09-10-2014 01

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index fb68702339997705b0dee698d5cd3d1d9c011785..e7e7e0db029a6ef0637ea4f6739dc80848572548 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -354,6 +354,7 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{multirow}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{multirow}

+\usepackage{graphicx}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -583,8 +584,7 @@ performances.
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
 a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
 a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some

-convergence  results  on this  method.   In Section~\ref{sec:05},  parallization
-details  of  TSARM  are  given.  Section~\ref{sec:06}  shows  some  experimental
+convergence  results  on this  method.   Section~\ref{sec:05}  shows  some  experimental

 results  obtained on large  clusters of  our algorithm  using routines  of PETSc
 toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 %%%*********************************************************
 results  obtained on large  clusters of  our algorithm  using routines  of PETSc
 toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 %%%*********************************************************


@@ -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
 
 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
 
 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


@@ -680,18 +680,6 @@ To summarize, the important parameters of TSARM are:
 \item $\epsilon_{ls}$ the threshold to stop the least-square method
 \end{itemize}
 
 \item $\epsilon_{ls}$ the threshold to stop the least-square method
 \end{itemize}
 

-%%%*********************************************************
-%%%*********************************************************
-
-\section{Convergence results}
-\label{sec:04}
-
-
-
-%%%*********************************************************
-%%%*********************************************************
-\section{Parallelization}
-\label{sec:05}

 
 The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-square step,  all the other  parts are
 
 The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-square step,  all the other  parts are


@@ -733,10 +721,21 @@ In each iteration  of CGLS, there is two  matrix-vector multiplications and some
 classical operations:  dots, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.
 
 classical operations:  dots, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.
 

+
+
+%%%*********************************************************
+%%%*********************************************************
+
+\section{Convergence results}
+\label{sec:04}
+
+
+
+

 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}

-\label{sec:06}
+\label{sec:05}

 
 
 In order to see the influence of our algorithm with only one processor, we first
 
 
 In order to see the influence of our algorithm with only one processor, we first


@@ -814,17 +813,19 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
 
 
 

-Larger experiments ....\\

 
 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:
 \begin{itemize}
 
 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:
 \begin{itemize}

-\item  ex15   is  an  example  which   solves  in  parallel   a  2D  homogeneous
-  Laplacian. Thediagonal 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 propagation...
-\item
+\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 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 coefficient in embedded circle, it is called $\alpha$.

 \end{itemize}
 \end{itemize}

-
+For more technical details on  these applications, interested reader are invited
+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.

 
 
 
 
 
 


@@ -854,6 +855,17 @@ In the following we describe the applications of PETSc we have experimented. Tho
 \end{table*}
 
 
 \end{table*}
 
 

+\begin{figure}
+\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}}
+\label{fig:01}
+\end{figure}
+
+
+
+
+

 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 


@@ -911,7 +923,7 @@ In the following we describe the applications of PETSc we have experimented. Tho
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

-\label{sec:07}
+\label{sec:06}

 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************