X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/5e3a7342021a720e17be4f147b4e73e5ad5396b9..1a37f09d70ad11b86abe47b86d61cb8ba6f3bf12:/krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 91e4745..5a97c5f 100644
--- 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
-$v_l^k$  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}
@@ -160,7 +161,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 \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.
@@ -186,10 +187,33 @@ 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
-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. 
+
+It should be noted that the convergence of the inner iterative solver for the different
+linear sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the
+multisplitting method. It strongly depends on the properties of the sparse linear system
+to be solved and the computing environment~\cite{o1985multi,ref18}. Furthermore, the multisplitting
+of the linear system among several clusters of processors increases the spectral radius
+of the iteration matrix, thereby slowing the convergence. In this paper, we based on the
+work presented in~\cite{huang1993krylov} to increase the convergence and improve the
+scalability of the multisplitting methods. 
+
+In order to accelerate the convergence, we implement the outer iteration of the multisplitting
+solver as a Krylov subspace method which minimizes some error function over a Krylov subspace~\cite{S96}.
+The Krylov space of the method that we used is spanned by a basis composed of the solutions issued from
+solving the $L$ splittings~(\ref{sec03:eq03})
+\begin{equation}
+\{x^1,x^2,\ldots,x^s\},~s\ll n,
+\label{sec03:eq04}
+\end{equation}
+where for $k\in\{1,\ldots,s\}$, $x^k=[X_1^k,\ldots,X_L^k]$ is a solution of the global linear
+system. 
+%The advantage such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis.
+The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations
+between the different clusters to generate this basis.