\end{abstract}
\begin{IEEEkeywords}
-Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir...
+Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir...
\end{IEEEkeywords}
The present paper is organized as follows. In Section~\ref{sec:02} some related
works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
a least-square residual minimization. Section~\ref{sec:04} describes some
-convergence results on this method. Section~\ref{sec:05} shows some
-experimental results obtained on large clusters of our algorithm using routines
-of PETSc toolkit. Finally Section~\ref{sec:06} concludes and gives some
-perspectives.
+convergence results on this method. In Section~\ref{sec:05}, parallization
+details of TSARM are given. Section~\ref{sec:06} shows some experimental
+results obtained on large clusters of our algorithm using routines of PETSc
+toolkit. Finally Section~\ref{sec:06} concludes and gives some perspectives.
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-\section{A Krylov two-stage algorithm}
+\section{Two-stage algorithm with least-square residuals minimization}
\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
\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x^0$
- \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
- \State $x^k=Solve(A,b,x^{k-1},m)$ \label{algo:solve}
+ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{kryl}$)} \label{algo:conv}
+ \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}
- \If {$k$ mod $s=0$ {\bf and} not convergence}
- \State $R=AS$ \Comment{compute dense matrix}
- \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+ \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
+ \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
\State $x^k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
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
-called for a maximum of $m$ iterations. In practice, we suggest to choose $m$
-equals to the restart number of the GMRES like method. 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 $S$. After the minimization, the matrix $S$ is reused with
-the new values of the residuals.
+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
+threshold must be specified for the solver. In practice, this threshold must be
+much smaller than the convergence threshold of the TSARM algorithm (i.e.
+$\epsilon$). 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 $S$. 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
+method.
+
+To summarize, the important parameters of TSARM are:
+\begin{itemize}
+\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov 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 to stop the least-square method
+\end{itemize}
%%%*********************************************************
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\section{Convergence results}
\label{sec:04}
+
+
%%%*********************************************************
%%%*********************************************************
-\section{Experiments using petsc}
+\section{Parallelization}
\label{sec:05}
+The parallelisation of TSARM relies on the parallelization of all its
+parts. More precisely, except the least-square 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.
+
+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.
+
+\begin{algorithm}[t]
+\caption{CGLS}
+\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 $\beta=g/g_{old}$
+ \EndFor
+\end{algorithmic}
+\label{algo:02}
+\end{algorithm}
+
+
+In each iteration of CGLS, there is two matrix-vector multiplications and some
+classical operations: dots, norm, multiplication and addition on vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Experiments using petsc}
+\label{sec:06}
+
In order to see the influence of our algorithm with only one processor, we first
show a comparison with the standard version of GMRES and our algorithm. In
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\section{Conclusion}
-\label{sec:06}
+\label{sec:07}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
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future plan : \\
- study other kinds of matrices, problems, inner solvers\\
+- test the influence of all the parameters\\
- adaptative number of outer iterations to minimize\\
- other methods to minimize the residuals?\\
- implement our solver inside PETSc
