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[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 4cd16c3355ff4b1402e9093542b26d33c6606af7..e3d19ec0d604e10229be70a5643a4b4c08aeb92b 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -369,7 +369,7 @@
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-\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
+\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}

 
 
 
 
 
 


@@ -581,7 +581,7 @@ performances.
 
 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
 
 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
+a  least-squares  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
 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


@@ -604,7 +604,7 @@ is summarized while intended perspectives are provided.
 
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-\section{Two-stage iteration with least-square residuals minimization algorithm}
+\section{Two-stage iteration with least-squares residuals minimization algorithm}

 \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


@@ -615,7 +615,7 @@ 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 
 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.
+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 $m$
 
 At each outer iteration, the sparse linear system $Ax=b$ is partially 
 solved using only $m$


@@ -636,7 +636,7 @@ with $R=AS$. Then the new solution $x$ is 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  $\mathbb{R}^{n\times s}$,

-with $s\ll n$.   In order  to minimize~\eqref{eq:01}, a  least-square method  such as
+with $s\ll n$.   In order  to minimize~\eqref{eq:01}, a  least-squares 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.
 
 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.
 


@@ -680,19 +680,19 @@ Let us summarize the most important parameters of TSIRM:
 \item $\epsilon_{tsirm}$: the threshold to stop the TSIRM 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 $\epsilon_{tsirm}$: the threshold to stop the TSIRM 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.
+\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
+\item $\epsilon_{ls}$: the threshold used to stop the least-squares method.

 \end{itemize}
 
 
 The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
 \end{itemize}
 
 
 The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its

-parts. More  precisely, except  the least-square step,  all the other  parts are
+parts. More  precisely, except  the least-squares step,  all the other  parts are

 obvious to  achieve out in parallel. In  order to develop a  parallel version of
 our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
 line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
 efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
 colums in  practice. As explained  previously, at least  two methods seem  to be
 obvious to  achieve out in parallel. In  order to develop a  parallel version of
 our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
 line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
 efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
 colums in  practice. As explained  previously, at least  two methods seem  to be

-interesting to solve the least-square minimization, CGLS and LSQR.
+interesting to solve the least-squares minimization, CGLS and LSQR.

 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or
 less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or
 less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.


@@ -897,7 +897,7 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}

-\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}

 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 


@@ -965,6 +965,13 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 \end{center}
 \end{table*}
 
 \end{center}
 \end{table*}
 

+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
+\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
+\label{fig:02}
+\end{figure}
+

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