X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/df562f71fc6dcfdbb4e0b77f138977ca7219df6f..26d38e217c09735a23eb667846b3869559154681:/IJHPCN/paper.tex

diff --git a/IJHPCN/paper.tex b/IJHPCN/paper.tex
index 0c88f29..2e4cfb6 100644
--- a/IJHPCN/paper.tex
+++ b/IJHPCN/paper.tex
@@ -49,9 +49,7 @@
 \makeatletter
 \def\theequation{\arabic{equation}}
 
-%\JOURNALNAME{\TEN{\it Int. J. System Control and Information
-%Processing,
-%Vol. \theVOL, No. \theISSUE, \thePUBYEAR\hfill\thepage}}%
+\JOURNALNAME{\TEN{\it International Journal of High Performance Computing and Networking}}
 %
 %\def\BottomCatch{%
 %\vskip -10pt
@@ -73,10 +71,9 @@
 
 \setcounter{page}{1}
 
-\LRH{F. Wang et~al.}
+\LRH{R. Couturier, L. Ziane Khodja and C. Guyeux}
 
-\RRH{Metadata Based Management and Sharing of Distributed Biomedical
-Data}
+\RRH{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm}
 
 \VOL{x}
 
@@ -86,7 +83,7 @@ Data}
 
 \BottomCatch
 
-\PUBYEAR{2012}
+\PUBYEAR{2015}
 
 \subtitle{}
 
@@ -109,19 +106,25 @@ Data}
 
 
 \begin{abstract}
-In  this article, a  two-stage iterative  algorithm is  proposed to  improve the
+In  this paper,  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
+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 least-squares minimization step is applied on the matrix composed by the saved
-residuals, in order  to 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  runs around  5 or  7 times  faster than
-GMRES.
+residuals, in order to  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.
+%%NEW
+Several experiments  have been performed
+with  the PETSc  solver  with  linear and  nonlinear  problems.  They show  good
+speedups   compared  to   GMRES  with   up  to   16,394  cores   with  different
+preconditioners.
+%%ENDNEW
 \end{abstract}
 
+
+
 \KEYWORD{Iterative Krylov methods; sparse linear and non linear systems; two stage iteration; least-squares residual minimization; PETSc.}
 
 %\REF{to this paper should be made as follows: Rodr\'{\i}guez
@@ -131,28 +134,11 @@ GMRES.
 %Semantics and Ontologies}, Vol. x, No. x, pp.xxx\textendash xxx.}
 
 \begin{bio}
-Manuel Pedro Rodr\'iguez Bol\'ivar received his PhD in Accounting at
-the University of Granada. He is a Lecturer at the Department of
-Accounting and Finance, University of Granada. His research
-interests include issues related to conceptual frameworks of
-accounting, diffusion of financial information on Internet, Balanced
-Scorecard applications and environmental accounting. He is author of
-a great deal of research studies published at national and
-international journals, conference proceedings as well as book
-chapters, one of which has been edited by Kluwer Academic
-Publishers.\vs{9}
-
-\noindent Bel\'en Sen\'es Garc\'ia received her PhD in Accounting at
-the University of Granada. She is a Lecturer at the Department of
-Accounting and Finance, University of Granada. Her research
-interests are related to cultural, institutional and historic
-accounting and in environmental accounting. She has published
-research papers at national and international journals, conference
-proceedings as well as chapters of books.\vs{8}
-
-\noindent Both authors have published a book about environmental
-accounting edited by the Institute of Accounting and Auditing,
-Ministry of Economic Affairs, in Spain in October 2003.
+RaphaÃ«l Couturier ....
+
+\noindent Lilia Ziane Khodja ...
+
+\noindent Christophe Guyeux ...
 \end{bio}
 
 
@@ -511,7 +497,7 @@ 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]
+\begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|r|r|r|} 
 \hline
@@ -528,7 +514,7 @@ 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*}
 Chosen parameters  are detailed below.   
 We have  stopped  the  GMRES every  30
 iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default 
@@ -550,7 +536,7 @@ fact this also depends on two parameters: the number of iterations before stoppi
 and the number of iterations to perform the minimization.
 
 
-\begin{table}[htbp]
+\begin{table*}[htbp]
 \begin{center}
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
@@ -571,7 +557,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
 \label{tab:02}
 \end{center}
-\end{table}
+\end{table*}
 
 
 
@@ -638,7 +624,7 @@ preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc 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*}
@@ -710,7 +696,7 @@ interesting.
 \hline
 
 \end{tabular}
-\caption{Comparison of FGMRES  and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
+\caption{Comparison of FGMRES  and TSIRM with FGMRES algorithms for ex54 of PETSc/KSP (both with the MG preconditioner) with 25,000 components per core on Curie ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
 \label{tab:04}
 \end{center}
 \end{table*}
@@ -769,7 +755,7 @@ taken into account with TSIRM.
 \hline
 
 \end{tabular}
-\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.}
+\caption{Comparison of FGMRES  and TSIRM for ex54 of PETSc/KSP (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*}
@@ -784,7 +770,7 @@ taken into account with TSIRM.
 
 Concerning the  experiments some  other remarks are  interesting.
 \begin{itemize}
-\item We have tested other examples  of PETSc (ex29, ex45, ex49).  For all these
+\item We have tested other examples  of PETSc/KSP (ex29, ex45, ex49).  For all these
   examples,  we have also  obtained similar  gains 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.
@@ -805,6 +791,82 @@ Concerning the  experiments some  other remarks are  interesting.
 %%%*********************************************************
 
 
+%%NEW
+\begin{table*}[htbp]
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores   & \multicolumn{2}{c|}{FGMRES/ASM} & \multicolumn{2}{c|}{TSIRM CGLS/ASM} & gain& \multicolumn{2}{c|}{FGMRES/HYPRE}   \\ 
+\cline{2-5} \cline{7-8}
+                    & Time  & \# Iter.  & Time  & \# Iter. &        & Time  & \# Iter.   \\\hline \hline
+   512              & 5.54      & 685    & 2.5 &       570 & 2.21   & 128.9 & 9     \\
+   2048             & 14.95     & 1,560  &  4.32 &     746 & 3.48   & 335.7 & 9 \\
+   4096             & 25.13    & 2,369   & 5.61 &   859    & 4.48   & >1000  & -- \\
+   8192             & 44.35   & 3,197   &  7.6  &  1083    &  5.84  & >1000 &  --   \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES  and TSIRM for ex45 of PETSc/KSP with two preconditioner (ASM and HYPRE)  having 25,000 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:06}
+\end{center}
+\end{table*}
+
+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex45_curie}
+\caption{Number of iterations per second with ex45 and the same parameters as in Table~\ref{tab:06} (weak scaling)}
+\label{fig:03}
+\end{figure}
+
+
+
+\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   \\ 
+\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\\
+
+\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.}
+\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  \\ 
+\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 \\
+
+\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.}
+\label{tab:08}
+\end{center}
+\end{table*}
+
+
+%%ENDNEW
 
 %%%*********************************************************
 %%%*********************************************************
