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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} LTAS-Mécanique numérique non linéaire, University of Liege, Belgium\\ Email: l.zianekhodja@ulg.ac.be}
%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 best  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.

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



%%%*********************************************************
%%%*********************************************************
\section{Related works}
\label{sec:02} 
Krylov subspace iteration methods have increasingly become key
techniques  for  solving  linear and nonlinear systems,  or  eigenvalue  problems,
especially      since       the      increasing      development       of      
preconditioners~\cite{Saad2003,Meijerink77}.  One reason  for  the popularity  of
these methods is their generality, simplicity, and efficiency to solve systems of
equations arising from very large and complex problems.

GMRES is one of the most  widely used Krylov iterative method for solving sparse
and   large  linear   systems.  It   has   been  developed   by  Saad   \emph{et
  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,  this  algorithm minimizes  the  residual over  the
current Krylov subspace  until convergence in at most  $n$ iterations, where $n$
is the size  of the sparse matrix.   Full GMRES is however 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,
GMRES is  restarted in  practice 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  $m$  value~\cite{Huang89}.  Therefore  in  most  cases,  a  preconditioning
technique  is applied  to the  restarted GMRES  method in  order to  improve its
convergence.

To enhance the robustness of Krylov iterative solvers, some techniques have been
proposed allowing the use of different preconditioners, if necessary, within the
iteration  itself   instead  of  restarting.   Those  techniques   may  lead  to
considerable  savings  in  CPU  time  and memory  requirements.  Van  der  Vorst
in~\cite{Vorst94} has for  instance proposed variants of the  GMRES algorithm in
which a  different preconditioner is applied  in each iteration,  leading to the
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  is referred to as  inner iterations.
Saad in~\cite{Saad:1993}  has proposed Flexible GMRES (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, 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.

Recently,  communication-avoiding  methods have  been  developed  to reduce  the
communication overheads in Krylov subspace iterative solvers. On modern computer
architectures,   communications  between   processors  are   much   slower  than
floating-point        arithmetic       operations        on        a       given
processor.   Communication-avoiding  techniques  reduce   either  communications
between processors or data movements  between levels of the memory hierarchy, by
reformulating the communication-bound kernels (more frequently SpMV kernels) and
the orthogonalization  operations within the Krylov  iterative solver. Different
works have  studied the communication-avoiding techniques for  the GMRES method,
so-called     CA-GMRES,     on     multicore    processors     and     multi-GPU
machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}.

Compared  to all these  works and  to all  the other  works on  Krylov iterative
methods,  the originality of  our work  is to  build a  second iteration  over a
Krylov  iterative method  and to  minimize  the residuals  with a  least-squares
method after a given number of outer iterations.

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



%%%*********************************************************
%%%*********************************************************
\section{TSIRM: 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.

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.  The 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. CGLS has recently been used to improve the performance of multisplitting algorithms \cite{cz15:ij}.



\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  ($max\_iter_{ls}$) and the threshold to stop
  the method ($\epsilon_{ls}$).

Let us summarize the most important parameters of TSIRM:
\begin{itemize}
\item $\epsilon_{tsirm}$: the threshold that stops 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}.    In
line~\ref{algo:matrix_mul}, the matrix-matrix  multiplication is implemented and
efficient since the matrix $A$ is sparse and 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,  the CGLS  and the  LSQR
methods.

In Algorithm~\ref{algo:02} 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 are 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.  It should be noticed that LSQR follows the same principle, it is a
little bit longer but it performs more or less the same operations.


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

\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 a 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}

Remark that a similar proposition can be formulated at each time
the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases 
regarding $A$, 
that the proposed TSIRM converges while the GMRES($m$) does not.

%%%*********************************************************
%%%*********************************************************
\section{Experiments using PETSc}
\label{sec:05}


In order to see the behavior of our approach 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.   
We have  stopped  the  GMRES every  30
iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default 
setting of GMRES restart parameter.  The parameter $s$ has been set to 8. CGLS 
 minimizes  the   least-squares  problem   with  parameters
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to
$\epsilon_{tsirm}=1e-10$.  These  experiments have been performed  on an Intel(R)
Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version  of PETSc.


