modif de christophe

diff --git a/paper.tex b/paper.tex
index 112b322324c88803dcc327629623965b330b3276..4f1da9bb181cbd6d6eeeceed136bd45adc874f86 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -348,7 +348,8 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 

-
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}

 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{amsmath}
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{amsmath}


@@ -362,7 +363,7 @@
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 

-
+\newtheorem{proposition}{Proposition}

 
 \begin{document}
 %
 
 \begin{document}
 %


@@ -382,7 +383,7 @@
 % use a multiple column layout for up to two different
 % affiliations
 
 % use a multiple column layout for up to two different
 % affiliations
 

-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2} and Christophe Guyeux\IEEEauthorrefmark{1}}
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}

 \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
 \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\


@@ -427,16 +428,16 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-In  this paper  we propose  a  two stage  iterative method  which increases  the
-convergence of Krylov iterative methods,  typically those of GMRES variants. The
-principle of  our approach  is to  build an external  iteration over  the Krylov
-method  and to  save  the current  residual  frequently (for  example, for  each
-restart of GMRES). Then after a given number of outer iterations, a minimization
-step  is applied  on the  matrix composed  of the  saved residuals  in  order to
-compute a better solution and make  a new iteration if necessary.  We prove that
-our method has  the same convergence property than the  inner method used.  Some
-experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
-can be around 7 times faster.
+In  this article,  a  two-stage  iterative method is proposed to improve  the
+convergence of Krylov based iterative ones,  typically those of GMRES variants. The
+principle of  the proposed approach  is to  build an external  iteration over  the Krylov
+method, and to  frequently store its current  residual   (at  each
+GMRES restart for instance). After a given number of outer iterations, a minimization
+step  is applied  on the  matrix composed by the  saved residuals,  in  order to
+compute a better solution while making  new iterations if required.  It is proven that
+the proposal has  the same convergence properties than the  inner embedded method itself. 
+Experiments using up  to 16,394 cores also show that the proposed algorithm
+run around 7 times faster than GMRES.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -548,45 +549,47 @@ Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
 
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
 

-Iterative methods  became more attractive than  direct ones to  solve very large
-sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
-context,  with  thousands  of  cores,  and  require  less  memory  and  arithmetic
-operations than direct  methods. A number of iterative  methods are proposed and
-adapted by many researchers and the increased need for solving very large sparse
-linear  systems  triggered the  development  of  efficient iterative  techniques
-suitable for the parallel processing.
-
-Most of the successful iterative methods currently available are based on Krylov
-subspaces which  consist in forming a  basis of a sequence  of successive matrix
-powers times an initial vector for example the residual. These methods are based
-on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
-systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
-Gradient method and GMRES method (generalized minimal residual).
+Iterative methods have recently become more attractive than  direct ones to  solve very large
+sparse  linear systems.  They are more  efficient  in a  parallel
+context,  supporting  thousands  of  cores,  and they require  less  memory  and  arithmetic
+operations than direct  methods. This is why new iterative  methods are frequently 
+proposed or adapted by researchers, and the increasing need to solve very large sparse
+linear  systems  has triggered the  development  of such efficient iterative  techniques
+suitable for parallel processing.
+
+Most of the successful iterative methods currently available are based on so-called ``Krylov
+subspaces''. They  consist in forming a  basis of successive matrix
+powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov  subspace  basis in order to solve  linear
+systems.  The  most known iterative  Krylov subspace methods  are conjugate
+gradient and GMRES ones (Generalized Minimal RESidual).
+

 
 However,  iterative  methods suffer  from scalability  problems  on parallel
 
 However,  iterative  methods suffer  from scalability  problems  on parallel

-computing  platforms  with many  processors  due  to  their need  for  reduction
-operations    and   collective    communications   to    perform   matrix-vector
+computing  platforms  with many  processors, due  to  their need  of  reduction
+operations, and to  collective    communications   to  achive   matrix-vector

 multiplications. The  communications on large  clusters with thousands  of cores
 multiplications. The  communications on large  clusters with thousands  of cores

-and  large  sizes of  messages  can  significantly  affect the  performances  of
-iterative methods. In practice, Krylov subspace iteration methods are often used
-with preconditioners in order to increase their convergence and accelerate their
+and  large  sizes of  messages  can  significantly  affect the  performances  of these
+iterative methods. As a consequence, Krylov subspace iteration methods are often used
+with preconditioners in practice, to increase their convergence and accelerate their

 performances.  However, most  of the  good preconditioners  are not  scalable on
 large clusters.
 
 performances.  However, most  of the  good preconditioners  are not  scalable on
 large clusters.
 

