\algnewcommand\Output{\item[\algorithmicoutput]}
\newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
\begin{document}
%
% 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}
+\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}
% 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\\
\begin{abstract}
-In this article, a two-stage iterative algorithm is proposed to improve the
-convergence of Krylov based iterative methods, 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 and to make 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
-runs around 5 or 7 times faster than GMRES.
+In this article, a two-stage iterative algorithm is proposed to improve the
+convergence of Krylov based iterative methods, 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 least-squares minimization step is applied on the matrix composed by the saved
+residuals, in order to compute a better solution and to make 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 runs around 5 or 7 times faster than
+GMRES.
\end{abstract}
\begin{IEEEkeywords}
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
%%%*********************************************************
%%%*********************************************************
-\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 $m$
$S$ 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.
\begin{algorithmic}[1]
\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
- \State Set the initial guess $x^0$
+ \State Set the initial guess $x_0$
\For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
- \State $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$ \label{algo:solve}
+ \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}
+ \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 $x^k=S\alpha$ \Comment{compute new solution}
+ \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
+ \State $x_k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
\end{algorithmic}
\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{algorithmic}[1]
\Input $A$ (matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
- \State $r=b-Ax$
- \State $p=A'r$
- \State $s=p$
- \State $g=||s||^2_2$
- \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
- \State $q=Ap$
- \State $\alpha=g/||q||^2_2$
- \State $x=x+alpha*p$
- \State $r=r-alpha*q$
- \State $s=A'*r$
- \State $g_{old}=g$
- \State $g=||s||^2_2$
- \State $\beta=g/g_{old}$
+ \State Let $x_0$ be an initial approximation
+ \State $r_0=b-Ax_0$
+ \State $p_1=A^Tr_0$
+ \State $s_0=p_1$
+ \State $\gamma=||s_0||^2_2$
+ \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
+ \State $q_k=Ap_k$
+ \State $\alpha_k=\gamma/||q_k||^2_2$
+ \State $x_k=x_{k-1}+\alpha_kp_k$
+ \State $r_k=r_{k-1}-\alpha_kq_k$
+ \State $s_k=A^Tr_k$
+ \State $\gamma_{old}=\gamma$
+ \State $\gamma=||s_k||^2_2$
+ \State $\beta_k=\gamma/\gamma_{old}$
+ \State $p_{k+1}=s_k+\beta_kp_k$
\EndFor
\end{algorithmic}
\label{algo:02}
\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}
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+\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 $k \rightarrow +\infty$.
+Each step of the TSIRM algorithm
+\end{proof}
%%%*********************************************************
%%%*********************************************************
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
- representing the neighbors in each directions is equal to -1. This example is
+ difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
+ representing the neighbors in each directions are equal to -1. This example is
used in many physical phenomena, for example, heat and fluid flow, wave
- propagation...
+ propagation, etc.
\item ex54 is another example based on 2D problem discretized with quadrilateral
finite elements. For this example, the user can define the scaling of material
- coefficient in embedded circle, it is called $\alpha$.
+ coefficient in embedded circle called $\alpha$.
\end{itemize}
-For more technical details on these applications, interested reader are invited
-to read the codes available in the PETSc sources. Those problem have been
-chosen because they are scalable with many cores. We have tested other problem
-but they are not scalable with many cores.
+For more technical details on these applications, interested readers are invited
+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...}\\
\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}
+
%%%*********************************************************
%%%*********************************************************
% that's all folks
\end{document}
-
-
