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[GMRES2stage.git] / paper.tex

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@@ -618,15 +618,14 @@ It can be summarized as follows: the
 inner solver is a Krylov based one. In order to accelerate its convergence, the 
 outer solver periodically applies a least-squares minimization  on the residuals computed by  the inner one. %Tsolver which does not required to be changed.
 
 inner solver is a Krylov based one. In order to accelerate its convergence, the 
 outer solver periodically applies a least-squares minimization  on the residuals computed by  the inner one. %Tsolver which does not required to be changed.
 

-At each outer iteration, the sparse linear system $Ax=b$ is partially 
-solved using only $m$
-iterations of an iterative method, this latter being initialized with the 
-last obtained approximation. 
-GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
-inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
-$S$, which is composed by the $s$ last solutions that have been computed during 
-the inner iterations phase.
-In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. 
+At each  outer iteration,  the sparse linear  system $Ax=b$ is  partially solved
+using only $m$ iterations of  an iterative method, this latter being initialized
+with the last obtained approximation.  GMRES method~\cite{Saad86}, or any of its
+variants, can potentially be used  as inner solver. The current approximation of
+the Krylov  method is then  stored inside  a $n \times  s$ matrix $S$,  which is
+composed by  the $s$  last solutions  that have been  computed during  the inner
+iterations phase.   In the remainder,  the $i$-th column  vector of $S$  will be
+denoted by $S_i$.

 
 At each $s$ iterations, another kind of minimization step is applied in order to
 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
 
 At each $s$ iterations, another kind of minimization step is applied in order to
 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by


@@ -652,9 +651,8 @@ appropriate than a single direct method in a parallel context.
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x_0$
   \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x_0$
   \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \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  $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$   \label{algo:solve}
+    \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column (k mod s) of S}

     \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
             \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
     \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
             \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}


@@ -665,19 +663,22 @@ appropriate than a single direct method in a parallel context.
 \label{algo:01}
 \end{algorithm}
 
 \label{algo:01}
 \end{algorithm}
 

-Algorithm~\ref{algo:01}  summarizes  the principle  of  the proposed  method.  The  outer
-iteration is  inside the \emph{for}  loop. Line~\ref{algo:solve}, the Krylov  method is
-called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
-equal to  the restart  number in 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 TSIRM  algorithm  (\emph{i.e.},
-$\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k \mod s}=x^k$ consists in copying the
-solution  $x_k$  into the  column  $k \mod s$ of $S$.
-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 iterations and the threshold to stop the
-method.
+Algorithm~\ref{algo:01} summarizes  the principle  of the proposed  method.  The
+outer iteration is inside the \emph{for} loop. Line~\ref{algo:solve}, the Krylov
+method is called  for a maximum of $max\_iter_{kryl}$  iterations.  In practice,
+we suggest to  set this parameter equal to the restart  number in 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  TSIRM algorithm  (\emph{i.e.}, $\epsilon_{tsirm}$).  We also  consider that
+after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
+which is defined by $||Ax^k-b||_2$.
+
+  Line~\ref{algo:store},
+$S_{k \mod  s}=x^k$ consists in  copying the solution  $x_k$ into the  column $k
+\mod s$ of $S$.   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 iterations
+and the threshold to stop the method.

 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}


@@ -799,10 +800,12 @@ than the one of the GMRES method.
 
 
 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

-show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
-Table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
-characteristics. For all  the matrices, the name, the field,  the number of rows
-and the number of nonzero elements are given.
+show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01},
+we  show the  matrices we  have  used and  some of  them characteristics.  Those
+matrices  are   chosen  from   the  Davis  collection   of  the   University  of
+Florida~\cite{Dav97}. They are matrices arising in real-world applications.  For
+all the  matrices, the name,  the field,  the number of  rows and the  number of
+nonzero elements are given.

 
 \begin{table}[htbp]
 \begin{center}
 
 \begin{table}[htbp]
 \begin{center}


@@ -871,7 +874,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
 
 
 

-In order to perform larger  experiments, we have tested some example applications
+In order to perform larger experiments, we have tested some example applications

 of PETSc. Those  applications are available in the ksp part  which is suited for
 scalable linear equations solvers:
 \begin{itemize}
 of PETSc. Those  applications are available in the ksp part  which is suited for
 scalable linear equations solvers:
 \begin{itemize}


@@ -884,9 +887,10 @@ scalable linear equations solvers:
   finite elements. For this example, the user can define the scaling of material
   coefficient in embedded circle called $\alpha$.
 \end{itemize}
   finite elements. For this example, the user can define the scaling of material
   coefficient in embedded circle called $\alpha$.
 \end{itemize}

-For more technical details on  these applications, interested readers are invited
-to  read the  codes available  in the  PETSc sources.   Those problems  have been
-chosen because they  are scalable with many cores which is not the case of other problems that we have tested.
+For more technical details on these applications, interested readers are invited
+to read  the codes  available in  the PETSc sources.   Those problems  have been
+chosen because they are scalable with many  cores which is not the case of other
+problems that we have tested.

 
 In  the  following   larger  experiments  are  described  on   two  large  scale
 architectures:  Curie and  Juqeen.  Both  these architectures  are supercomputer
 
 In  the  following   larger  experiments  are  described  on   two  large  scale
 architectures:  Curie and  Juqeen.  Both  these architectures  are supercomputer


@@ -897,7 +901,8 @@ Partnership  for Advanced  Computing in  Europe)  which aims  at proposing  high
 performance supercomputing architecture to enhance research in Europe. The Curie
 architecture is composed of Intel E5-2680  processors at 2.7 GHz with 2Gb memory
 by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with
 performance supercomputing architecture to enhance research in Europe. The Curie
 architecture is composed of Intel E5-2680  processors at 2.7 GHz with 2Gb memory
 by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with

-1Gb memory per core.
+1Gb memory per  core. Both those architecture are equiped  with a dedicated high
+speed network.