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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}
%où
%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
%où
%\title{???}





% author names and affiliations
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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 Comte, 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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%Homer Simpson\IEEEauthorrefmark{2},
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%Montgomery Scott\IEEEauthorrefmark{3} and
%Eldon Tyrell\IEEEauthorrefmark{4}}
%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
%Georgia Institute of Technology,
%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
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%Telephone: (800) 555--1212, Fax: (888) 555--1212}
%\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 paper  we propose  a  two stage  iterative method  which increases  the
convergence of Krylov iterative methods,  typically those of GMRES variants. The
principle of  our approach  is to  build an external  iteration over  the Krylov
method  and to  save  the current  residual  frequently (for  example, for  each
restart of GMRES). Then after a given number of outer iterations, a minimization
step  is applied  on the  matrix composed  of the  saved residuals  in  order to
compute a better solution and make  a new iteration if necessary.  We prove that
our method has  the same convergence property than the  inner method used.  Some
experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
can be around 7 times faster.
\end{abstract}

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


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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  became more attractive than  direct ones to  solve very large
sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
context,  with  thousands  of  cores,  and  require  less  memory  and  arithmetic
operations than direct  methods. A number of iterative  methods are proposed and
adapted by many researchers and the increased need for solving very large sparse
linear  systems  triggered the  development  of  efficient iterative  techniques
suitable for the parallel processing.

Most of the successful iterative methods currently available are based on Krylov
subspaces which  consist in forming a  basis of a sequence  of successive matrix
powers times an initial vector for example the residual. These methods are based
on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
Gradient method and GMRES method (generalized minimal residual).

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

In this  paper we propose a  two-stage algorithm based on  two nested iterations
called inner-outer  iterations.  This algorithm  consists in solving  the sparse
linear system iteratively  with a small number of  inner iterations and restarts
the outer  step with a  new solution minimizing  some error functions  over some
previous residuals. This algorithm is iterative and easy to parallelize on large
clusters   and  the   minimization  technique   improves  its   convergence  and
performances.

The present paper is organized  as follows. In Section~\ref{sec:02} some related
works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
convergence  results  on this  method.   Section~\ref{sec:05}  shows  some  experimental
results  obtained on large  clusters of  our algorithm  using routines  of PETSc
toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
%%%*********************************************************
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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 algorithm with least-square residuals minimization}
\label{sec:03}
A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
points of our solver are given in Algorithm~\ref{algo:01}.

In order to accelerate the convergence, the outer iteration periodically applies
a least-square minimization  on the residuals computed by  the inner solver. The
inner solver is based on a Krylov method which does not require to be changed.

At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
iterations, using an iterative method restarting with the previous solution. For
example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
inner solver. The current solution of the Krylov method is saved inside a matrix
$S$ composed of successive solutions computed by the inner iteration.

Periodically, every $s$ iterations, the minimization step is applied in order to
compute a new  solution $x$. For that, the previous  residuals are computed with
$(b-AS)$. The minimization of the residuals is obtained by 
\begin{equation}
   \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\end{equation}
with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.


In  practice, $R$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
$s\ll n$.   In order  to minimize~(\ref{eq:01}), a  least-square method  such as
CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Those  methods are more
appropriate than a direct method in a parallel context.

\begin{algorithm}[t]
\caption{TSARM}
\begin{algorithmic}[1]
  \Input $A$ (sparse matrix), $b$ (right-hand side)
  \Output $x$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \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$>\epsilon_{tsarm}$}
      \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
      \State $x^k=S\alpha$  \Comment{compute new solution}
    \EndIf
  \EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}

Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
threshold must be specified for the  solver. In practice, this threshold must be
much  smaller  than the  convergence  threshold  of  the TSARM  algorithm  (i.e.
$\epsilon_{tsarm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
solution  $x_k$  into the  column  $k~  mod~ s$ of  the  matrix  $S$. After  the
minimization, the matrix $S$ is reused with the new values of the residuals.  To
solve the minimization problem, an  iterative method is used. Two parameters are
required for that: the maximum number of iteration and the threshold to stop the
method.

To summarize, the important parameters of TSARM are:
\begin{itemize}
\item $\epsilon_{tsarm}$ the threshold to stop the TSARM method
\item $max\_iter_{kryl}$ the maximum number of iterations for the krylov method
\item $s$ the number of outer iterations before applying the minimization step
\item $max\_iter_{ls}$ the maximum number of iterations for the iterative least-square method
\item $\epsilon_{ls}$ the threshold to stop the least-square method
\end{itemize}


The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
parts. More  precisely, except  the least-square step,  all the other  parts are
obvious to  achieve out in parallel. In  order to develop a  parallel version of
our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
colums in  practice. As explained  previously, at least  two methods seem  to be
interesting to solve the least-square minimization, CGLS and LSQR.

