X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/d1ede5c6359bdeda060c8b0336f5db31e226a285..b599aea7bd0cb52057a014508bf1bccdd41c7770:/paper.tex

diff --git a/paper.tex b/paper.tex
index e626ba0..112b322 100644
--- a/paper.tex
+++ b/paper.tex
@@ -615,7 +615,7 @@ points of our solver are given in Algorithm~\ref{algo:01}.
 
 In order to accelerate the convergence, the outer iteration periodically applies
 a least-square minimization  on the residuals computed by  the inner solver. The
-inner solver is a Krylov based solver which does not required to be changed.
+inner solver is based on a Krylov method which does not require to be changed.
 
 At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
 iterations, using an iterative method restarting with the previous solution. For
@@ -768,22 +768,20 @@ the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
 the GMRES every 30 iterations, $max\_iter_{kryl}=30$).  $s$ 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
-$1e-10$  (i.e. ).   Those experiments
-have been  performed on  a Intel(R)  Core(TM) i7-3630QM CPU  @ 2.40GHz  with the
-version 3.5.1 of PETSc.
+$\epsilon_{tsarm}=1e-10$.  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 out 2
 stage algorithm are  given. In the second column, it can  be noticed that either
 gmres or fgmres is used to  solve the linear system.  According to the matrices,
-different preconditioner is  used.  With the 2 stage  algorithm, the same solver
-and  the same  preconditionner  is used.   This  Table shows  that  the 2  stage
-algorithm  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  and   the  number  of  iterations  to  perform  the
-minimization.
+different  preconditioner is used.   With TSARM,  the same  solver and  the same
+preconditionner is used.  This Table shows that TSARM 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  and  the number  of
+iterations to perform the minimization.
 
 
 \begin{table}
@@ -813,14 +811,14 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
-In   the   following  we   describe   the   applications   of  PETSc   we   have
-experimented. Those applications  are available in the ksp  part which is suited
-for scalable linear equations solvers:
+In order to perform larger  experiments, we have tested some example application
+of PETSc. Those  applications are available in the 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  equals to  4  and  4  extra-diagonals
   representing the neighbors in each directions  is equal to -1. This example is
-  used  in many  physical phenomena  , for  exemple, heat  and fluid  flow, wave
+  used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
   propagation...
 \item ex54 is another example based on 2D problem discretized with quadrilateral
   finite elements. For this example, the user can define the scaling of material
@@ -831,15 +829,17 @@ to  read the  codes available  in the  PETSc sources.   Those problem  have been
 chosen because they  are scalable with many cores. We  have tested other problem
 but they are not scalable with many cores.
 
+In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
 
 
+{\bf Description of preconditioners}
 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores & precond   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
+  nb. cores & precond   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM 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 \\
@@ -853,11 +853,23 @@ but they are not scalable with many cores.
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSARM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
 \label{tab:03}
 \end{center}
 \end{table*}
 
+Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
+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.
+
+In this Table, we  can notice that TSARM is always faster  than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSARM according to
+the minimization procedure: CGLS or LSQR.
+
 
 \begin{figure}
 \centering
