X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/9fda9aa7b779d8ce482f4ecf1a6d1623cee17fc2..ad91b931741db537cefe135f277f1fd5065db21f:/paper.tex

diff --git a/paper.tex b/paper.tex
index 4f1da9b..6292705 100644
--- a/paper.tex
+++ b/paper.tex
@@ -241,7 +241,7 @@
 % quality.
 
 
-%\usepackage{eqparbox}
+\usepackage{eqparbox}
 % Also of notable interest is Scott Pakin's eqparbox package for creating
 % (automatically sized) equal width boxes - aka "natural width parboxes".
 % Available at:
@@ -369,7 +369,7 @@
 %
 % paper title
 % can use linebreaks \\ within to get better formatting as desired
-\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}
+\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
 %oÃ¹
 %\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
 %oÃ¹
@@ -428,16 +428,16 @@ Email: lilia.ziane@inria.fr}
 
 
 \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}
@@ -646,12 +646,12 @@ appropriate than a single direct method in a parallel context.
 
 
 \begin{algorithm}[t]
-\caption{TSARM}
+\caption{TSIRM}
 \begin{algorithmic}[1]
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
-  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \label{algo:conv}
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
     \State  $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$   \label{algo:solve}
     \State retrieve error
     \State $S_{k \mod s}=x^k$ \label{algo:store}
@@ -670,17 +670,17 @@ iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
 called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
 equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
 threshold must be specified for the  solver. In practice, this threshold must be
-much  smaller  than the  convergence  threshold  of  the TSARM  algorithm  (\emph{i.e.}
-$\epsilon_{tsarm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
-solution  $x_k$  into the  column  $k~  mod~Â s$ of  the  matrix  $S$. After  the
+much  smaller  than the  convergence  threshold  of  the TSIRM  algorithm  (\emph{i.e.}
+$\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+solution  $x_k$  into the  column  $k~ mod~ s$ of  the  matrix  $S$. After  the
 minimization, the matrix $S$ is reused with the new values of the residuals.  To
 solve the minimization problem, an  iterative method is used. Two parameters are
 required for that: the maximum number of iteration and the threshold to stop the
 method.
 
-Let us summarize the most important parameters of TSARM:
+Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
-\item $\epsilon_{tsarm}$: the threshold to stop the TSARM method;
+\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
 \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
 \item $s$: the number of outer iterations before applying the minimization step;
 \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-square method;
@@ -688,7 +688,7 @@ Let us summarize the most important parameters of TSARM:
 \end{itemize}
 
 
-The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
+The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-square step,  all the other  parts are
 obvious to  achieve out in parallel. In  order to develop a  parallel version of
 our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
@@ -759,16 +759,16 @@ Table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
 characteristics. For all  the matrices, the name, the field,  the number of rows
 and the number of nonzero elements is given.
 
-\begin{table*}
+\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
 
@@ -776,14 +776,14 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \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
 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
-$\epsilon_{tsarm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
+$\epsilon_{tsirm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
 Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
 
 
@@ -791,20 +791,20 @@ 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 TSARM,  the same  solver and  the same
-preconditionner is used.  This Table shows that TSARM can drastically reduce the
+different  preconditioner is used.   With TSIRM,  the same  solver and  the same
+preconditionner is 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 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}
+\begin{table}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
- \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ 
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\ 
 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
@@ -849,12 +849,12 @@ In the following larger experiments are described on two large scale architectur
 
 {\bf Description of preconditioners}
 
-\begin{table*}
+\begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores & precond   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
+  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 \\
@@ -868,7 +868,7 @@ In the following larger experiments are described on two large scale architectur
 \hline
 
 \end{tabular}
-\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.}
+\caption{Comparison of FGMRES and TSIRM 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*}
@@ -877,16 +877,27 @@ 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}
+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
+the minimization  procedure: CGLS or  LSQR. Even if  we have computed  the worst
+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.  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
   \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
 \caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
@@ -894,15 +905,26 @@ the minimization procedure: CGLS or LSQR.
 \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.
 
 
 
-\begin{table*}
+
+
+
+\begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
+  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \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      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
@@ -921,15 +943,15 @@ the minimization procedure: CGLS or LSQR.
 \end{table*}
 
 
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
 
 
-
-\begin{table*}
+\begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
 \cline{2-7} \cline{9-11}
                     & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
    512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
