X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/26d38e217c09735a23eb667846b3869559154681..1e098dfc32858d5c40fdc47bec94526503edf207:/IJHPCN/paper.tex

diff --git a/IJHPCN/paper.tex b/IJHPCN/paper.tex
index 2e4cfb6..9c7ff0c 100644
--- a/IJHPCN/paper.tex
+++ b/IJHPCN/paper.tex
@@ -488,6 +488,13 @@ that the proposed TSIRM converges while the GMRES($m$) does not.
 \section{Experiments using PETSc}
 \label{sec:05}
 
+%%NEW
+In this section four kinds of experiments have been performed. First, some experiments on real matrices issued from the sparse matrix florida have been achieved out. Second, some experiments in parallel with some linear problems are reported and analyzed. Third, some experiments in parallÃ¨le with som nonlinear problems are illustrated. Finally some parameters of TSIRM are studied in order to understand their influences.
+
+
+\subsection{Real matrices}
+%%ENDNEW
+
 
 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
@@ -560,8 +567,9 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \end{table*}
 
 
-
-
+%%NEW
+\subsection{Parallel linear problems}
+%%ENDNEW
 
 In order to perform larger experiments, we have tested some example applications
 of  PETSc. These  applications are  available in  the \emph{ksp}  part,  which is
@@ -768,52 +776,6 @@ taken into account with TSIRM.
 \end{figure}
 
 
-Concerning the  experiments some  other remarks are  interesting.
-\begin{itemize}
-\item We have tested other examples  of PETSc/KSP (ex29, ex45, ex49).  For all these
-  examples,  we have also  obtained similar  gains between  GMRES and  TSIRM but
-  those  examples are  not scalable  with many  cores. In  general, we  had some
-  problems with more than $4,096$ cores.
-\item We have tested many iterative  solvers available in PETSc.  In fact, it is
-  possible to use most of them with TSIRM. From our point of view, the condition
-  to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
-  feature. More precisely,  the solver must support to  be stopped and restarted
-  without decreasing its convergence. That is  why with GMRES we stop it when it
-  is  naturally restarted (\emph{i.e.}   with $m$  the restart  parameter).  The
-  Conjugate Gradient (CG) and all its variants do not have ``restarted'' version
-  in PETSc,  so they are not efficient.   They will converge with  TSIRM but not
-  quickly because  if we  compare a  normal CG with  a CG  which is  stopped and
-  restarted every  16 iterations (for example),  the normal CG will  be far more
-  efficient.   Some  restarted  CG or  CG  variant  versions  exist and  may  be
-  interesting to study in future works.
-\end{itemize}
-%%%*********************************************************
-%%%*********************************************************
-
-
-%%NEW
-\begin{table*}[htbp]
-\begin{center}
-\begin{tabular}{|r|r|r|r|r|r|r|r|} 
-\hline
-
-  nb. cores   & \multicolumn{2}{c|}{FGMRES/ASM} & \multicolumn{2}{c|}{TSIRM CGLS/ASM} & gain& \multicolumn{2}{c|}{FGMRES/HYPRE}   \\ 
-\cline{2-5} \cline{7-8}
-                    & Time  & \# Iter.  & Time  & \# Iter. &        & Time  & \# Iter.   \\\hline \hline
-   512              & 5.54      & 685    & 2.5 &       570 & 2.21   & 128.9 & 9     \\
-   2048             & 14.95     & 1,560  &  4.32 &     746 & 3.48   & 335.7 & 9 \\
-   4096             & 25.13    & 2,369   & 5.61 &   859    & 4.48   & >1000  & -- \\
-   8192             & 44.35   & 3,197   &  7.6  &  1083    &  5.84  & >1000 &  --   \\
-
-\hline
-
-\end{tabular}
-\caption{Comparison of FGMRES  and TSIRM for ex45 of PETSc/KSP with two preconditioner (ASM and HYPRE)  having 25,000 components per core on Curie ($\epsilon_{tsirm}=1e-10$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
-\label{tab:06}
-\end{center}
-\end{table*}
-
-
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex45_curie}
@@ -822,6 +784,19 @@ Concerning the  experiments some  other remarks are  interesting.
 \end{figure}
 
