From ee6ff2cef0b50051cbcf657aa4f3fc61bc050472 Mon Sep 17 00:00:00 2001
From: lilia <lilia@mondomaine.fr>
Date: Wed, 8 Jan 2014 00:32:05 +0100
Subject: [PATCH] 07-01-2014 V2

---
 krylov_multi.tex | 26 ++++++++++++++++++--------
 1 file changed, 18 insertions(+), 8 deletions(-)

diff --git a/krylov_multi.tex b/krylov_multi.tex
index e61890d..963ce3e 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -80,22 +80,22 @@ of such a method is dependent on the condition
 The advantage of  the multisplitting method is that  at each iteration
 $k$ there are $L$ different linear sub-systems
 \begin{equation}
-y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
 \label{eq04}
 \end{equation}
 to be solved  independently by a direct or  an iterative method, where
-$y_l^k$  is the solution  of the  local sub-system.   A multisplitting
+$v_l^k$  is the solution  of the  local sub-system.   A multisplitting
 method  using   an  iterative  method  for  solving   the  $L$  linear
 sub-systems is  called an inner-outer iterative method  or a two-stage
-method.   The   results    $y_l^k$   obtained   from   the   different
+method.   The   results    $v_l^k$   obtained   from   the   different
 splittings~(\ref{eq04}) are combined to  compute the solution $x^k$ of
 the linear system by using the diagonal weighting matrices
 \begin{equation}
-x^k = \displaystyle\sum^L_{l=1} E_l y_l^k,
+x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
 \label{eq05}
 \end{equation}    
 In the case where the diagonal weighting matrices $E_l$ have only zero
-and   one   factors  (i.e.   $y_l^k$   are   disjoint  vectors),   the
+and   one   factors  (i.e.   $v_l^k$   are   disjoint  vectors),   the
 multisplitting method is non-overlapping  and corresponds to the block
 Jacobi method.
 %%%%%%%%%%%%%%%%%%%%%%%
@@ -162,8 +162,10 @@ So, the multisplitting format of the linear system is defined as follows:
 \end{equation} 
 where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular matrix $A_l$, $X_i\neq X_l$
 is a sub-vector of size $n_i$ of the solution vector $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$,
-for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
-the following spare sub-linear system: 
+for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. 
+
+The multisplitting method proceeds by iteration for solving the linear system in such a
+way each sub-system
 \begin{equation}
 \left\{
 \begin{array}{l}
@@ -173,7 +175,15 @@ Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
 \right.
 \label{sec03:eq03}
 \end{equation}
-where the sub-vectors $X_i$ define the data dependencies between the cluster $l$ and other clusters.
+is solved independently by a cluster of processors and communication are required to
+update the right-hand side vectors $Y_l$, such that the vectors $X_i$ represent the data
+dependencies between the clusters. In this case, the parallel GMRES method is used
+as an inner iteration method for solving the linear sub-systems~(\ref{sec03:eq03}).  
+
+
+
+
+
 
 
 
-- 
2.39.5