ex14}\r
\label{fig:snes_ex14} \r
\end{figure}\r
\r
+\begin{figure}[htbp]\r
+\centering\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
+\r
\r
\r
\r