-In this  paper we propose a  two-stage algorithm based on  two nested iterations
-called inner-outer  iterations.  This algorithm  consists in solving  the sparse
-linear system iteratively  with a small number of  inner iterations and restarts
+In this research work, a two-stage algorithm based on  two nested iterations
+called inner-outer  iterations is proposed.  This algorithm  consists in solving  the sparse
+linear system iteratively  with a small number of  inner iterations, and restarting

 the outer  step with a  new solution minimizing  some error functions  over some
 previous residuals. This algorithm is iterative and easy to parallelize on large
 the outer  step with a  new solution minimizing  some error functions  over some
 previous residuals. This algorithm is iterative and easy to parallelize on large

-clusters   and  the   minimization  technique   improves  its   convergence  and
+clusters. Furthermore,  the   minimization  technique   improves  its   convergence  and

 performances.
 
 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
-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.
+The present  article is  organized as follows.   Related works are  presented in
+Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
+a  least-square  residual   minimization,  while  Section~\ref{sec:04}  provides
+convergence  results  regarding this  method.   Section~\ref{sec:05} shows  some
+experimental  results  obtained  on  large  clusters  using  routines  of  PETSc
+toolkit. This research work ends by  a conclusion section, in which the proposal
+is summarized while intended perspectives are provided.
+

 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************
 


@@ -608,22 +611,24 @@ toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 \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

-nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
-$b\in\mathbb{R}^n$ 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}.
-
-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 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
-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
+nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector,   and
+$b\in\mathbb{R}^n$ is  the right-hand side.  As explained previously, 
+the algorithm is implemented  as an
+inner-outer iteration  solver based  on iterative Krylov  methods. The  main 
+key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
+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-square 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 
+best known approximation previously obtained. 
+GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
+inner solver. The current approximation of the Krylov method is then stored inside a matrix
+$S$ composed by the successive solutions that are computed during inner iterations.
+
+At each $s$ 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}
 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}


@@ -633,10 +638,12 @@ $(b-AS)$. The minimization of the residuals is obtained by
 with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
 
 
 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.
+In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
+with $s\ll n$.   In order  to minimize~(\eqref{eq:01}), a  least-square method  such as
+CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Remark that these  methods are more
+appropriate than a single direct method in a parallel context.
+
+

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


@@ -647,10 +654,10 @@ appropriate than a direct method in a parallel context.
   \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
   \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}
-    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$}
+    \State $S_{k \mod s}=x^k$ \label{algo:store}
+    \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}

       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
       \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 Solve least-square 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
       \State $x^k=S\alpha$  \Comment{compute new solution}
     \EndIf
   \EndFor


@@ -663,7 +670,7 @@ iteration is  inside the 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
 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
 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
 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.
+much  smaller  than the  convergence  threshold  of  the TSARM  algorithm  (\emph{i.e.}

 $\epsilon_{tsarm}$).  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
 $\epsilon_{tsarm}$).  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


@@ -671,13 +678,13 @@ 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.
 
 required for that: the maximum number of iteration and the threshold to stop the
 method.
 

-To summarize, the important parameters of TSARM are:
+Let us summarize the most important parameters of TSARM:

 \begin{itemize}
 \begin{itemize}

-\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
+\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 used to stop the least-square method.

 \end{itemize}
 
 
 \end{itemize}
 
 


@@ -728,19 +735,27 @@ these operations are easy to implement in PETSc or similar environment.
 
 \section{Convergence results}
 \label{sec:04}
 
 \section{Convergence results}
 \label{sec:04}

-
+Let us recall the following result, see~\cite{Saad86}.
+\begin{proposition}
+Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
+\begin{equation}
+||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+\end{equation}
+where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves 
+the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
+\end{proposition}

 
 
 
 %%%*********************************************************
 %%%*********************************************************
 
 
 
 %%%*********************************************************
 %%%*********************************************************

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

 \label{sec:05}
 
 
 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
 \label{sec:05}
 
 
 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
+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 is given.
 
 characteristics. For all  the matrices, the name, the field,  the number of rows
 and the number of nonzero elements is given.