modif

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index deff6f33cbf2c9afed920cdf992094cfc9a4e149..e626ba05e614f730fb273627854e9d11c06ad6dc 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.
 %%%*********************************************************


@@ -644,11 +644,11 @@ appropriate than a direct method in a parallel context.
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
   \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$<\epsilon_{kryl}$)} \label{algo:conv}
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \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}
     \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}

-    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
+    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$}

       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
       \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
       \State $x^k=S\alpha$  \Comment{compute new solution}
       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
       \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
       \State $x^k=S\alpha$  \Comment{compute new solution}


@@ -664,7 +664,7 @@ called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  sugges
 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  (i.e.
 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  (i.e.

-$\epsilon$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+$\epsilon_{tsarm}$).  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
 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


@@ -673,25 +673,13 @@ method.
 
 To summarize, the important parameters of TSARM are:
 \begin{itemize}
 
 To summarize, the important parameters of TSARM are:
 \begin{itemize}

-\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov method
+\item $\epsilon_{tsarm}$ the threshold to stop the TSARM 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
 \item $\epsilon_{ls}$ the threshold to stop the least-square method
 \end{itemize}
 
 \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
 \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


@@ -766,13 +765,12 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 
 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 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     (line     \ref{algo:solve}    in
-Algorithm~\ref{algo:01}).   $s$ is  set to  8. CGLS  is chosen  to  minimize the
-least-squares  problem.  Two  conditions  are  used to  stop  CGLS,  either  the
-precision is under $1e-40$ or the  number of iterations is greater to $20$.  The
-external   precision    is   set    to   $1e-10$   (line    \ref{algo:conv}   in
-Algorithm~\ref{algo:01}).  Those  experiments have been performed  on a Intel(R)
-Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+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
+$1e-10$  (i.e. ).   Those experiments
+have been  performed on  a Intel(R)  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
 
 
 In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear


@@ -814,9 +812,27 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
 
 
 

-Larger experiments ....\\

 
 

-Describe the problems ex15 and ex54
+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 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}
+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.
+
+
+

 
 \begin{table*}
 \begin{center}
 
 \begin{table*}
 \begin{center}


@@ -843,6 +859,17 @@ Describe the problems ex15 and ex54
 \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|} 


@@ -900,7 +927,7 @@ Describe the problems ex15 and ex54
 %%%*********************************************************
 %%%*********************************************************
 \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
 %%%*********************************************************
 %%%*********************************************************