10-01-2014

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 48d9e2e191998300b47a9534c27a1dd89c622534..0ea51d8b74048cccfd43792b6bcb1b64c194206a 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -161,7 +161,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 \right.
 \label{sec03:eq01}
 \end{equation}  
 \right.
 \label{sec03:eq01}
 \end{equation}  

-where for $l\in\{1,\ldots,L\}$ $A_l$ 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.


@@ -192,6 +192,29 @@ iteration method for solving the sub-systems~(\ref{sec03:eq03}). It is a well-kn
 iterative method which gives good performances for solving sparse linear systems in
 parallel on a cluster of processors. 
 
 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 method is that the Krylov subspace need neither to be spanned by an orthogonal
+basis nor synchronizations between the different clusters to generate this basis. 
+