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% paper title
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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}
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
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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
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 $m$
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.
\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
-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
-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.
\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}
\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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