TRuc

[GMRES2stage.git] / paper.tex

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\begin{document}
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% paper title
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\title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}






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\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche-Comt\'e, France\\
Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
Email: lilia.ziane@inria.fr}
}



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%Eldon Tyrell\IEEEauthorrefmark{4}}
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%Georgia Institute of Technology,
%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
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%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}




% use for special paper notices
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% make the title area
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\begin{abstract}
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 least-squares minimization step is applied on the matrix composed by the saved
residuals, in order  to 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  runs around  5 or  7 times  faster than
GMRES.
\end{abstract}

\begin{IEEEkeywords}
Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir... 
\end{IEEEkeywords}


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%%%*********************************************************
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\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 have recently become more attractive than  direct ones to  solve very large
sparse  linear systems.  They are more  efficient  in a  parallel
context,  supporting  thousands  of  cores,  and they require  less  memory  and  arithmetic
operations than direct  methods. This is why new iterative  methods are frequently 
proposed or adapted by researchers, and the increasing need to solve very large sparse
linear  systems  has triggered the  development  of such efficient iterative  techniques
suitable for parallel processing.

Most of the successful iterative methods currently available are based on so-called ``Krylov
subspaces''. They  consist in forming a  basis of successive matrix
powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov  subspace  basis in order to solve  linear
systems.  The  most known iterative  Krylov subspace methods  are conjugate
gradient and GMRES ones (Generalized Minimal RESidual).


However,  iterative  methods suffer  from scalability  problems  on parallel
computing  platforms  with many  processors, due  to  their need  of  reduction
operations, and to  collective    communications   to  achieve   matrix-vector
multiplications. The  communications on large  clusters with thousands  of cores
and  large  sizes of  messages  can  significantly  affect the  performances  of these
iterative methods. As a consequence, Krylov subspace iteration methods are often used
with preconditioners in practice, to increase their convergence and accelerate their
performances.  However, most  of the  good preconditioners  are not  scalable on
large clusters.

In this research work, a two-stage algorithm based on  two nested iterations
called inner-outer  iterations is proposed.  This algorithm  consists in solving  the sparse
linear system iteratively  with a small number of  inner iterations, and restarting
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. Furthermore,  the   minimization  technique   improves  its   convergence  and
performances.

The present  article is  organized as follows.   Related works are  presented in
Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
a  least-squares  residual   minimization,  while  Section~\ref{sec:04}  provides
convergence  results  regarding this  method.   Section~\ref{sec:05} shows  some
experimental  results  obtained  on  large  clusters  using  routines  of  PETSc
toolkit. This research work ends by  a conclusion section, in which the proposal
is summarized while intended perspectives are provided.

%%%*********************************************************
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%%%*********************************************************
%%%*********************************************************
\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 iteration with least-squares 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
nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector,   and
$b\in\mathbb{R}^n$ is  the right-hand side.  As explained previously, 
the algorithm is implemented  as an
inner-outer iteration  solver based  on iterative Krylov  methods. The  main 
key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
It can be summarized as follows: the
inner solver is a Krylov based one. In order to accelerate its convergence, the 
outer solver periodically applies a least-squares minimization  on the residuals computed by  the inner one. %Tsolver which does not required to be changed.

At each outer iteration, the sparse linear system $Ax=b$ is partially 
solved using only $m$
iterations of an iterative method, this latter being initialized with the 
last obtained approximation. 
GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
$S$, which is composed by the $s$ last solutions that have been computed during 
the inner iterations phase.
In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. 

$\|r_n\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{n/2} \|r_0\|,$
In the general case, where A is not positive definite, we have

$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, \,$


At each $s$ iterations, another kind of minimization step is applied in order to
compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
the inner iterations 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 $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.


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-squares 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{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$<\epsilon_{tsirm}$)} \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}
    \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
      \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
            \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \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  the proposed  method.  The  outer
iteration is  inside the \emph{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
equal to  the restart  number in 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 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 $S$.
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 iterations and the threshold to stop the
method.

Let us summarize the most important parameters of TSIRM:
\begin{itemize}
\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-squares method;
\item $\epsilon_{ls}$: the threshold used to stop the least-squares method.
\end{itemize}


The  parallelization  of  TSIRM  relies   on  the  parallelization  of  all  its
parts. More  precisely, except  the least-squares 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
columns in  practice. As explained  previously, at least  two methods seem  to be
interesting to solve the least-squares minimization, CGLS and LSQR.

