09-01-2014

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 91e47458d85749235dbe5d7e5106fecee13efcf2..48d9e2e191998300b47a9534c27a1dd89c622534 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -89,12 +89,13 @@ 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
 \label{eq04}
 \end{equation}
 to be solved  independently by a direct or  an iterative method, where

-$v_l^k$  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    $v_l^k$   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
+$v_l^k$  is   the  solution  of   the  local  sub-system.   Thus,  the
+calculations  of $v_l^k$  may be  performed in  parallel by  a  set of
+processors.   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 $v_l^k$  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 v_l^k,
 \label{eq05}
 \begin{equation}
 x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
 \label{eq05}


@@ -160,7 +161,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 \right.
 \label{sec03:eq01}
 \end{equation}  
 \right.
 \label{sec03:eq01}
 \end{equation}  

-where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
+where for $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$

 and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
 case, we use a row-by-row splitting without overlapping in such a way that successive
 rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.
 and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
 case, we use a row-by-row splitting without overlapping in such a way that successive
 rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.


@@ -186,10 +187,10 @@ Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
 \end{equation}
 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
 \end{equation}
 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}).  
-
-
+dependencies between the clusters. In this work, we use the GMRES method as an inner
+iteration method for solving the sub-systems~(\ref{sec03:eq03}). It is a well-known
+iterative method which gives good performances for solving sparse linear systems in
+parallel on a cluster of processors.