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[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index bfb7aa224beeacd558e81a59adba22ba7b4cf474..c8e503de6328e577ff31281def7a76c4f5440151 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -855,12 +855,12 @@ which concludes the induction and the proof.
 \label{sec:05}
 
 
 \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]
 Florida~\cite{Dav97}.
 
 \begin{table}[htbp]


@@ -881,12 +881,10 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
 \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.
 $\epsilon_{ls}=1e-40$ 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.


@@ -894,13 +892,13 @@ 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
 
 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
 (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
 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.
 
 
 and the number of iterations to perform the minimization.
 
 


@@ -932,8 +930,8 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 In order to perform larger experiments, we have tested some example applications
 
 
 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
 \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


@@ -946,20 +944,19 @@ scalable linear equations solvers:
 \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
 \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
 
 In  the  following   larger  experiments  are  described  on   two  large  scale

-architectures:  Curie and  Juqeen.  Both  these architectures  are supercomputer
-composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
-are respectively hosted  by GENCI in France and  Jülich Supercomputing Centre in
-Germany. They belongs with other similar architectures of the PRACE initiative (
-Partnership  for Advanced  Computing in  Europe)  which aims  at proposing  high
-performance supercomputing architecture to enhance research in Europe. The Curie
-architecture is composed of Intel E5-2680  processors at 2.7 GHz with 2Gb memory
-by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with
-1Gb memory per  core. Both those architecture are equiped  with a dedicated high
-speed network.
+architectures: Curie  and Juqueen.   Both these architectures  are supercomputer
+respectively  composed  of  80,640  cores   for  Curie  and  458,752  cores  for
+Juqueen. Those  machines are respectively hosted  by GENCI in  France and Jülich
+Supercomputing Centre in Germany.  They belongs with other similar architectures
+of the  PRACE initiative (Partnership  for Advanced Computing  in Europe) which
+aims  at  proposing  high  performance supercomputing  architecture  to  enhance
+research  in  Europe.  The  Curie  architecture is  composed  of  Intel  E5-2680
+processors  at 2.7  GHz with  2Gb memory  by core.  The Juqueen  architecture is
+composed of  IBM PowerPC  A2 at  1.6 GHz with  1Gb memory  per core.  Both those
+architecture are equiped with a dedicated high speed network.

 
 
 In  many situations, using  preconditioners is  essential in  order to  find the
 
 
 In  many situations, using  preconditioners is  essential in  order to  find the


@@ -967,7 +964,7 @@ solution of a linear system.  There are many preconditioners available in PETSc.
 For parallel applications all  the preconditioners based on matrix factorization
 are  not  available. In  our  experiments, we  have  tested  different kinds  of
 preconditioners, however  as it is  not the subject  of this paper, we  will not
 For parallel applications all  the preconditioners based on matrix factorization
 are  not  available. In  our  experiments, we  have  tested  different kinds  of
 preconditioners, however  as it is  not the subject  of this paper, we  will not

-present results with many preconditioners. In  practise, we have chosen to use a
+present results with many preconditioners. In  practice, we have chosen to use a

 multigrid (mg)  and successive  over-relaxation (sor). For  more details  on the
 preconditioner in PETSc please consult~\cite{petsc-web-page}.
 
 multigrid (mg)  and successive  over-relaxation (sor). For  more details  on the
 preconditioner in PETSc please consult~\cite{petsc-web-page}.
 


@@ -999,26 +996,29 @@ preconditioner in PETSc please consult~\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
 
 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 unknowns  of the
-problems)  per core  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 Table~\ref{tab:03}, 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
-the minimization  procedure: CGLS or  LSQR. Even if  we have computed  the worst
-case  between CGLS  and LSQR,  it is  clear that  TSIRM is  always  faster than
-FGMRES. For this example, the  multigrid preconditioner is faster than SOR. The
-gain  between   TSIRM  and  FGMRES  is   more  or  less  similar   for  the  two
+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
+unknowns  of  the problems)  per  core  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. Other parameters  for this application  are described in
+the legend of this Table.
+
+
+
+In  Table~\ref{tab:03},  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 the minimization  procedure: CGLS or  LSQR. Even if  we have
+computed the worst case between CGLS and  LSQR, it is clear that TSIRM is always
+faster than  FGMRES. For  this example, the  multigrid preconditioner  is faster
+than SOR. The gain between TSIRM and  FGMRES is more or less similar for the two

 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
 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.
+recorded  but they  are  time-consuming.  In  general each  $max\_iter_{kryl}*s$
+iterations which corresponds to 30*12, there are $max\_iter_{ls}$ iterations for
+the least-squares method which corresponds to 15.

 
 \begin{figure}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -1071,10 +1071,10 @@ architecture are reported.  For this  application, we fixed $\alpha=0.6$.  As it
 can be seen in that Table, the size of the problem has a strong influence on the
 number of iterations to reach the  convergence. That is why we have preferred to
 change the threshold.  If we set  it to $1e-3$ as with the previous application,
 can be seen in that Table, the size of the problem has a strong influence on the
 number of iterations to reach the  convergence. That is why we have preferred to
 change the threshold.  If we set  it to $1e-3$ as with the previous application,

