\label{tab:01}
\end{center}
\end{table}
-Chosen parameters are detailed below. As by default the restart of GMRES is
-performed every 30 iterations, we have chosen to stop the GMRES every 30
-iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen
-to minimize the least-squares problem with the following parameters:
+Chosen parameters are detailed below.
+We have stopped the GMRES every 30
+iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default
+setting of GMRES. $s$, for its part, has been set to 8. CGLS
+ minimizes the least-squares problem with parameters
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
-$\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R)
-Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+$\epsilon_{tsirm}=1e-10$. These experiments have been performed on an Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version of PETSc.
-In Table~\ref{tab:02}, some experiments comparing the solving of the linear
-systems obtained with the previous matrices with a GMRES variant and with TSIRM
-are given. In the second column, it can be noticed that either GMRES or FGMRES
-(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system. According
-to the matrices, different preconditioners are used. With TSIRM, the same
-solver and the same preconditionner are used. This Table shows that TSIRM can
+Experiments comparing
+a GMRES variant with TSIRM in the resolution of linear systems are given in Table~\ref{tab:02}.
+The second column describes whether GMRES or FGMRES
+(Flexible GMRES~\cite{Saad:1993}) has been used for linear systems solving.
+Different preconditioners have been used according to the matrices. With TSIRM, the same
+solver and the same preconditionner are used. This table shows that TSIRM can
drastically reduce the number of iterations to reach the convergence when the
number of iterations for the normal GMRES is more or less greater than 500. In
fact this also depends on two parameters: the number of iterations to stop GMRES
\hline
\end{tabular}
-\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
\label{tab:02}
\end{center}
\end{table}
In order to perform larger experiments, we have tested some example applications
-of PETSc. Those applications are available in the \emph{ksp} part which is
+of PETSc. Those applications are available in the \emph{ksp} part, which is
suited for scalable linear equations solvers:
\begin{itemize}
-\item ex15 is an example which solves in parallel an operator using a finite
+\item ex15 is an example that solves in parallel an operator using a finite
difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
representing the neighbors in each directions are equal to -1. This example is
used in many physical phenomena, for example, heat and fluid flow, wave
For future work, the authors' intention is to investigate other kinds of
-matrices, problems, and inner solvers. The influence of all parameters must be
+matrices, problems, and inner solvers. In particular, the possibility
+to obtain a convergence of TSIRM in situations where the GMRES is divergent will be
+investigated. The influence of all parameters must be
tested too, while other methods to minimize the residuals must be regarded. The
number of outer iterations to minimize should become adaptative to improve the
overall performances of the proposal. Finally, this solver will be implemented
-inside PETSc. This would be very interesting because it would allow us to test
+inside PETSc, which would be of interest as it would allow us to test
all the non-linear examples and compare our algorithm with the other algorithm
implemented in PETSc.
