new

diff --git a/paper.tex b/paper.tex
index acb46bbfc4d18076e7ec6a4f85b32c3c5df5e3c8..f2e06211923bfa7caa6e5ad1954c79dcbe2c081c 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -583,10 +583,10 @@ 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.    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.
+convergence  results  on this  method.   In Section~\ref{sec:05},  parallization
+details  of  TSARM  are  given.  Section~\ref{sec:06}  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.

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@@ -604,7 +604,7 @@ perspectives.
 
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-\section{A Krylov two-stage algorithm}
+\section{Two-stage algorithm with least-square residuals minimization}

 \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
 \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


@@ -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$<\epsilon$)} \label{algo:conv}
-    \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{kryl}$)} \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 retrieve error
     \State $S_{k~mod~s}=x^k$ \label{algo:store}

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

       \State $R=AS$ \Comment{compute dense matrix}
       \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}
       \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}


@@ -660,19 +660,24 @@ appropriate than a direct method in a parallel context.
 
 Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
 iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
 
 Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
 iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is

-called for a  maximum of $m$ iterations.  In practice, we  suggest to choose $m$
+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
 equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance

-threshold must be specified for the  solver. In practise, 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 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. % à continuer Line
+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
+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.

 
 To summarize, the important parameters of are:
 \begin{itemize}
 
 To summarize, the important parameters of are:
 \begin{itemize}

-\item $\epsilon$ the threshold to stop the method
-\item $m$ the number of iterations for the krylov method
+\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov 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 $s$ 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}
 
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 \end{itemize}
 
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@@ -681,11 +686,18 @@ To summarize, the important parameters of are:
 \section{Convergence results}
 \label{sec:04}
 
 \section{Convergence results}
 \label{sec:04}
 

+
+

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-\section{Experiments using petsc}
+\section{Parallelization}

 \label{sec:05}
 
 \label{sec:05}
 

+%%%*********************************************************
+%%%*********************************************************
+\section{Experiments using petsc}
+\label{sec:06}
+

 
 In order to see the influence of our algorithm with only one processor, we first
 show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
 
 In order to see the influence of our algorithm with only one processor, we first
 show  a comparison  with the  standard version  of GMRES  and our  algorithm. In


@@ -819,7 +831,7 @@ Larger experiments ....
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 \section{Conclusion}
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 \section{Conclusion}

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

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