we suggest to set this parameter equal to the restart number in the GMRES-like
method. Moreover, a tolerance threshold must be specified for the solver. In
practice, this threshold must be much smaller than the convergence threshold of
-the TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that
-after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
-which is defined by $||Ax_k-b||_2$.
+the TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that
+after the call of the $Solve$ function, we obtain the vector $x_k$ and the
+$error$ which is defined by $||Ax_k-b||_2$.
- Line~\ref{algo:store},
-$S_{k \mod s}=x_k$ consists in copying the solution $x_k$ into the column $k
-\mod s$ of $S$. After the minimization, the matrix $S$ is reused with the new
-values of the residuals. To solve the minimization problem, an iterative method
-is used. Two parameters are required for that: the maximum number of iterations
-and the threshold to stop the method.
+ Line~\ref{algo:store}, $S_{k \mod s}=x_k$ consists in copying the solution
+ $x_k$ into the column $k \mod s$ of $S$. After the minimization, the matrix
+ $S$ is reused with the new values of the residuals. To solve the minimization
+ problem, an iterative method is used. Two parameters are required for that:
+ the maximum number of iterations ($max\_iter_{ls}$) and the threshold to stop
+ the method ($\epsilon_{ls}$).
Let us summarize the most important parameters of TSIRM:
\begin{itemize}
columns in practice. As explained previously, at least two methods seem to be
interesting to solve the least-squares minimization, CGLS and LSQR.
-In the following we remind the CGLS algorithm. The LSQR method follows more or
-less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows
+more or less the same principle but it takes more place, so we briefly explain
+the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dot product, norm, multiplication and addition on vectors. All
-these operations are easy to implement in PETSc or similar environment.
-
+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.
%%%*********************************************************
\label{sec:05}
-In order to see the behavior of the proposal when considering only one processor, a first
-comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented.
-Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in
-Table~\ref{tab:01}. These latter, which are real-world applications matrices,
-have been extracted
- from the Davis collection, University of
+In order to see the behavior of our approach when considering only one processor,
+a first comparison with GMRES or FGMRES and the new algorithm detailed
+previously has been experimented. Matrices that have been used with their
+characteristics (names, fields, rows, and nonzero coefficients) are detailed in
+Table~\ref{tab:01}. These latter, which are real-world applications matrices,
+have been extracted from the Davis collection, University of
Florida~\cite{Dav97}.
\begin{table}[htbp]
\label{tab:01}
\end{center}
\end{table}
-Chosen parameters are detailed below.
-%The following parameters have been chosen for our experiments.
-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. 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:
$\epsilon_{ls}=1e-40$ 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.
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 GRMES or FGMRES
+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 preconditioner is used. With TSIRM, the same solver
-and the same preconditionner are used. This Table shows that TSIRM can
+to the matrices, different preconditioners are used. 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 tow parameters: the number of iterations to stop GMRES
+fact this also depends on two parameters: the number of iterations to stop GMRES
and the number of iterations to perform the minimization.
In order to perform larger experiments, we have tested some example applications
-of PETSc. Those applications are available in the ksp part which is suited for
-scalable linear equations solvers:
+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
difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
\end{itemize}
For more technical details on these applications, interested readers are invited
to read the codes available in the PETSc sources. Those problems have been
-chosen because they are scalable with many cores which is not the case of other
-problems that we have tested.
+chosen because they are scalable with many cores.
In the following larger experiments are described on two large scale
architectures: Curie and Juqeen. Both these architectures are supercomputer