 
+%%NEW
+
+\subsection{Parallel nonlinear problems}
+
+With  PETSc,  linear  solvers  are  used inside  nonlinear  solvers.   The  SNES
+(Scalable Nonlinear  Equations Solvers) module  in PETSc implements easy  to use
+methods,  like  Newton-type, quasi-Newton  or  full  approximation scheme  (FAS)
+multigrid to solve systems of nonlinears equations.  As the SNES is based on the
+Krylov methods of PETSc, it is interesting to investigate if the TSIRM method is
+also efficient and scalable with non linear problems.
+
+
+
 
 \begin{table*}[htbp]
 \begin{center}
@@ -866,8 +841,39 @@ Concerning the  experiments some  other remarks are  interesting.
 \end{table*}
 
 
+\subsection{Influence of parameters for TSIRM}
+
+
+
+
+
+\subsection{Experiments conclusions }
+
+{\bf A refaire}
+
+Concerning the  experiments some  other remarks are  interesting.
+\begin{itemize}
+\item We have tested other examples  of PETSc/KSP (ex29, ex45, ex49).  For all these
+  examples,  we have also  obtained similar  gains between  GMRES and  TSIRM but
+  those  examples are  not scalable  with many  cores. In  general, we  had some
+  problems with more than $4,096$ cores.
+\item We have tested many iterative  solvers available in PETSc.  In fact, it is
+  possible to use most of them with TSIRM. From our point of view, the condition
+  to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
+  feature. More precisely,  the solver must support to  be stopped and restarted
+  without decreasing its convergence. That is  why with GMRES we stop it when it
+  is  naturally restarted (\emph{i.e.}   with $m$  the restart  parameter).  The
+  Conjugate Gradient (CG) and all its variants do not have ``restarted'' version
+  in PETSc,  so they are not efficient.   They will converge with  TSIRM but not
+  quickly because  if we  compare a  normal CG with  a CG  which is  stopped and
+  restarted every  16 iterations (for example),  the normal CG will  be far more
+  efficient.   Some  restarted  CG or  CG  variant  versions  exist and  may  be
+  interesting to study in future works.
+\end{itemize}
+
 %%ENDNEW
 
+
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
@@ -876,28 +882,20 @@ Concerning the  experiments some  other remarks are  interesting.
 %%%*********************************************************
 %%%*********************************************************
 
-A new two-stage iterative  algorithm TSIRM has been proposed in this article,
-in order to accelerate the convergence of Krylov iterative  methods.
-Our TSIRM proposal acts as a merger between Krylov based solvers and
-a least-squares minimization step.
-The convergence of the method has been proven in some situations, while 
-experiments up to 16,394 cores have been led to verify that TSIRM runs
-5 or  7 times  faster than GMRES.
+%%NEW
+In this paper a new two-stage algorithm TSIRM has been described. This method allows us to improve the convergence of  Krylov iterative  methods. It is based
+on a least-squares minimization step which uses the  Krylov residuals.
 
 
-For  future  work, the  authors'  intention is  to  investigate  other kinds  of
-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  adaptive to improve the
-overall performances of the proposal.   Finally, this solver will be implemented
-inside PETSc, which would be of interest as it would  allows us to test
-all the non-linear  examples and compare our algorithm  with the other algorithm
-implemented in PETSc.
+We have implemented our code in PETSc in order to show that it is efficient and scalable. Some experiments with classical examples of PETSc for linear and nonlinear problems have been performed. We observed that TSIRM outperforms GMRES variants when the number of iterations is large. TSIRM is also scalable since we made some experiments with up to 16,394 cores.
 
+We also observed that TSIRM is efficient with different preconditioners. The hypre preconditioner that is globally very efficient for many problems is also very time consuming. Consequently, sometimes using a less performent preconditioners may be a better solution. In that case, TSIRM is also more efficient than traditional Krylov methods.
 
-% conference papers do not normally have an appendix
+{\bf A CHECKER !!}
+The influence of some important parameters of TSIRM have been studied. It can be noticed that they have a strong influence on the convergence speed
+
+In future works, we plan to study other problems coming from different research areas. Other efficient Krylov optimisation methods as communication avoiding technique may be interesting to be investigated
+%%ENDNEW
 
 
 
