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-%\usepackage{eqparbox}
+\usepackage{eqparbox}
% Also of notable interest is Scott Pakin's eqparbox package for creating
% (automatically sized) equal width boxes - aka "natural width parboxes".
% Available at:
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
-\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}
-%où
-%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
-%où
-%\title{???}
+\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
+
\begin{abstract}
-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
+In this article, a two-stage iterative algorithm is proposed to improve the
+convergence of Krylov based iterative methods, 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
+compute a better solution and to make 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.
+runs around 5 or 7 times faster than GMRES.
\end{abstract}
\begin{IEEEkeywords}
%%%*********************************************************
%%%*********************************************************
-\section{Two-stage algorithm with least-square residuals minimization}
+\section{Two-stage iteration with least-square residuals minimization algorithm}
\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
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
+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}
+\caption{TSIRM}
\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 (error$<\epsilon_{tsarm}$)} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \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}
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 (\emph{i.e.}
-$\epsilon_{tsarm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+much smaller than the convergence threshold of the TSIRM algorithm (\emph{i.e.}
+$\epsilon_{tsirm}$). 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
+required for that: the maximum number of iterations and the threshold to stop the
method.
-Let us summarize the most important parameters of TSARM:
+Let us summarize the most important parameters of TSIRM:
\begin{itemize}
-\item $\epsilon_{tsarm}$: the threshold to stop the TSARM method;
+\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM 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;
\end{itemize}
-The parallelisation of TSARM relies on the parallelization of all its
+The parallelisation of TSIRM 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
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.
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dots, norm, multiplication and addition on vectors. All
+classical operations: dot product, norm, multiplication and addition on vectors. All
these operations are easy to implement in PETSc or similar environment.
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
characteristics. For all the matrices, the name, the field, the number of rows
-and the number of nonzero elements is given.
+and the number of nonzero elements are given.
-\begin{table*}
+\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|r|r|r|}
\hline
Matrix name & Field &\# Rows & \# Nonzeros \\\hline \hline
crashbasis & Optimization & 160,000 & 1,750,416 \\
-parabolic\_fem & Computational fluid dynamics & 525,825 & 2,100,225 \\
+parabolic\_fem & Comput. fluid dynamics & 525,825 & 2,100,225 \\
epb3 & Thermal problem & 84,617 & 463,625 \\
-atmosmodj & Computational fluid dynamics & 1,270,432 & 8,814,880 \\
-bfwa398 & Electromagnetics problem & 398 & 3,678 \\
+atmosmodj & Comput. fluid dynamics & 1,270,432 & 8,814,880 \\
+bfwa398 & Electromagnetics pb & 398 & 3,678 \\
torso3 & 2D/3D problem & 259,156 & 4,429,042 \\
\hline
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
-\end{table*}
+\end{table}
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, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
+the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ 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)
+$\epsilon_{tsirm}=1e-10$. 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 TSARM, the same solver and the same
-preconditionner is used. This Table shows that TSARM can drastically reduce the
+different preconditioner is used. With TSIRM, the same solver and the same
+preconditionner are used. This Table shows that TSIRM 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{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|r|r|r|r|}
\hline
- \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\
+ \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\
\cline{3-6}
& precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
-In order to perform larger experiments, we have tested some example application
+In order to perform larger experiments, we have tested some example applications
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
+ difference scheme. The diagonal is equal 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...
{\bf Description of preconditioners}
-\begin{table*}
+\begin{table*}[htbp]
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\
+ nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 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.}
+\caption{Comparison of FGMRES and TSIRM 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*}
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}
+problems) per processor is fixed to 25,000, also called weak scaling. 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 TSIRM is always faster than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSIRM according to
+the minimization procedure: CGLS or LSQR. Even if we have computed the worst
+case between CGLS and LSQR, it is clear that TSIRM is alsways faster than
+FGMRES. For this example, the multigrid preconditionner is faster than SOR. The
+gain between TSIRM and FGMRES is more or less similar for the two
+preconditioners. Looking at the number of iterations to reach the convergence,
+it is obvious that TSIRM allows the reduction of the number of iterations. It
+should be noticed that for TSIRM, in those experiments, only the iterations of
+the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
+corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
+
+\begin{figure}[htbp]
\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}}
\end{figure}
+In Figure~\ref{fig:01}, the number of iterations per second corresponding to
+Table~\ref{tab:01} is displayed. It can be noticed that the number of
+iterations per second of FMGRES is constant whereas it decrease with TSIRM with
+both preconditioner. This can be explained by the fact that when the number of
+core increases the time for the minimization step also increases but, generally,
+when the number of cores increases, the number of iterations to reach the
+threshold also increases, and, in that case, TSIRM is more efficient to reduce
+the number of iterations. So, the overall benefit of using TSIRM is interesting.
+
+
-\begin{table*}
+
+\begin{table*}[htbp]
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\
+ nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 \\
\end{table*}
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported
-
-\begin{table*}
+\begin{table*}[htbp]
\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} \\
+ nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 \\
