\hyphenation{op-tical net-works semi-conduc-tor}
-
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
\usepackage{algorithm}
\usepackage{algpseudocode}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multirow}
+\usepackage{graphicx}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
\algnewcommand\Output{\item[\algorithmicoutput]}
-
+\newtheorem{proposition}{Proposition}
\begin{document}
%
% 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 paper we propose a two stage iterative method which increases the
-convergence of Krylov iterative methods, typically those of GMRES variants. The
-principle of our approach is to build an external iteration over the Krylov
-method and to save the current residual frequently (for example, for each
-restart of GMRES). Then after a given number of outer iterations, a minimization
-step is applied on the matrix composed of the saved residuals in order to
-compute a better solution and make a new iteration if necessary. We prove that
-our method has the same convergence property than the inner method used. Some
-experiments using up to 16,394 cores show that compared to GMRES our algorithm
-can be around 7 times faster.
+In this article, a two-stage iterative method is proposed to improve the
+convergence of Krylov based iterative ones, 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 while making 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
+run around 7 times faster than GMRES.
\end{abstract}
\begin{IEEEkeywords}
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
-Iterative methods became more attractive than direct ones to solve very large
-sparse linear systems. Iterative methods are more effecient in a parallel
-context, with thousands of cores, and require less memory and arithmetic
-operations than direct methods. A number of iterative methods are proposed and
-adapted by many researchers and the increased need for solving very large sparse
-linear systems triggered the development of efficient iterative techniques
-suitable for the parallel processing.
-
-Most of the successful iterative methods currently available are based on Krylov
-subspaces which consist in forming a basis of a sequence of successive matrix
-powers times an initial vector for example the residual. These methods are based
-on orthogonality of vectors of the Krylov subspace basis to solve linear
-systems. The most well-known iterative Krylov subspace methods are Conjugate
-Gradient method and GMRES method (generalized minimal residual).
+Iterative methods have recently become more attractive than direct ones to solve very large
+sparse linear systems. They are more efficient in a parallel
+context, supporting thousands of cores, and they require less memory and arithmetic
+operations than direct methods. This is why new iterative methods are frequently
+proposed or adapted by researchers, and the increasing need to solve very large sparse
+linear systems has triggered the development of such efficient iterative techniques
+suitable for parallel processing.
+
+Most of the successful iterative methods currently available are based on so-called ``Krylov
+subspaces''. They consist in forming a basis of successive matrix
+powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov subspace basis in order to solve linear
+systems. The most known iterative Krylov subspace methods are conjugate
+gradient and GMRES ones (Generalized Minimal RESidual).
+
However, iterative methods suffer from scalability problems on parallel
-computing platforms with many processors due to their need for reduction
-operations and collective communications to perform matrix-vector
+computing platforms with many processors, due to their need of reduction
+operations, and to collective communications to achive matrix-vector
multiplications. The communications on large clusters with thousands of cores
-and large sizes of messages can significantly affect the performances of
-iterative methods. In practice, Krylov subspace iteration methods are often used
-with preconditioners in order to increase their convergence and accelerate their
+and large sizes of messages can significantly affect the performances of these
+iterative methods. As a consequence, Krylov subspace iteration methods are often used
+with preconditioners in practice, to increase their convergence and accelerate their
performances. However, most of the good preconditioners are not scalable on
large clusters.
-In this paper we propose a two-stage algorithm based on two nested iterations
-called inner-outer iterations. This algorithm consists in solving the sparse
-linear system iteratively with a small number of inner iterations and restarts
+In this research work, a two-stage algorithm based on two nested iterations
+called inner-outer iterations is proposed. This algorithm consists in solving the sparse
+linear system iteratively with a small number of inner iterations, and restarting
the outer step with a new solution minimizing some error functions over some
previous residuals. This algorithm is iterative and easy to parallelize on large
-clusters and the minimization technique improves its convergence and
+clusters. Furthermore, the minimization technique improves its convergence and
performances.
-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. 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.
+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
+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
+is summarized while intended perspectives are provided.
+
%%%*********************************************************
%%%*********************************************************
\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
-nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector and
-$b\in\mathbb{R}^n$ is the right-hand side. The algorithm is implemented as an
-inner-outer iteration solver based on iterative Krylov methods. The main key
-points of our solver are given in Algorithm~\ref{algo:01}.
