archi

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 063cbf9acbeb195e2500a1bbacaa905fa9327143..d00cbe105f66a3782e53dde816ca0e89af594740 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -380,8 +380,8 @@
 % 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}}
-\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
+\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-Comt\'e, France\\

 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
 Email: lilia.ziane@inria.fr}
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
 Email: lilia.ziane@inria.fr}


@@ -564,7 +564,7 @@ gradient and GMRES ones (Generalized Minimal RESidual).
 
 However,  iterative  methods suffer  from scalability  problems  on parallel
 computing  platforms  with many  processors, due  to  their need  of  reduction
 
 However,  iterative  methods suffer  from scalability  problems  on parallel
 computing  platforms  with many  processors, due  to  their need  of  reduction

-operations, and to  collective    communications   to  achive   matrix-vector
+operations, and to  collective    communications   to  achieve   matrix-vector

 multiplications. The  communications on large  clusters with thousands  of cores
 and  large  sizes of  messages  can  significantly  affect the  performances  of these
 iterative methods. As a consequence, Krylov subspace iteration methods are often used
 multiplications. The  communications on large  clusters with thousands  of cores
 and  large  sizes of  messages  can  significantly  affect the  performances  of these
 iterative methods. As a consequence, Krylov subspace iteration methods are often used


@@ -621,19 +621,21 @@ outer solver periodically applies a least-squares minimization  on the residuals
 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 
 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
+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
 the inner iterations 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}
 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
 the inner iterations 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}

-with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.

 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,


@@ -663,14 +665,15 @@ 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  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  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
 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
+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
 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.}
+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
 $\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k \mod s}=x^k$ consists in copying the

-solution  $x_k$  into the  column  $k \mod s$ of  the  matrix  $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. After  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
 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


@@ -735,37 +738,59 @@ 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}.
+Let us recall the following result, see~\cite{Saad86} for further readings.

 \begin{proposition}
 \begin{proposition}

+\label{prop:saad}

 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}
 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 
+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}
 
 
 We can now claim that,
 \begin{proposition}
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 
 
 We can now claim that,
 \begin{proposition}

-If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, 
+let $r_k$ be the
+$k$-th residue of TSIRM, then
+we still have:
+\begin{equation}
+||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
+\end{equation}
+where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.

 \end{proposition}
 
 \begin{proof}
 \end{proposition}
 
 \begin{proof}

-Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
-$k$-th iterate of TSIRM.
-We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
+$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$

 
 

-Each step of the TSIRM algorithm \\
+The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.
+
+Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
+We will show that the statement holds too for $r_k$. Two situations can occur:
+\begin{itemize}
+\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis.
+\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
+Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\

 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}

-& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
-& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
-& \leqslant ||b-Ax_{k-1}||
+& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\
+& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
+& \leqslant ||b-Ax_{k}||_2\\
+& = ||r_k||_2\\
+& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||,

 \end{array}$
 \end{array}$

+\end{itemize}
+which concludes the induction and the proof.

 \end{proof}
 
 \end{proof}
 

+We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
+than the one of the GMRES method.

 
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@@ -863,7 +888,17 @@ For more technical details on  these applications, interested readers are invite
 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.
 
 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... {\bf description...}\\
+In  the  following   larger  experiments  are  described  on   two  large  scale
+architectures:  Curie and  Juqeen.  Both  these architectures  are supercomputer
+composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
+are respectively hosted  by GENCI in France and  Jülich Supercomputing Centre in
+Germany. They belongs with other similar architectures of the PRACE initiative (
+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
+1Gb memory per core.
+

 
 
 {\bf Description of preconditioners}\\
 
 
 {\bf Description of preconditioners}\\


@@ -1007,13 +1042,22 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
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+A novel two-stage iterative  algorithm has been proposed in this article,
+in order to accelerate the convergence Krylov iterative  methods.
+Our TSIRM proposal acts as a merger between Krylov based solvers and
+a least-squares minimization step.
+The convergence of the method has been proven in some situations, while 
+experiments up to 16,394 cores have been led to verify that TSIRM runs
+5 or  7 times  faster than GMRES.

 
 

-future plan : \\
-- study other kinds of matrices, problems, inner solvers\\
-- test the influence of all parameters\\
-- adaptative number of outer iterations to minimize\\
-- other methods to minimize the residuals?\\
-- implement our solver inside PETSc
+
+For future work, the authors' intention is to investigate 
+other kinds of matrices, problems, and inner solvers. The 
+influence of all parameters must be tested too, while 
+other methods to minimize the residuals must be regarded.
+The number of outer iterations to minimize should become 
+adaptative to improve the overall performances of the proposal.
+Finally, this solver will be implemented inside PETSc.

 
 
 % conference papers do not normally have an appendix
 
 
 % conference papers do not normally have an appendix


@@ -1068,4 +1112,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}

-