In the following  we remind the CGLS algorithm. The LSQR  method follows more or
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}
\begin{algorithmic}[1]
  \Input $A$ (matrix), $b$ (right-hand side)
  \Output $x$ (solution vector)\vspace{0.2cm}
  \State Let $x_0$ be an initial approximation
  \State $r_0=b-Ax_0$
  \State $p_1=A^Tr_0$
  \State $s_0=p_1$
  \State $\gamma=||s_0||^2_2$
  \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
    \State $q_k=Ap_k$
    \State $\alpha_k=\gamma/||q_k||^2_2$
    \State $x_k=x_{k-1}+\alpha_kp_k$
    \State $r_k=r_{k-1}-\alpha_kq_k$
    \State $s_k=A^Tr_k$
    \State $\gamma_{old}=\gamma$
    \State $\gamma=||s_k||^2_2$
    \State $\beta_k=\gamma/\gamma_{old}$
    \State $p_{k+1}=s_k+\beta_kp_k$
  \EndFor
\end{algorithmic}
\label{algo:02}
\end{algorithm}


In each iteration  of CGLS, there is two  matrix-vector multiplications and some
classical operations:  dot product, norm, multiplication  and addition on  vectors. All
these operations are easy to implement in PETSc or similar environment.



%%%*********************************************************
%%%*********************************************************

\section{Convergence results}
\label{sec:04}
Let us recall the following result, see~\cite{Saad86} for further readings.
\begin{proposition}
\label{prop:saad}
Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
\begin{equation}
||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
\end{equation}
where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves 
the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
\end{proposition}


We can now claim that,
\begin{proposition}
If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, 
let $r_k$ be the
$k$-th residue of TSIRM, then
we still have:
\begin{equation}
||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
\end{equation}
where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.
\end{proposition}

\begin{proof}
We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$

The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.

Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
We will show that the statement holds too for $r_k$. Two situations can occur:
\begin{itemize}
\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis.
\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\
$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$

$\begin{array}{ll}
& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\
& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
& \leqslant ||b-Ax_{k}||_2\\
& = ||r_k||_2\\
& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||,
\end{array}$
\end{itemize}
which concludes the induction and the proof.
\end{proof}

We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
than the one of the GMRES method.

%%%*********************************************************
%%%*********************************************************
\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 are given.

\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     & Comput. fluid dynamics  & 525,825 & 2,100,225 \\
epb3               & Thermal problem   & 84,617  & 463,625  \\
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

\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 (\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
$\epsilon_{tsirm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.


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 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}[htbp]
\begin{center}
\begin{tabular}{|c|c|r|r|r|r|} 
\hline

 \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM 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 TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
\label{tab:02}
\end{center}
\end{table}





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  equal to  4  and  4  extra-diagonals
  representing the neighbors in each directions  are equal to -1. This example is
  used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
  propagation, etc.
\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 called $\alpha$.
\end{itemize}
For more technical details on  these applications, interested readers are invited
to  read the  codes available  in the  PETSc sources.   Those problems  have been
chosen because they  are scalable with many cores which is not the case of other problems that we have tested.

In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\


{\bf Description of preconditioners}\\

\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|}{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 \\
  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 TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (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*}

Table~\ref{tab:03} shows  the execution  times and the  number of  iterations of
example ex15  of PETSc on the  Juqueen architecture. Different  numbers of cores
are  studied ranging  from  2,048  up-to 16,383.   Two  preconditioners have  been
tested: {\it mg} and {\it sor}.   For those experiments,  the number  of components  (or unknowns  of the
problems)  per core  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 Table~\ref{tab:03}, 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  always  faster than
FGMRES. For this example, the  multigrid preconditioner 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} (weak scaling)}
\label{fig:01}
\end{figure}


In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
Table~\ref{tab:03}  is  displayed.   It  can  be  noticed  that  the  number  of
iterations per second of FMGRES is  constant whereas it decreases with TSIRM with
both preconditioners. This  can be explained by the fact that  when the number of
cores increases the time for the least-squares 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*}[htbp]
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores & threshold   & \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      & 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 TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12),  time is expressed in seconds.}
\label{tab:04}
\end{center}
\end{table*}


In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.


\begin{table*}[htbp]
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores   & \multicolumn{2}{c|}{FGMRES} & \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. &   & FGMRES & 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 TSIRM with FGMRES 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, threshold 5e-5),  time is expressed in seconds.}
\label{tab:05}
\end{center}
\end{table*}

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
\label{fig:02}
\end{figure}

%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************

A novel two-stage iterative  algorithm has been proposed in this article,
in order to accelerate the convergence Krylov iterative  methods.
Our TSIRM proposal acts as a merger between Krylov based solvers and
a least-squares minimization step.
The convergence of the method has been proven in some situations, while 
experiments up to 16,394 cores have been led to verify that TSIRM runs
5 or  7 times  faster than GMRES.


For future work, the authors' intention is to investigate 
other kinds of matrices, problems, and inner solvers. The 
influence of all parameters must be tested too, while 
other methods to minimize the residuals must be regarded.
The number of outer iterations to minimize should become 
adaptative to improve the overall performances of the proposal.
Finally, this solver will be implemented inside PETSc.


% conference papers do not normally have an appendix



% use section* for acknowledgement
%%%*********************************************************
%%%*********************************************************
\section*{Acknowledgment}
This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resources
Curie and Juqueen respectively based in France and Germany.



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% can use a bibliography generated by BibTeX as a .bbl file
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%% \bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.

%% \bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.

%% \bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.

%% \bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
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