In the following  we remind the CGLS algorithm. The LSQR  method follows more or
less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.

\begin{algorithm}[t]
\caption{CGLS}
\begin{algorithmic}[1]
  \Input $A$ (matrix), $b$ (right-hand side)
  \Output $x$ (solution vector)\vspace{0.2cm}
  \State $r=b-Ax$
  \State $p=A'r$
  \State $s=p$
  \State $g=||s||^2_2$
  \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
    \State $q=Ap$
    \State $\alpha=g/||q||^2_2$
    \State $x=x+alpha*p$
    \State $r=r-alpha*q$
    \State $s=A'*r$
    \State $g_{old}=g$
    \State $g=||s||^2_2$
    \State $\beta=g/g_{old}$
  \EndFor
\end{algorithmic}
\label{algo:02}
\end{algorithm}


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



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

\section{Convergence results}
\label{sec:04}




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


In order to see the influence of our algorithm with only one processor, we first
show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
characteristics. For all  the matrices, the name, the field,  the number of rows
and the number of nonzero elements is given.

\begin{table*}
\begin{center}
\begin{tabular}{|c|c|r|r|r|} 
\hline
Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
crashbasis         & Optimization      & 160,000  &  1,750,416  \\
parabolic\_fem     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
epb3               & Thermal problem   & 84,617  & 463,625  \\
atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
bfwa398            & Electromagnetics problem & 398 & 3,678 \\
torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
\hline

\end{tabular}
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
\end{table*}

The following  parameters have been chosen  for our experiments.   As by default
the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
the GMRES every 30 iterations, $max\_iter_{kryl}=30$).  $s$ is set to 8. CGLS is
chosen  to minimize  the least-squares  problem with  the  following parameters:
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to
$\epsilon_{tsarm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
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 TSARM,  the same  solver and  the same
preconditionner is used.  This Table shows that TSARM can drastically reduce the
number of iterations to reach the  convergence when the number of iterations for
the normal GMRES is more or less  greater than 500. In fact this also depends on
tow  parameters: the  number  of iterations  to  stop GMRES  and  the number  of
iterations to perform the minimization.


\begin{table}
\begin{center}
\begin{tabular}{|c|c|r|r|r|r|} 
\hline

 \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ 
\cline{3-6}
       &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline

crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
parabolic\_fem     & gmres / ilu           & 1009.94   & 7573 & 401.52 & 2970 \\
epb3               & fgmres / sor             &  8.67     & 600  &  8.21 & 540  \\
atmosmodj          &  fgmres / sor & 104.23  & 451 & 88.97 & 366  \\
bfwa398            & gmres / none  & 1.42 & 9612 & 0.28 & 1650 \\
torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
\hline

\end{tabular}
\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
\label{tab:02}
\end{center}
\end{table}





In order to perform larger  experiments, we have tested some example application
of PETSc. Those  applications are available in the ksp part  which is suited for
scalable linear equations solvers:
\begin{itemize}
\item ex15  is an example  which solves in  parallel an operator using  a finite
  difference  scheme.   The  diagonal  is  equals to  4  and  4  extra-diagonals
  representing the neighbors in each directions  is equal to -1. This example is
  used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
  propagation...
\item ex54 is another example based on 2D problem discretized with quadrilateral
  finite elements. For this example, the user can define the scaling of material
  coefficient in embedded circle, it is called $\alpha$.
\end{itemize}
For more technical details on  these applications, interested reader are invited
to  read the  codes available  in the  PETSc sources.   Those problem  have been
chosen because they  are scalable with many cores. We  have tested other problem
but they are not scalable with many cores.

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


{\bf Description of preconditioners}

\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores & precond   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
\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 TSARM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
\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. Differents  number of cores
are  studied rangin  from  2,048  upto 16,383.   Two  preconditioners have  been
tested.   For those experiments,  the number  of components  (or unknown  of the
problems)  per processor is  fixed to  25,000. This  number can  seem relatively
small. In fact, for  some applications that need a lot of  memory, the number of
components per processor requires sometimes to be small.

In this Table, we  can notice that TSARM is always faster  than FGMRES. The last
column shows the ratio between FGMRES and the best version of TSARM according to
the minimization procedure: CGLS or LSQR.


\begin{figure}
\centering
  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
\label{fig:01}
\end{figure}





\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
\cline{3-8}
             &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
  2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
  2,048      & 6e-5                  & 194.01 & 30,270  & 35.50  &  5,430  & 27.74  & 4,350   & 6.99 \\
  4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
  4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
  8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
  8,192      & 5e-5                  & 785.04 & 109,590 & 76.07  &  10,470  & 69.42 & 9,030  & 11.30 \\
  16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
\hline

\end{tabular}
\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12),  time is expressed in seconds.}
\label{tab:04}
\end{center}
\end{table*}





\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
\hline

  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
\cline{2-7} \cline{9-11}
                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
   512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
   1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
   2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\
   4096             & 405.60    & 28,380 & 111.67 & 7,590  & 91.72  & 6,510 & 4.42  & 1.22   &  .79     &   .84 \\
   8192             & 785.04   & 109,590 & 76.07  & 10,470 & 69.42 & 9,030  & 11.30 &   .32  &   .58    &  .56 \\

\hline

\end{tabular}
\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5),  time is expressed in seconds.}
\label{tab:05}
\end{center}
\end{table*}

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



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


future plan : \\
- study other kinds of matrices, problems, inner solvers\\
- test the influence of all the parameters\\
- adaptative number of outer iterations to minimize\\
- other methods to minimize the residuals?\\
- implement our solver 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 resource
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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