%
% paper title
% can use linebreaks \\ within to get better formatting as desired
-\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
-%où
-%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
-%où
-%\title{???}
+\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 algorithm with least-square residuals minimization}
+\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 $m$
$S$ 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
+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}
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.
\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 Solve least-square problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
+ \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
\State $x^k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
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
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
+required for that: the maximum number of iterations and the threshold to stop the
method.
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 $max\_iter_{ls}$: 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 take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dots, norm, multiplication and addition on vectors. All
+classical operations: dot product, norm, multiplication and addition on vectors. All
these operations are easy to implement in PETSc or similar environment.
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 is given.
+and the number of nonzero elements are given.
\begin{table}[htbp]
\begin{center}
The following parameters have been chosen for our experiments. As by default
the restart of GMRES is performed every 30 iterations, we have chosen to stop
-the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
+the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
chosen to minimize the least-squares problem with the following parameters:
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
$\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R)
stage algorithm are given. In the second column, it can be noticed that either
gmres or fgmres is used to solve the linear system. According to the matrices,
different preconditioner is used. With TSIRM, the same solver and the same
-preconditionner is used. This Table shows that TSIRM can drastically reduce the
+preconditionner are used. This Table shows that TSIRM can drastically reduce the
number of iterations to reach the convergence when the number of iterations for
the normal GMRES is more or less greater than 500. In fact this also depends on
tow parameters: the number of iterations to stop GMRES and the number of
-In order to perform larger experiments, we have tested some example application
+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}
\item ex15 is an example which solves in parallel an operator using a finite
- difference scheme. The diagonal is equals to 4 and 4 extra-diagonals
+ difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
representing the neighbors in each directions is equal to -1. This example is
used in many physical phenomena, for example, heat and fluid flow, wave
propagation...
\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{table*}
-In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
\begin{table*}[htbp]
\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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