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. In Section~\ref{sec:05}, parallization
-details of TSARM are given. Section~\ref{sec:06} shows some experimental
+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.
%%%*********************************************************
\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}
+ \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
required for that: the maximum number of iteration and the threshold to stop the
method.
-To summarize, the important parameters of are:
+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 $\epsilon_{ls}$ 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{Parallelization}
-\label{sec:05}
%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
-\label{sec:06}
+\label{sec:05}
In order to see the influence of our algorithm with only one processor, we first
\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 \\
\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*}
+
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:07}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************
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
