X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/1e098dfc32858d5c40fdc47bec94526503edf207..3f095ff7f3fff2553897be9e8dce25d2c3e9298f:/IJHPCN/paper.tex

diff --git a/IJHPCN/paper.tex b/IJHPCN/paper.tex
index 9c7ff0c..999ce37 100644
--- a/IJHPCN/paper.tex
+++ b/IJHPCN/paper.tex
@@ -608,7 +608,7 @@ However, for parallel applications, all  the preconditioners based on matrix fac
 are  not  available. In  our  experiments, we  have  tested  different kinds  of
 preconditioners, but  as it is  not the subject  of this paper, we  will not
 present results with many preconditioners. In  practice, we have chosen to use a
-multigrid (mg)  and successive  over-relaxation (sor). For  further details  on the
+multigrid (MG)  and successive  over-relaxation (SOR). For  further details  on the
 preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
 
 
@@ -621,18 +621,18 @@ preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
   nb. cores & precond   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ 
 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
-  2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
-  2,048      & sor                   & 745.37   & 57,060    & 87.31  & 6,150   & 104.21 & 7,230  & 8.53 \\
-  4,096      & mg                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
-  4,096      & sor                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
-  8,192      & mg                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
-  8,192      & sor                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
-  16,384     & mg                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
-  16,384     & sor                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
+  2,048      & MG                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
+  2,048      & SOR                   & 745.37   & 57,060    & 87.31  & 6,150   & 104.21 & 7,230  & 8.53 \\
+  4,096      & MG                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
+  4,096      & SOR                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
+  8,192      & MG                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
+  8,192      & SOR                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
+  16,384     & MG                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
+  16,384     & SOR                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc/KSP with two preconditioners (mg and sor) having 25,000 components per core on Juqueen ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc/KSP with two preconditioners (MG and SOR) having 25,000 components per core on Juqueen ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
 \label{tab:03}
 \end{center}
 \end{table*}
@@ -640,7 +640,7 @@ preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
 Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
 example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
 are studied  ranging from 2,048 up-to  16,383 with the  two preconditioners {\it
-  mg}  and {\it  sor}.   For those  experiments,  the number  of components  (or
+  MG}  and {\it  SOR}.   For those  experiments,  the number  of components  (or
 unknowns  of  the problems)  per  core  is fixed  at  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
@@ -791,56 +791,65 @@ taken into account with TSIRM.
 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
+multigrid to  solve systems of  nonlinears equations.  As  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.
-
-
+also efficient  and scalable with non  linear problems. In PETSc,  some examples
+are provided.  An important criteria is the scalability of the initial code with
+classical solvers. Consequently, we have chosen  two of these examples: ex14 and
+ex20.  In ex14, the code solves the  Bratu (SFI - solid fuel ignition) nonlinear
+partial  difference equations  in 3  dimension.  In  ex20, the  code solves  a 3
+dimension radiative transport test problem.  For more details on these examples,
+interested readers are invited to see the code in the PETSc examples.
 
+In Table~\ref{tab:07} we  report the result of our experiments  for the example
+ex14. 
 
 \begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores   & \multicolumn{2}{c|}{FGMRES/BJAC} & \multicolumn{2}{c|}{TSIRM CGLS/BJAC} & gain   \\ 
+  nb. cores   & \multicolumn{2}{c|}{FGMRES/BJAC} & \multicolumn{2}{c|}{TSIRM CGLS/BJAC} & gain  \\ 
 \cline{2-5}
-                    & Time         & \# Iter.  & Time   & \# Iter.  &  \\\hline \hline
-   1024              & 667.92      & 48,732    & 81.65  &     5,087 &  8.18 \\
-   2048             & 966.87       & 77,177    &  90.34 &     5,716 &  10.70\\
-   4096             & 1,742.31     & 124,411   &  119.21 &   6,905  & 14.61\\
-   8192             & 2,739.21     & 187,626   &  168.9  &  9,000   & 16.22\\
+                    & Time         & \# Iter.  & Time   & \# Iter. &  \\\hline \hline
+   1024              & 159.52      & 11,584    &  26.34  &     1,563  &  6.06  \\
+   2048             & 226.24       & 16,459    &  37.23 &     2,248   &  6.08\\
+   4096             & 391.21     & 27,794   &  50.93 &   2,911  &  7.69\\
+   8192             & 543.23     & 37,770   &  79.21  &  4,324  & 6.86 \\
 
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES  and TSIRM for ex20 of PETSc/SNES with a Block Jacobi  preconditioner  having 100,000 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.}
+\caption{Comparison of FGMRES  and TSIRM for ex14 of PETSc/SNES with a Block Jacobi  preconditioner  having 100,000 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:07}
 \end{center}
 \end{table*}
 
+
 \begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores   & \multicolumn{2}{c|}{FGMRES/BJAC} & \multicolumn{2}{c|}{TSIRM CGLS/BJAC} & gain  \\ 
+  nb. cores   & \multicolumn{2}{c|}{FGMRES/BJAC} & \multicolumn{2}{c|}{TSIRM CGLS/BJAC} & gain   \\ 
 \cline{2-5}
-                    & Time         & \# Iter.  & Time   & \# Iter. &  \\\hline \hline
-   1024              & 159.52      & 11,584    &  26.34  &     1,563  &  6.06  \\
-   2048             & 226.24       & 16,459    &  37.23 &     2,248   &  6.08\\
-   4096             & 391.21     & 27,794   &  50.93 &   2,911  &  7.69\\
-   8192             & 543.23     & 37,770   &  79.21  &  4,324  & 6.86 \\
+                    & Time         & \# Iter.  & Time   & \# Iter.  &  \\\hline \hline
+   1024              & 667.92      & 48,732    & 81.65  &     5,087 &  8.18 \\
+   2048             & 966.87       & 77,177    &  90.34 &     5,716 &  10.70\\
+   4096             & 1,742.31     & 124,411   &  119.21 &   6,905  & 14.61\\
+   8192             & 2,739.21     & 187,626   &  168.9  &  9,000   & 16.22\\
 
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES  and TSIRM for ex14 of PETSc/SNES with a Block Jacobi  preconditioner  having 100,000 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.}
+\caption{Comparison of FGMRES  and TSIRM for ex20 of PETSc/SNES with a Block Jacobi  preconditioner  having 100,000 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:08}
 \end{center}
 \end{table*}
 
 
+
+
 \subsection{Influence of parameters for TSIRM}
 
 
