X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/eab5f4a9438f75792a4317e54757ff950cd04faf..ea5e4dbcd3255d09e763fb4d496a01ff90cde5f9:/paper.tex

diff --git a/paper.tex b/paper.tex
index d0fac64..f2114e5 100644
--- a/paper.tex
+++ b/paper.tex
@@ -601,7 +601,21 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \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.
+GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers. 
+
+The next two chapters explore a few methods which are considered currently to be among the
+most important iterative techniques available for solving large linear systems. These techniques
+are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which
+are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these
+techniques approximate A â1 b by p(A)b, where p is a âgoodâ polynomial. This chapter covers
+methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers
+methods based on Lanczos biorthogonalization.
+
+Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.
+
+Preconditioned Krylov-subspace iterations are a key ingredient in
+many modern linear solvers, including in solvers that employ support
+preconditioners. 
 %%%*********************************************************
 %%%*********************************************************
 
@@ -654,10 +668,10 @@ appropriate than a single direct method in a parallel context.
   \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}
+  \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 $S_{k \mod s}=x_k$ \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}
@@ -675,10 +689,10 @@ 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 $||Ax^k-b||_2$.
+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 $k
+$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
@@ -790,7 +804,12 @@ Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\foral
 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 $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
+\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,\\
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
@@ -801,14 +820,16 @@ $\begin{array}{ll}
 & \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||,
+& \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.
+%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
+%than the one of the GMRES method.
 
 %%%*********************************************************
 %%%*********************************************************
@@ -816,13 +837,13 @@ than the one of the GMRES method.
 \label{sec:05}
 
 
-In order to see the influence of our algorithm with only one processor, we first
-show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01},
-we  show the  matrices we  have  used and  some of  them characteristics.  Those
-matrices  are   chosen  from   the  Davis  collection   of  the   University  of
-Florida~\cite{Dav97}. They are matrices arising in real-world applications.  For
-all the  matrices, the name,  the field,  the number of  rows and the  number of
-nonzero elements are given.
+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}
@@ -842,8 +863,9 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
-
-The following  parameters have been chosen  for our experiments.   As by default
+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:
@@ -923,8 +945,16 @@ by core. The Juqueen architecture is composed  of IBM PowerPC A2 at 1.6 GHz with
 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}.
+
 
-{\bf Description of preconditioners}\\
 
 \begin{table*}[htbp]
 \begin{center}
