Let us summarize the most important parameters of TSIRM:
\begin{itemize}
-\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $\epsilon_{tsirm}$: the threshold that stops the TSIRM method;
\item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
\item $s$: the number of outer iterations before applying the minimization step;
\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
The parallelization of TSIRM relies on the parallelization of all its
parts. More precisely, except the least-squares step, all the other parts are
obvious to achieve out in parallel. In order to develop a parallel version of
-our code, we have chosen to use PETSc~\cite{petsc-web-page}. For
-line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and
-efficient since the matrix $A$ is sparse and since the matrix $S$ contains few
+our code, we have chosen to use PETSc~\cite{petsc-web-page}. In
+line~\ref{algo:matrix_mul}, the matrix-matrix multiplication is implemented and
+efficient since the matrix $A$ is sparse and the matrix $S$ contains few
columns in practice. As explained previously, at least two methods seem to be
interesting to solve the least-squares minimization, CGLS and LSQR.
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dot product, norm, multiplication and addition on
+classical operations: dot product, norm, multiplication, and addition on
vectors. All these operations are easy to implement in PETSc or similar
environment. It should be noticed that LSQR follows the same principle, it is a
little bit longer but it performs more or less the same operations.
which concludes the induction and the proof.
\end{proof}
-%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower
-%than the one of the GMRES method.
+Remark that a similar proposition can be formulated at each time
+the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
+with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases
+regarding $A$,
+that the proposed TSIRM converges while the GMRES($m$) does not.
%%%*********************************************************
%%%*********************************************************
\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.
