\begin{abstract}
-In this article, a two-stage iterative method is proposed to improve the
-convergence of Krylov based iterative ones, typically those of GMRES variants. The
+In this article, a two-stage iterative algorithm is proposed to improve the
+convergence of Krylov based iterative methods, typically those of GMRES variants. The
principle of the proposed approach is to build an external iteration over the Krylov
method, and to frequently store its current residual (at each
GMRES restart for instance). After a given number of outer iterations, a minimization
step is applied on the matrix composed by the saved residuals, in order to
-compute a better solution while making new iterations if required. It is proven that
+compute a better solution and to make new iterations if required. It is proven that
the proposal has the same convergence properties than the inner embedded method itself.
Experiments using up to 16,394 cores also show that the proposed algorithm
-run around 7 times faster than GMRES.
+runs around 5 or 7 times faster than GMRES.
\end{abstract}
\begin{IEEEkeywords}
characteristics. For all the matrices, the name, the field, the number of rows
and the number of nonzero elements is given.
-\begin{table*}[htbp]
+\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|r|r|r|}
\hline
Matrix name & Field &\# Rows & \# Nonzeros \\\hline \hline
crashbasis & Optimization & 160,000 & 1,750,416 \\
-parabolic\_fem & Computational fluid dynamics & 525,825 & 2,100,225 \\
+parabolic\_fem & Comput. fluid dynamics & 525,825 & 2,100,225 \\
epb3 & Thermal problem & 84,617 & 463,625 \\
-atmosmodj & Computational fluid dynamics & 1,270,432 & 8,814,880 \\
-bfwa398 & Electromagnetics problem & 398 & 3,678 \\
+atmosmodj & Comput. fluid dynamics & 1,270,432 & 8,814,880 \\
+bfwa398 & Electromagnetics pb & 398 & 3,678 \\
torso3 & 2D/3D problem & 259,156 & 4,429,042 \\
\hline
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
-\end{table*}
+\end{table}
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
example ex15 of PETSc on the Juqueen architecture. Differents number of cores
are studied rangin from 2,048 upto 16,383. Two preconditioners have been
tested. For those experiments, the number of components (or unknown of the
-problems) per processor is fixed to 25,000. This number can seem relatively
-small. In fact, for some applications that need a lot of memory, the number of
-components per processor requires sometimes to be small.
+problems) per processor is fixed to 25,000, also called weak scaling. This
+number can seem relatively small. In fact, for some applications that need a lot
+of memory, the number of components per processor requires sometimes to be
+small.
+
+
In this Table, we can notice that TSIRM is always faster than FGMRES. The last
column shows the ratio between FGMRES and the best version of TSIRM according to
case between CGLS and LSQR, it is clear that TSIRM is alsways faster than
FGMRES. For this example, the multigrid preconditionner is faster than SOR. The
gain between TSIRM and FGMRES is more or less similar for the two
-preconditioners
-
+preconditioners. Looking at the number of iterations to reach the convergence,
+it is obvious that TSIRM allows the reduction of the number of iterations. It
+should be noticed that for TSIRM, in those experiments, only the iterations of
+the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
+corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
\begin{figure}[htbp]
\centering
\end{figure}
+In Figure~\ref{fig:01}, the number of iterations per second corresponding to
+Table~\ref{tab:01} is displayed. It can be noticed that the number of
+iterations per second of FMGRES is constant whereas it decrease with TSIRM with
+both preconditioner. This can be explained by the fact that when the number of
+core increases the time for the minimization step also increases but, generally,
+when the number of cores increases, the number of iterations to reach the
+threshold also increases, and, in that case, TSIRM is more efficient to reduce
+the number of iterations. So, the overall benefit of using TSIRM is interesting.
+
+
+
\end{table*}
-
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
\begin{table*}[htbp]
