X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/d6d5ef6890f9b888cf59c00e8eacc5f2863b1458..2fc464daa5a04774f5fa10bd96d36c53cc908a4a:/paper.tex

diff --git a/paper.tex b/paper.tex
index 4f9f60e..54e35d3 100644
--- a/paper.tex
+++ b/paper.tex
@@ -370,10 +370,7 @@
 % paper title
 % can use linebreaks \\ within to get better formatting as desired
 \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
-%\title{???}
+
 
 
 
@@ -428,16 +425,16 @@ 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}
@@ -607,7 +604,7 @@ is summarized while intended perspectives are provided.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Two-stage algorithm with least-square residuals minimization}
+\section{Two-stage iteration with least-square residuals minimization algorithm}
 \label{sec:03}
 A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
@@ -639,7 +636,7 @@ with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
-with $s\ll n$.   In order  to minimize~(\eqref{eq:01}), a  least-square method  such as
+with $s\ll n$.   In order  to minimize~\eqref{eq:01}, a  least-square method  such as
 CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Remark that these  methods are more
 appropriate than a single direct method in a parallel context.
 
@@ -675,7 +672,7 @@ $\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in cop
 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
+required for that: the maximum number of iterations and the threshold to stop the
 method.
 
 Let us summarize the most important parameters of TSIRM:
@@ -698,7 +695,7 @@ colums in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-square minimization, CGLS and LSQR.
 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or
-less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
 
 \begin{algorithm}[t]
 \caption{CGLS}
@@ -725,7 +722,7 @@ less the same principle but it take more place, so we briefly explain the parall
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
-classical operations:  dots, norm, multiplication  and addition on  vectors. All
+classical operations:  dot product, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.
 
 
@@ -759,16 +756,16 @@ 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*}[htbp]
+\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
 
@@ -776,7 +773,7 @@ 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
@@ -877,9 +874,12 @@ 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.
+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
@@ -887,8 +887,12 @@ 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
-
+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
@@ -898,6 +902,17 @@ preconditioners
 \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.
+
+
+
 
 
 
@@ -925,7 +940,7 @@ preconditioners
 \end{table*}
 
 
-
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported
 
 
 \begin{table*}[htbp]
