Sec 04: multisplitting mehods

diff --git a/paper.tex b/paper.tex
index 77dde186a7f9569f7752000f218829a036ba1fd8..3120437ca7154ffc12866986118ef10f350600f0 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -84,6 +84,8 @@
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
 }
 
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
 }
 

+%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
+

 \begin{abstract}
 ABSTRACT
 \end{abstract}
 \begin{abstract}
 ABSTRACT
 \end{abstract}


@@ -105,17 +107,34 @@ ABSTRACT
 \label{sec:04}
 \subsection{Multisplitting methods for sparse linear systems}
 \label{sec:04.01}
 \label{sec:04}
 \subsection{Multisplitting methods for sparse linear systems}
 \label{sec:04.01}

-Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$

 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}
 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}

-where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows: 
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows

 \begin{equation}
 x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
 \label{eq:02}
 \end{equation}
 \begin{equation}
 x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
 \label{eq:02}
 \end{equation}

-where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix.
+where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or a group of processors is responsible for solving one splitting as a linear sub-system   
+\begin{equation}
+M_\ell y_\ell^{k+1} = R_\ell^k,\mbox{~such that~} R_\ell^k = N_\ell x^k_\ell + b,
+\label{eq:03}
+\end{equation}
+then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y^{k+1}_\ell.
+\label{eq:04}
+\end{equation}
+The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors. It is dependent on the condition  
+\begin{equation}
+\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
+\label{eq:05}
+\end{equation}
+where $\rho$ is the spectral radius of the square matrix. The different linear splittings~(\ref{eq:03}) arising from the multisplitting of matrix $A$can be solved exactly with a direct method or approximated with an iterative method. When the inner method used to solve the linear sub-systems is iterative, the multisplitting method is called {\it inner-outer iterative method} or {\it two-stage multisplitting method}.
+
+In this paper we are focused on two-stage multisplitting methods where the well-known iterative method GMRES is used as an inner iteration. 

 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 
 
 \subsection{Simulation of two-stage methods using SimGrid framework}