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[GMRES2stage.git] / paper.tex

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@@ -613,57 +613,67 @@ $b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
 inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
 points of our solver are given in Algorithm~\ref{algo:01}.
 
 inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
 points of our solver are given in Algorithm~\ref{algo:01}.
 

-In order to accelerate the convergence, the outer iteration applies a least-square minimization on the residuals computed by the inner   some  error functions  over a  Krylov
-subspace~\cite{Saad2003}. At  each iteration, the  sparse linear system  $Ax=b$ is
-solved   iteratively    with   an   iterative   method,    for   example   GMRES
-method~\cite{Saad86} or  some of its variants,  and the Krylov  subspace that we
-used is spanned by a basis  $S$ composed of successive solutions issued from the
-inner iteration
-\begin{equation}
-  S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
-\end{equation} 
-The advantage  of such a Krylov subspace  is that we neither  need an orthogonal
-basis nor  any synchronization  between processors to  generate this  basis. The
-algorithm  is periodically  restarted every  $s$ iterations  with a  new initial
-guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
-subspace spanned by  vectors of $S$, where $\alpha$ is a  solution of the normal
-equations
-\begin{equation}
-  R^TR\alpha = R^Tb,
-\end{equation}
-which is associated with the least-squares problem
+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.
+
+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
+example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
+inner solver. The current solution of the Krylov method is saved inside a matrix
+$S$ composed of successive solutions computed by the inner iteration.
+
+Periodically, every $s$ iterations, the minimization step is applied in order to
+compute a new  solution $x$. For that, the previous  residuals are computed with
+$(b-AS)$. The minimization of the residuals is obtained by 

 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}
 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}

-such  that $R=AS$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
-$s\ll n$,  and $R^T$ denotes  the transpose of  matrix $R$. We use  an iterative
-method   to  solve   the  least-squares   problem~(\ref{eq:01})  such   as  CGLS
-~\cite{Hestenes52}  or LSQR~\cite{Paige82}  which  are more  appropriate than  a
-direct method in the parallel context.
+with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+
+
+In  practice, $R$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
+$s\ll n$.   In order  to minimize~(\ref{eq:01}), a  least-square method  such as
+CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Those  methods are more
+appropriate than a direct method in a parallel context.

 
 \begin{algorithm}[t]
 
 \begin{algorithm}[t]

-\caption{A Krylov two-stage algorithm}
+\caption{TSARM}

 \begin{algorithmic}[1]
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
 \begin{algorithmic}[1]
   \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} \label{algo:conv}
-    \State Solve iteratively $Ax^k=b$  \label{algo:solve}
-    \State $S_{k~mod~s}=x^k$ 
-    \If {$k$ mod $s=0$ {\bf and} not convergence}
-      \State Compute dense matrix $R=AS$
-      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
-      \State Compute minimizer $x^k=S\alpha$
+  \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}
+    \State retrieve error
+    \State $S_{k~mod~s}=x^k$ \label{algo:store}
+    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon$}
+      \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}

     \EndIf
   \EndFor
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 
     \EndIf
   \EndFor
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 

-Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ 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.
+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 $m$
+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
+
+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 $s$ the number of outer iterations before applying the minimization step
+\end{itemize}

 
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