X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/0e974d092288389bf1f266982be7f40dceca2f29..cce18db44812b9a67dd4d30ca37ea74fd46f16b1:/paper.tex

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@@ -601,21 +601,13 @@ is summarized while intended perspectives are provided.
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 \section{Related works}
 \label{sec:02} 
-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. 
+%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 −1 b by p(A)b, where p is a “good” polynomial. This chapter covers
-methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers
-methods based on Lanczos biorthogonalization.
+%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 −1 b by p(A)b, where p is a “good” 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.
+%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. 
+%Preconditioned Krylov-subspace iterations are a key ingredient in many modern linear solvers, including in solvers that employ support preconditioners. 
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@@ -1114,11 +1106,25 @@ taken into account with TSIRM.
 \end{figure}
 
 
-Concerning the  experiments some  other remarks are  interesting. 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. 
+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}
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