+%%%*********************************************************
+%%%*********************************************************
+\section{Introduction}
+% no \IEEEPARstart
+% 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).
+
+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
+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
+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
+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
+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. 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.
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Related works}
+\label{sec:02}
+%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\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
+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 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
+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}
+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{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 (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}
+
+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}
+\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
+characteristics. For all the matrices, the name, the field, the number of rows
+and the number of nonzero elements is given.
+
+\begin{table*}
+\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 \\
+epb3 & Thermal problem & 84,617 & 463,625 \\
+atmosmodj & Computational fluid dynamics & 1,270,432 & 8,814,880 \\
+bfwa398 & Electromagnetics problem & 398 & 3,678 \\
+torso3 & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\end{center}
+\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 (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)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+
+
+In Table~\ref{tab:02}, some experiments comparing the solving of the linear
+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.
+
+
+\begin{table}
+\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} \\
+\cline{3-6}
+ & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
+
+crashbasis & gmres / none & 15.65 & 518 & 14.12 & 450 \\
+parabolic\_fem & gmres / ilu & 1009.94 & 7573 & 401.52 & 2970 \\
+epb3 & fgmres / sor & 8.67 & 600 & 8.21 & 540 \\
+atmosmodj & fgmres / sor & 104.23 & 451 & 88.97 & 366 \\
+bfwa398 & gmres / none & 1.42 & 9612 & 0.28 & 1650 \\
+torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{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:
+\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} & \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 \\
+ 2,048 & sor & 745.37 & 57,060 & 87.31 & 6,150 & 104.21 & 7,230 & 8.53 \\
+ 4,096 & mg & 562.25 & 25,170 & 97.23 & 3,990 & 89.71 & 3,630 & 6.27 \\
+ 4,096 & sor & 912.12 & 70,194 & 145.57 & 9,750 & 168.97 & 10,980 & 6.26 \\
+ 8,192 & mg & 917.02 & 40,290 & 148.81 & 5,730 & 143.03 & 5,280 & 6.41 \\
+ 8,192 & sor & 1,404.53 & 106,530 & 212.55 & 12,990 & 180.97 & 10,470 & 7.76 \\
+ 16,384 & mg & 1,430.56 & 63,930 & 237.17 & 8,310 & 244.26 & 7,950 & 6.03 \\
+ 16,384 & sor & 2,852.14 & 216,240 & 418.46 & 21,690 & 505.26 & 23,970 & 6.82 \\
+\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.}
+\label{tab:03}
+\end{center}
+\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} & \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 \\
+ 2,048 & 6e-5 & 194.01 & 30,270 & 35.50 & 5,430 & 27.74 & 4,350 & 6.99 \\
+ 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 & 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
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
+\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*}
+
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