-
-In order to accelerate the convergence, the outer iteration periodically applies
-a least-square minimization on the residuals computed by the inner solver. The
-inner solver is a Krylov based solver which does not required to be changed.
-
-At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
-iterations, using an iterative method restarting with the previous solution. For
-example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
-inner solver. The current solution of the Krylov method is saved inside a matrix
-$S$ composed of successive solutions computed by the inner iteration.
-
-Periodically, every $s$ iterations, the minimization step is applied in order to
+nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector, and
+$b\in\mathbb{R}^n$ is the right-hand side. As explained previously,
+the algorithm is implemented as an
+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
+outer solver periodically applies a least-square 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 $m$
+iterations of an iterative method, this latter being initialized with the
+best known approximation previously obtained.
+GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
+inner solver. The current approximation of the Krylov method is then stored inside a matrix
+$S$ 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
\begin{equation}
with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
-In practice, $R$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$,
-$s\ll n$. In order to minimize~(\ref{eq:01}), a least-square method such as
-CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Those methods are more
-appropriate than a direct method in a parallel context.
+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
+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{algorithm}[t]
\caption{TSARM}
\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 (error$<\epsilon_{kryl}$)} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \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} error$>\epsilon_{kryl}$}
+ \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-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
+ \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}
\EndIf
\EndFor
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
+much smaller than the convergence threshold of the TSARM algorithm (\emph{i.e.}
+$\epsilon_{tsarm}$). 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:
+Let us summarize the most important parameters of TSARM:
\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
+\item $\epsilon_{tsarm}$: the threshold to stop the TSARM 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.
\end{itemize}
-%%%*********************************************************
-%%%*********************************************************
-
-\section{Convergence results}
-\label{sec:04}
-
-
-
-%%%*********************************************************
-%%%*********************************************************
-\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
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}
+
+\section{Convergence results}
+\label{sec:04}
+Let us recall the following result, see~\cite{Saad86}.
+\begin{proposition}
+Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
+\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
+the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
+\end{proposition}
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Experiments using PETSc}
+\label{sec:05}
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
-table~\ref{tab:01}, we show the matrices we have used and some of them
+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.
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 (line \ref{algo:solve} in
-Algorithm~\ref{algo:01}). $s$ is set to 8. CGLS is chosen to minimize the
-least-squares problem. Two conditions are used to stop CGLS, either the
-precision is under $1e-40$ or the number of iterations is greater to $20$. The
-external precision is set to $1e-10$ (line \ref{algo:conv} in
-Algorithm~\ref{algo:01}). Those experiments have been performed on a Intel(R)
+the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ 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
+$\epsilon_{tsarm}=1e-10$. Those experiments have been performed on a Intel(R)
Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
systems obtained with the previous matrices with a GMRES variant and with out 2
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 the 2 stage algorithm, the same solver
-and the same preconditionner is used. This Table shows that the 2 stage
-algorithm 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 iterations to perform the
-minimization.
+different preconditioner is used. With TSARM, the same solver and the same
+preconditionner is used. This Table shows that TSARM 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
+iterations to perform the minimization.
\begin{table}
-In the following we describe the applications of PETSc we have experimented. Those applications are available in the ksp part which is suited for scalable linear equations solvers:
+In order to perform larger experiments, we have tested some example application
+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 used in many physical phenomena , for exemple, heat and
- fluid flow, wave propagation...
-\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 $\alpha$.
+\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
+ used in many physical phenomena, for example, heat and fluid flow, wave
+ propagation...
+\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 $\alpha$.
\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.
+In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
+{\bf Description of preconditioners}
\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & precond & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\
+ nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\
\cline{3-8}
& & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\
\hline
\end{tabular}
-\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSARM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
\label{tab:03}
\end{center}
\end{table*}
+Table~\ref{tab:03} shows the execution times and the number of iterations of
+example ex15 of PETSc on the Juqueen architecture. Differents number of cores
+are studied rangin from 2,048 upto 16,383. Two preconditioners have been
+tested. For those experiments, the number of components (or unknown of the
+problems) per processor is fixed to 25,000. This number can seem relatively
+small. In fact, for some applications that need a lot of memory, the number of
+components per processor requires sometimes to be small.
+
+In this Table, we can notice that TSARM is always faster than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSARM according to
+the minimization procedure: CGLS or LSQR.
+
+
+\begin{figure}
+\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}}
+\label{fig:01}
+\end{figure}
+
+
+
+
\begin{table*}
\begin{center}
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:07}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************
