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.
%%%%%%%%%%%%%%%%%%%%%%%
\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}
\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}).
+
+
+
+
+
