\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multirow}
+\usepackage{graphicx}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
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 save 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
+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.
\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. 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.
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
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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
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 applies a least-square minimization on the residuals computed by the inner some error functions over a Krylov
-subspace~\cite{Saad2003}. At each iteration, the sparse linear system $Ax=b$ is
-solved iteratively with an iterative method, for example GMRES
-method~\cite{Saad86} or some of its variants, and the Krylov subspace that we
-used is spanned by a basis $S$ composed of successive solutions issued from the
-inner iteration
-\begin{equation}
- S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
-\end{equation}
-The advantage of such a Krylov subspace is that we neither need an orthogonal
-basis nor any synchronization between processors to generate this basis. The
-algorithm is periodically restarted every $s$ iterations with a new initial
-guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
-subspace spanned by vectors of $S$, where $\alpha$ is a solution of the normal
-equations
-\begin{equation}
- R^TR\alpha = R^Tb,
-\end{equation}
-which is associated with the least-squares problem
+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
+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}
\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\end{equation}
-such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$,
-$s\ll n$, and $R^T$ denotes the transpose of matrix $R$. We use an iterative
-method to solve the least-squares problem~(\ref{eq:01}) such as CGLS
-~\cite{Hestenes52} or LSQR~\cite{Paige82} which are more appropriate than a
-direct method in the parallel context.
+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.
\begin{algorithm}[t]
-\caption{A Krylov two-stage algorithm}
+\caption{TSARM}
\begin{algorithmic}[1]
\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x^0$
- \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
- \State Solve iteratively $Ax^k=b$ \label{algo:solve}
- \State $S_{k~mod~s}=x^k$
- \If {$k$ mod $s=0$ {\bf and} not convergence}
- \State Compute dense matrix $R=AS$
- \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
- \State Compute minimizer $x^k=S\alpha$
+ \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} 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
\end{algorithmic}
\label{algo:01}
\end{algorithm}
-Operation $S_{k~ mod~ s}=x^k$ consists in copying the residual $x_k$ 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.
+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 $max\_iter_{kryl}$ 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}
+
+
+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{Convergence results}
\label{sec:04}
+
+
+
%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
\begin{tabular}{|c|c|r|r|r|r|}
\hline
- \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\
+ \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\
\cline{3-6}
& precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
-Larger experiments ....
+
+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:
+\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 $\alpha$.
+\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.
+
+
+
\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & precond & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\
+ nb. cores & precond & \multicolumn{2}{c|}{GMRES} & \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 \\
\end{table*}
+\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}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & threshold & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\
+ nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \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 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
4,096 & 7e-5 & 160.59 & 22,530 & 35.15 & 5,130 & 29.21 & 4,350 & 5.49 \\
4,096 & 6e-5 & 249.27 & 35,520 & 52.13 & 7,950 & 39.24 & 5,790 & 6.35 \\
8,192 & 6e-5 & 149.54 & 17,280 & 28.68 & 3,810 & 29.05 & 3,990 & 5.21 \\
- 8,192 & 5e-5 & 792.11 & 109,590 & 76.83 & 10,470 & 65.20 & 9,030 & 12.14 \\
+ 8,192 & 5e-5 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 \\
16,384 & 4e-5 & 718.61 & 86,400 & 98.98 & 10,830 & 131.86 & 14,790 & 7.26 \\
\hline
\label{tab:04}
\end{center}
\end{table*}
+
+
+
+
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
+\hline
+
+ nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\
+\cline{2-7} \cline{9-11}
+ & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline
+ 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\
+ 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\
+ 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\
+ 4096 & 405.60 & 28,380 & 111.67 & 7,590 & 91.72 & 6,510 & 4.42 & 1.22 & .79 & .84 \\
+ 8192 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 & .32 & .58 & .56 \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.}
+\label{tab:05}
+\end{center}
+\end{table*}
+
%%%*********************************************************
%%%*********************************************************
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