-only one iteration is necessray  to reach the convergence. So Table~\ref{tab:04}
-shows the results  of differents executions with differents  number of cores and
-differents thresholds. As  with the previous example, we  can observe that TSIRM
-is faster than FGMRES. The ratio greatly depends on the number of iterations for
+only one iteration is necessary  to reach the convergence. So Table~\ref{tab:04}
+shows the  results of  different executions with  different number of  cores and
+different thresholds. As with the previous example, we can observe that TSIRM is
+faster than  FGMRES. The ratio greatly  depends on the number  of iterations for

 FMGRES to reach the threshold. The greater the number of iterations to reach the
 convergence is, the  better the ratio between our algorithm  and FMGRES is. This
 experiment is  also a  weak scaling with  approximately $25,000$  components per
 FMGRES to reach the threshold. The greater the number of iterations to reach the
 convergence is, the  better the ratio between our algorithm  and FMGRES is. This
 experiment is  also a  weak scaling with  approximately $25,000$  components per


@@ -1086,19 +1086,20 @@ Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
 Curie  architecture. So  in  this case,  the  number of  unknownws  is fixed  to
 $204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
 of two.  The  threshold is fixed to $5e-5$ and only  the $mg$ preconditioner has
 Curie  architecture. So  in  this case,  the  number of  unknownws  is fixed  to
 $204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
 of two.  The  threshold is fixed to $5e-5$ and only  the $mg$ preconditioner has

-been tested. Here  again we can see that TSIRM is  faster that FGMRES. Efficiecy
-of each algorithms is reported. It  can be noticed that FGMRES is more efficient
-than TSIRM except with $8,192$ cores and that its efficiency is greater that one
-whereas the  efficiency of TSIRM is  lower than one. Nevertheless,  the ratio of
-TSIRM  with any  version  of the  least-squares  method is  always faster.  With
-$8,192$ cores when the number of iterations is far more important for FGMRES, we
-can see that it is only slightly more important for TSIRM.
+been tested. Here again we can  see that TSIRM is faster that FGMRES. Efficiency
+of each algorithm  is reported. It can be noticed that  the efficiency of FGMRES
+is better than  the TSIRM one except with $8,192$ cores  and that its efficiency
+is  greater   that  one   whereas  the  efficiency   of  TSIRM  is   lower  than
+one.  Nevertheless, the ratio  of TSIRM  with any  version of  the least-squares
+method is  always faster.  With $8,192$  cores when the number  of iterations is
+far  more important  for  FGMRES,  we can  see  that it  is  only slightly  more
+important for TSIRM.

 
 In  Figure~\ref{fig:02}  we report  the  number  of  iterations per  second  for
 
 In  Figure~\ref{fig:02}  we report  the  number  of  iterations per  second  for

-experiments  reported in  Table~\ref{tab:05}.  This Figure  highlights that  the
-number of iterations per  seconds is more of less the same  for FGMRES and TSIRM
+experiments  reported in  Table~\ref{tab:05}.  This  Figure highlights  that the
+number of iterations  per second is more  of less the same for  FGMRES and TSIRM

 with a little advantage for FGMRES. It  can be explained by the fact that, as we
 with a little advantage for FGMRES. It  can be explained by the fact that, as we

-have previously explained, that the iterations of the least-sqaure steps are not
+have previously explained, that the iterations of the least-squares steps are not

 taken into account with TSIRM.
 
 \begin{table*}[htbp]
 taken into account with TSIRM.
 
 \begin{table*}[htbp]


@@ -1118,7 +1119,7 @@ taken into account with TSIRM.
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of FGMRES  and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
+\caption{Comparison of FGMRES  and TSIRM for ex54 of PETSc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}

 \label{tab:05}
 \end{center}
 \end{table*}
 \label{tab:05}
 \end{center}
 \end{table*}


@@ -1133,16 +1134,16 @@ taken into account with TSIRM.
 
 Concerning the  experiments some  other remarks are  interesting.
 \begin{itemize}
 
 Concerning the  experiments some  other remarks are  interesting.
 \begin{itemize}

-\item We  can tested other examples of  PETSc (ex29, ex45, ex49).  For all these
+\item We  have tested other examples of  PETSc (ex29, ex45, ex49).  For all these

   examples,  we also obtained  similar gain  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.
   examples,  we also obtained  similar gain  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 fast, it is
+\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
   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  stoped and restarted
+  feature. More  precisely, the solver must  support to be  stopped and restarted

   without decrease its  converge. That is why  with GMRES we stop it  when it is
   without decrease its  converge. That is why  with GMRES we stop it  when it is

-  naturraly  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate
+  naturally  restarted (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  for which  we stop it  each 16
   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  for which  we stop it  each 16