Experiments comparing 
a GMRES variant with TSIRM in the resolution of linear systems are given in  Table~\ref{tab:02}. 
The  second column describes whether GMRES or FGMRES has been used for linear systems solving.  
Different preconditioners  have been used according to the matrices.  With  TSIRM, the  same
solver and the  same preconditioner are used.  This table  shows that TSIRM can
drastically reduce  the number of iterations needed 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 two parameters: the number of iterations before stopping 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 between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
\label{tab:02}
\end{center}
\end{table}





In order to perform larger experiments, we have tested some example applications
of  PETSc. These  applications are  available in  the \emph{ksp}  part,  which is
suited for scalable linear equations solvers:
\begin{itemize}
\item ex15  is an example  that 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 a 2D problem discretized with quadrilateral
  finite elements. In 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.   These problems  have been
chosen because they are scalable with many  cores.

In  the  following,   larger  experiments  are  described  on   two  large  scale
architectures: Curie  and Juqueen.   Both these architectures  are supercomputers
respectively  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 Center in Germany.  They belong with other similar architectures
to 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,
for its part, is
composed by IBM PowerPC  A2 at  1.6 GHz with  1Gb memory  per core.  Both those
architectures are equipped 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.
However, for parallel applications, all  the preconditioners based on matrix factorization
are  not  available. In  our  experiments, we  have  tested  different kinds  of
preconditioners, but  as it is  not the subject  of this paper, we  will not
present results with many preconditioners. In  practice, we have chosen to use a
multigrid (mg)  and successive  over-relaxation (sor). For  further details  on the
preconditioners in PETSc, readers are referred to~\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) having 25,000 components per core on Juqueen ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  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  at  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. Other parameters  for this application  are described in
the legend of this table.



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  $max\_iter_{kryl}*s$
iterations which corresponds to 30*12, there are $max\_iter_{ls}$ iterations for
the least-squares method which corresponds to 15.

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex15_juqueen}
\caption{Number of iterations per second with ex15 and the same parameters as 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 & $\epsilon_{tsirm}$  & \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 ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  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 necessary  to reach the convergence. So Table~\ref{tab:04}
shows the  results of  different executions with  different number of  cores and
different 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} shows  a strong scaling  experiment with example ex54  on the
Curie  architecture. So,  in  this case,  the  number of  unknowns  is fixed  at
$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
of two.  The  threshold is fixed at $5e-5$ and only  the $mg$ preconditioner has
been  tested. Here  again  we can  see that  TSIRM  is faster  than FGMRES.  The
efficiency of each algorithm is reported.  It can be noticed that the efficiency
of FGMRES is  better than the TSIRM  one except with $8,192$ cores  and that its
efficiency is  greater than 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 the
experiments  reported in  Table~\ref{tab:05}.  This  figure highlights  that the
number of iterations  per second is more  or 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, the  iterations of the  least-squares 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 for ex54 of PETSc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$),  time is expressed in seconds.}
\label{tab:05}
\end{center}
\end{table*}

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex54_curie}
\caption{Number of iterations per second with ex54 and the same parameters as in Table~\ref{tab:05} (strong scaling)}
\label{fig:02}
\end{figure}


Concerning the  experiments some  other remarks are  interesting.
\begin{itemize}
\item We have tested other examples  of PETSc (ex29, ex45, ex49).  For all these
  examples,  we have also  obtained similar  gains 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 fact, 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 stopped and restarted
  without decreasing its convergence. That is  why with GMRES we stop it when it
  is  naturally restarted (\emph{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  which is  stopped and
  restarted every  16 iterations (for example),  the normal CG will  be far more
  efficient.   Some  restarted  CG or  CG  variant  versions  exist and  may  be
  interesting to study in future works.
\end{itemize}
%%%*********************************************************
%%%*********************************************************



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

A new two-stage iterative  algorithm TSIRM has been proposed in this article,
in order to accelerate the convergence of 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. In particular, the possibility 
to obtain a convergence of TSIRM in situations where the GMRES is divergent will be
investigated. 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  adaptive to improve the
overall performances of the proposal.   Finally, this solver will be implemented
inside PETSc, which would be of interest as it would  allows 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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