X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/a1ff3fa3715cb0844074d7927cd852f6e0cb6b3f..721c9ed5bc3d45a7fbb91e920b8de4f11e0ffb1f:/paper.tex

diff --git a/paper.tex b/paper.tex
index e93737c..bfb7aa2 100644
--- a/paper.tex
+++ b/paper.tex
@@ -601,9 +601,27 @@ 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.
+Krylov subspace iteration methods have increasingly become useful and successful
+techniques  for  solving  linear,  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.
+
+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
+noticed 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.  
 
@@ -621,7 +639,7 @@ a given number of outer iterations.
 
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 %%%*********************************************************
-\section{Two-stage iteration with least-squares residuals minimization algorithm}
+\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
@@ -685,16 +703,16 @@ 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 $||Ax_k-b||_2$.
+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.
+  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}
@@ -715,8 +733,9 @@ 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.
+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}
@@ -745,9 +764,10 @@ less the same principle but it takes more place, so we briefly explain the paral
 
 
 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.
-
+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.
 
 
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