12-10-2014 08

[GMRES2stage.git] / paper.tex

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\begin{document}
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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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%Georgia Institute of Technology,
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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; two stage iteration; least-squares residual minimization; PETSc
\end{IEEEkeywords}


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%%%*********************************************************
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\section{Introduction}
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% (should never be an issue)

Iterative methods have recently become more attractive than direct ones to solve
very large sparse  linear systems\cite{Saad2003}.  They are more  efficient in a
parallel context,  supporting thousands of  cores, and they require  less memory
and  arithmetic operations than  direct methods~\cite{bahicontascoutu}.  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. For  further information  on two-stage
iteration      methods,     interested      readers      are     invited      to
consult~\cite{Nichols:1973:CTS}. Two-stage algorithms are easy to parallelize on
large clusters.  Furthermore,  the least-squares 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} 
Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.

GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.

In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However in practice the good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.      

% two-stage, communication avoiding 

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\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 $S_i$.

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  $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$   \label{algo:solve}
    \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column ($k \mod s$) of $S$}
    \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.}, $\epsilon_{tsirm}$).  We also  consider that
after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
which is defined by $||Ax_k-b||_2$.

  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}


We can now claim that,
\begin{proposition}
\label{prop:saad}
If $A$ is either a definite positive or a positive 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 have the following boundaries:
\begin{itemize}
\item when $A$ is positive:
\begin{equation}
||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
\end{equation}
where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$;
\item when $A$ is positive definite:
\begin{equation}
\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|.
\end{equation}
\end{itemize}
%In the general case, where A is not positive definite, we have
%$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .$
\end{proposition}

\begin{proof}
Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows:
\begin{equation*}
\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{k/2} \|r_0\| .
\end{equation*}
Additionally, when $A$ 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 $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves 
the convergence of GMRES($m$) for all $m$ under such assumptions regarding $A$.
These well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.

We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite.

The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ that follows the inductive hypothesis due, to the results recalled above.

Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ in the definite positive one.
We will show that the statement holds too for $r_k$. Two situations can occur:
\begin{itemize}
\item If $k \not\equiv 0 ~(\textrm{mod}\ m)$, then the TSIRM algorithm consists in executing GMRES once. In that case and by using the inductive hypothesis, we obtain either $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ if $A$ is positive, or $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite case.
\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies:
\begin{itemize}
\item $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, 
\item $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite one,
\end{itemize}
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||, \textrm{ if $A$ is positive,}\\
& \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|, \textrm{ if $A$ is}\\
& \textrm{positive definite,} 
\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 behavior of the proposal when considering only one processor, a first
comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented. 
Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in 
Table~\ref{tab:01}.  These latter, which are real-world applications matrices, 
have been extracted 
 from   the  Davis  collection,   University  of
Florida~\cite{Dav97}.

\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}
Chosen parameters are detailed below.
%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
GRMES or  FGMRES (Flexible  GMRES)~\cite{Saad:1993} 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.  Both  these architectures  are supercomputer
composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
are respectively hosted  by GENCI in France and  Jülich Supercomputing Centre in
Germany. They belongs with other similar architectures of the PRACE initiative (
Partnership  for Advanced  Computing in  Europe)  which aims  at proposing  high
performance supercomputing architecture to enhance research in Europe. The Curie
architecture is composed of Intel E5-2680  processors at 2.7 GHz with 2Gb memory
by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with
1Gb memory per  core. Both those architecture are equiped  with a dedicated high
speed network.


In  many situations, using  preconditioners is  essential in  order to  find the
solution of a linear system.  There are many preconditioners available in PETSc.
For parallel applications all  the preconditioners based on matrix factorization
are  not  available. In  our  experiments, we  have  tested  different kinds  of
preconditioners, however  as it is  not the subject  of this paper, we  will not
present results with many preconditioners. In  practise, we have chosen to use a
multigrid (mg)  and successive  over-relaxation (sor). For  more details  on the
preconditioner in PETSc please consult~\cite{petsc-web-page}.



\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 with the two preconditioners {\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.  For this  application, we fixed $\alpha=0.6$.  As it
can be seen in that Table, the size of the problem has a strong influence on the
number of iterations to reach the  convergence. That is why we have preferred to
change the threshold.  If we set  it to $1e-3$ as with the previous application,
only one iteration is necessray  to reach the convergence. So Table~\ref{tab:04}
shows the results  of differents executions with differents  number of cores and
differents thresholds. As  with the previous example, we  can observe that TSIRM
is faster than FGMRES. The ratio greatly depends on the number of iterations for
FMGRES to reach the threshold. The greater the number of iterations to reach the
convergence is, the  better the ratio between our algorithm  and FMGRES is. This
experiment is  also a  weak scaling with  approximately $25,000$  components per
core. It can also  be observed that the difference between CGLS  and LSQR is not
significant. Both can be good but it seems not possible to know in advance which
one will be the best.

Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
Curie  architecture. So  in  this case,  the  number of  unknownws  is fixed  to
$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
of two.  The  threshold is fixed to $5e-5$ and only  the $mg$ preconditioner has
been tested. Here  again we can see that TSIRM is  faster that FGMRES. Efficiecy
of each algorithms is reported. It  can be noticed that FGMRES is more efficient
than TSIRM except with $8,192$ cores and that its efficiency is greater that one
whereas the  efficiency of TSIRM is  lower than one. Nevertheless,  the ratio of
TSIRM  with any  version  of the  least-squares  method is  always faster.  With
$8,192$ cores when the number of iterations is far more important for FGMRES, we
can see that it is only slightly more important for TSIRM.

In  Figure~\ref{fig:02}  we report  the  number  of  iterations per  second  for
experiments  reported in  Table~\ref{tab:05}.  This Figure  highlights that  the
number of iterations per  seconds is more of less the same  for FGMRES and TSIRM
with a little advantage for FGMRES. It  can be explained by the fact that, as we
have previously explained, that the iterations of the least-sqaure steps are not
taken into account with TSIRM.

\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}


Concerning the  experiments some  other remarks are  interesting.
\begin{itemize}
\item We  can tested other examples of  PETSc (ex29, ex45, ex49).  For all these
  examples,  we also obtained  similar gain  between GMRES  and TSIRM  but those
  examples are  not scalable with many  cores. In general, we  had some problems
  with more than $4,096$ cores.
\item We have tested many iterative  solvers available in PETSc.  In fast, it is
  possible to use most of them with TSIRM. From our point of view, the condition
  to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
  feature. More  precisely, the solver must  support to be  stoped and restarted
  without decrease its  converge. That is why  with GMRES we stop it  when it is
  naturraly  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate
  Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
  so they  are not  efficient.  They  will converge with  TSIRM but  not quickly
  because if  we compare  a normal CG  with a CG  for which  we stop it  each 16
  iterations  for example,  the  normal CG  will  be for  more efficient.   Some
  restarted CG  or CG variant versions exist  and may be interested  to study in
  future works.
\end{itemize}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\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. This  would be very interesting because it would  allow us to test
all the non-linear  examples and compare our algorithm  with the other algorithm
implemented in 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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%% \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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