\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}
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.
%%%*********************************************************
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.
+inner solver is based on a Krylov method which does not require 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
\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$>\epsilon_{kryl}$}
+ \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$}
\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}
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
+$\epsilon_{tsarm}$). 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
To summarize, the important parameters of TSARM are:
\begin{itemize}
-\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov 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 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{Convergence results}
+\label{sec:04}
+
+
+
+
%%%*********************************************************
%%%*********************************************************
\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
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}
\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 ....\\
-Describe the problems ex15 and ex54
+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 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.
+
+
+
\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
\begin{table*}
\begin{center}
-\begin{tabular}{|r|r|r|r|r|r|r|r|}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\
-\cline{2-7}
- & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
- 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & \\
- 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & \\
- 2048 & 919.62 & 31,470 & 237,52 & 8,040 & 194.22 & 6,510 & \\
- 4096 & 405.60 & 28,380 & 111.67 & 7,590 & 91.72 & 6,510 & \\
- 8192 & 785.04 & 109,590 & 76.07 & 10,470 & 69,42 & 9,030 & \\
+ 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
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:07}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************
